Study to improve the performance of interferometer with ultra-cold atoms
Xiangyu Dong, Shengjie Jin, Hongmian Shui, Peng Peng, Xiaoji Zhou
SStudy to improve the performance of interferometerwith ultra-cold atoms
Xiangyu Dong , Shengjie Jin , Hongmian Shui , Peng Peng ,Xiaoji Zhou ,(cid:63) State Key Laboratory of Advanced Optical Communication System and Network,Department of Electronics, Peking University, Beijing 100871, China
January 25, 2021
Ultra-cold atoms provide ideal platforms for interferometry. The macroscopicmatter-wave property of ultra-cold atoms leads to large coherent length andlong coherent time, which enable high accuracy and sensitivity to measure-ment. Here, we review our efforts to improve the performance of the interfer-ometer. We demonstrate a shortcut method for manipulating ultra-cold atomsin an optical lattice. Compared with traditional ones, this shortcut methodcan reduce manipulation time by up to three orders of magnitude. We con-struct a matter-wave Ramsey interferometer for trapped motional quantumstates and significantly increase its coherence time by one order of magnitudewith an echo technique based on this method. Efforts have also been made toenhance the resolution by multimode scheme. Application of a noise-resilientmulti-component interferometer shows that increasing the number of pathscould sharpen the peaks in the time-domain interference fringes, which leadsto a resolution nearly twice compared with that of a conventional double-pathtwo-mode interferometer. With the shortcut method mentioned above, im- a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n rovement of the momentum resolution could also be fulfilled, which leads toatomic momentum patterns less than 0.6 (cid:126) k L . To identify and remove system-atic noises, we introduce the methods based on the principal component anal-ysis (PCA) that reduce the noise in detection close to the / √ of the photon-shot noise and separate and identify or even eliminate noises. Furthermore, wegive a proposal to measure precisely the local gravity acceleration within a fewcentimeters based on our study of ultracold atoms in precision measurements.Keywords: Precision Measurement, Ultra-cold atoms, Atomic interferometer, Gravity mea-surements
PACS:
Precision measurement is the cornerstone of the development of modern physics. Atom-basedprecision measurement is an important part. In recent years, ultra-cold atoms have attractedextensive interest in different fields ( ), because of their macroscopic matter-wave prop-erty (
5, 6 ). This property will lead to large coherent length and long coherent time, whichenable high fringe contrast ( ). Hence, with these advantages, ultra-cold atoms provide idealstages for precision measurement (
10, 11 ) and have numerous applications (
1, 8, 12, 12–16 ),ranging from inertia measurements (
1, 8, 12 ) to precision time keeping (
14, 15 ). For example,a Bose-Einstein condensate is used as an atomic source for a high precision sensor, which isreleased into free fall for up to 750 ms and probed with a T = 130 ms Mach-Zehnder atominterferometer based on Bragg transitions ( ). A trapped geometry is realized to probe gravityby holding ultra-cold cesium atoms for 20 seconds ( ), which suppresses the phase variancedue to vibrations by three to four orders of magnitude, overcoming the dominant noise source2n atom-interferometric gravimeters. With ultra-cold atoms in an optical cavity, the detection ofweak force can achieve a sensitivity of 42 yN / √ Hz , which is a factor of 4 above the standardquantum limit ( ).With the superiority mentioned above, it is obvious to consider an interferometer by ultra-cold atoms to further precision measurements. Interferometers with atoms propagating in freefall are ideally suited for inertia measurements ( ). Meanwhile, with atoms held in tighttraps or guides, they are better to measure weak localized interactions. For example, a directmeasurement to the Casimir-Polder force is performed by I. Carusotto et al. in 2005, which isas large as − gravity ( ). However, ultra-cold atoms still get some imperfections neededto be surmounted when combining with interferometry. The macroscopic matter-wave propertyis generated simultaneously with non-linear atom-atom interactions. Phase diffusion caused byinteractions limits the coherence time, and ultimately restricts the sensitivity of interferometers.Besides the interrogation time, the momentum splitting as well as the path number also has animpact on the sensitivity. It has been demonstrated by experiments of multipath interferometersthat, interferometric fringes can be sharpened due to the higher-harmonic phase contributions ofthe multiple energetically equidistant Zeeman states (
25, 26 ), whereas a decrease in the averagenumber of atoms per path causes a greater susceptibility to shot noise. Equilibrium betweenthese parameters could lead to an optimal resolution. In addition, we should also pay attentionto the signal analysis procedure as the interferometric information is mainly extracted from thesignal detected. The resulting resolution severely relies on the probing system.In this review, we mainly introduce our experimental developments that study these fun-damental and important issues to improve the performance of interferometer with ultra-coldatoms. The main developments are concentrated in three aspects: increasing coherent time,using multimode scheme and reducing systematic noises.
A. Enhanced resolution by increasing coherent time.
We introduce an effective and fast3few microseconds) methods, for manipulating ultra-cold atoms in an optical lattice (OL), whichcan be used to construct the atomic interferometer and increase the coherent time to finally geta higher resolution. This shortcut loading method is a designed pulse sequence, which can beused for preparing and manipulating arbitrary pure states and superposition states. Another ad-vantage of this method is that the manipulation time is much shorter than traditional methods(100 ms → µ s ). Based on this shortcut method, we constructed an echo-Ramsey interfer-ometer (RI) with motional Bloch states (at zero quasi-momentum on S- and D-bands of anOL) ( ). Thanks to the rapidity of shortcut methods, more time could be used for the RIprocess. We identified the mechanisms that reduced the RI contrast, and greatly increased thecoherent time (1.3 ms → B. Enhanced resolution by multimode scheme.
Several efforts have been made to avoid-ing the decays of interferometric resolution because of the experimental noises. We demon-strated that the improvement of the phase resolution could be accomplished by a noise-resilientmulti-component interferometric scheme. With the relative phase of different components re-maining stable, increasing the number of paths could sharpen the peaks in the interferencefringes, which led to a resolution nearly twice compared with that of a conventional double-path two-mode interferometer. Moreover, improvement of the momentum resolution was ful-filled with optical lattice pulses. We got results of atomic momentum patterns with intervalsless than the double recoil momentum. The momentum pattern exhibited 10 main peaks.
C. Enhanced resolution by removing the systematic noise.
The method to identify andremove systematic noises for ultra-cold atoms is also introduced in this paper. For improving thequality of absorption image, which is the basic detection result in ultra-cold atoms experiments,we developed an optimized fringe removal algorithm (OFRA), making the noise close to thetheoretical limit as / √ of the photon-shot noise. Besides, for the absorption images after4reprocessing by OFRA, we applied the principal component analysis to successfully separateand identify noises from different origins of leading contribution, which helped to reduce oreven eliminate noises via corresponding data processing procedures.Furthermore, based on our study of ultracold atoms in precision measurements, we demon-strated a scheme for potential compact gravimeter with ultra-cold atoms in a small displacement.The text structure is as follows. In Sec.2, a shortcut method manipulating ultra-cold atomsin an optical lattice and an Echo-Ramsey interferometry with motional quantum states are in-troduced, which can increase the coherent time. In Secs.3, we prove that the resolution canbe increased using a double-path multimode interferometer with spinor Bose-Einstein conden-sates (BECs) or an optical pulse, both of which can be classified into multimode scheme. InSecs.4, methods for identifying and reducing the systematic noises for ultra-cold atoms aredemonstrated. Finally, we give a proposal on gravimeter with ultra-cold atoms in Secs.5. The macroscopic coherent properties of ultra-cold atoms ( ) are conducive to precise mea-surement. To make full use of the advantages of ultra-cold atomic coherence properties, onemethod is to reduce the manipulation time and another is to suppress the attenuation of coher-ence.Firstly, we demonstrated a shortcut process for manipulating BECs trapped in an OL ( ). By optimizing the parameters of the pulses, which constitute the sequence of the shortcutprocess, We can get extremely high fidelity and robustness for manipulating BECs into thedesired states, including the ground state, excited states, and superposition states of a one, twoor three-dimensional OLs. Another advantage of this method is that the manipulation time ismuch shorter than that in traditional methods (100 ms → µ s ).This shortcut is composed of optical lattice pulses and intervals that are imposed on the5 a) (b) x SFGPD -1 10 q/k weak harmonic trap E / E r Time sequencesstep:
11 12 2221 t t t t
Lattice pulse 1 pulse 2 (d)(c) (e)
Figure 1: Schematic diagram of the shortcut method (take ground state preparation as an ex-ample). (a) At the beginning, the BECs are formed in a weak harmonic trap. (b) Time se-quence of shortcut method. (c) Mapping the shortcut process onto the Bloch sphere. The track A → C → | S (cid:105) and track A → B → E → M → | S (cid:105) represent one pulse and two pulsesshortcut process, respectively. (d) After this shortcut process, the desired states of an 3D opti-cal lattice are prepared. (e) Band structure of 1D OL with different quasi-momentum q when V = 10 E r . Reproduced with permission from Ref. ( ).6ystem before the lattice is switched on. The time durations and intervals in the sequenceare optimized to transfer the initial state to the target state with high fidelity. This shortcutprocedure can be completed in several tens of microseconds, which is shorter than the traditionalmethod (usually hundreds of milliseconds). It can be applied to the fast manipulation of thesuperposition of Bloch states.Then, based on this method, we constructed an echo-Ramsey interferometer (RI) with mo-tional Bloch states (at zero quasi-momentum on S- and D-bands of an OL) ( ). The key torealizing a RI is to design effective π - and π/ pulses, which can be obtained by the shortcutmethod (
28, 42, 43 ). Thanks to the rapidity of shortcut methods, more time can be used for theRI process. We identified the mechanisms that reduced the RI contrast, and greatly increased thecoherent time (1.3 ms → Efficient and fast manipulation of BECs in OLs can be used for precise measurements, such asconstructing atom-based interferometers and increasing the coherent time of these interferome-ters. Here we demonstrate an effective and fast (around 100 µ s ) method for manipulating BECsfrom an arbitrary initial state to a desired OL state. This shortcut method is a designed pulsesequence, in which the parameters, such as duration and interval of each step, are optimized tomaximize fidelity and robustness of the final state. With this shortcut method, the pure Blochstates with even or odd parity and superposition states of OLs can be prepared and manipu-lated. In addition, the idea can be extended to the case of two- or three-dimensional OLs. Thismethod has been verified by experiments many times and is very consistent with the theoreticalanalysis (
27, 28, 42, 44–48 ).We used the simplest one-dimensional standing wave OL to demonstrate the design princi-7le of this method. The OL potential is V OL ( x ) = V cos kx , where V is used to characterizethe depth of the OL.Supposing that the target state | ψ a (cid:105) is in the OL with depth V , m -step preloading sequencehas been applied on the initial state | ψ i (cid:105) . The final states | ψ f (cid:105) is given by | ψ f (cid:105) = m (cid:89) j =1 ˆ Q j | ψ i (cid:105) , (1)where ˆ Q j = e − i ˆ H j t j is the evolution operator of the j th step. By maximizing the fidelity F idelity = |(cid:104) ψ a | ψ f (cid:105)| , (2)we can get the optimal parameters ˆ H j and t j .This preprocess is called a shortcut method, which can be used for loading atoms into differ-ent bands of an optical lattice. For example, the shortcut loading ultra-cold atoms into S-bandin a one-dimensional optical lattice is shown in Fig. 1.By setting different initial state and target state, different time sequences can be designedto manipulate atoms, to build different interferometers, which greatly saves the coherent time.Based on this shortcut method, we can prepare exotic quantum states (
3, 28, 49 ) and constructinterferometer with motional quantum states of ultra-cold atoms ( ). Suppressing the decoherence mechanism in the atomic interferometer is beneficial for increas-ing coherence time and improving the measurement accuracy. Here we demonstrated an echomethod can increase the coherent time for Ramsey interferometry with motional Bloch states (atzero quasi-momentum on S- and D-bands of an OL) of ultra-cold atoms ( ). The RI can be ap-plied to the measurement of quantum many-body effects. The key challenge for the constructionof this RI is to achieve π - and π/ -pulses, because there is no selection rule for Bloch states of8Ls. The π - or π/ -pulse sequences can be obtained by the shortcut method ( ), whichprecisely and rapidly manipulates the superposition of BECs at the zero quasi-momentum onthe 1st and 3rd Bloch bands. Retaining the OL, we observed the interference between statesand measured the decay of coherent oscillations.We identified the mechanisms that reduced the RI contrast: thermal fluctuations, laser in-tensity fluctuation, transverse expansion induced by atomic interaction, and the nonuniform OLdepth. Then, we greatly increased the coherent time (1.3 ms → This RI starts from BECs of Rb at the temperature 50 nK similar to our previous work ( ). Then a 1D standing wave OL is formed (Fig. 2). After a shortcut sequence, the BEC istransferred into the ground band of the OL, denoted as φ S, .Fig. 2(b) illustrates that the RI is constructed with Bloch states φ i,q , which includes theground band | S (cid:105) , the third band | D (cid:105) , and their superposition state ψ = a S | S (cid:105) + a D | D (cid:105) (denotedas (cid:0) a S a D (cid:1) ). It is difficult to realize the interferometer with this pseudo-spin system, because thereis no selection rule for Bloch states of OLs. However, thanks to the existence of a coherent,macroscopic matter-wave, a π/ -pulse for BECs in an OL can be obtained, where the Blochstates | S (cid:105) and | D (cid:105) are to be manipulated to | ψ (cid:105) = ( | S (cid:105) + | D (cid:105) ) / √ and | ψ (cid:105) = ( −| S (cid:105) + | D (cid:105) ) / √ ,respectively. Fig. 2(d) shows the π/ pulse we used for the RI (
28, 42, 43 ) with fidelities of . and . respectively (
27, 28, 42, 43 ).Fig. 2(c) illustrates the whole process of RI. First, the BEC is transferred to | S (cid:105) of an OL.Then, a π/ -pulse ˆ L ( π/ is applied to atoms, where ˆ L ( π/ (cid:0) (cid:1) = √ (cid:0) (cid:1) . After holding time t OL and another π/ -pulse, the state is: ψ f = ˆ L ( π/
2) ˆ Q ( t OL ) ˆ L ( π/ ψ i , (3)9 a) (b)(c)(d) Figure 2: Experimental configuration for a Ramsey interferometer in a V = 10 E r lattice: (a)The BEC is divided into discrete pancakes in yz plane by an 1-dimensional optical lattice along x axis with a lattice constant d = 426 nm. (b) Band energies for the S-band and the D-band.(c) Time sequences for the Ramsey interferometry. The atoms are first loaded into the S bandof OL, followed by the RI sequence: π/ pulse, holding time t OL , and the second π/ pulse.Finally band mapping is used to detect the atom number in the different bands. (d) The usedpulse sequences designed by an optimised shortcut method. Reproduced with permission fromRef. ( ). 10 (ms) p D (a)(b) Experiment with 50nkIdeal optical latticeReal inhomo. potentialTransverse expansionIntensity fluctuationThermal fluctuation (c) (ms) C ( ) Figure 3: (a) Change of p D , the population of atoms in the D-band, over time t OL with tem-perature T = 50 nK. (b) Influence of different mechanisms on the RI. (c) Characteristic time τ for the different number of π pulse n and different temperatures. The circles, squares, anddiamonds represent the experimental results and lines are fitting curves. Reproduced with per-mission from Ref. ( ). 11here ˆ L ( α ) = (cos α − i sin α )ˆ σ y . The operator ˆ Q ( t OL ) = (cos ωt + i sin ωt )ˆ σ z ( ω correspondsto the energy gap between S and D bands at zeros quasi-momentum).Fig. 3(a) depicts the results p D ( t OL ) = N D / ( N S + N D ) at different t OL for the RI process( ˆ R ( π/ − ˆ U ( t OL ) − ˆ R ( π/ ). N S ( N D ) represents the number of atoms in S-band (D-band).The period of the oscillation of p D is . ± . µs . This period related to the band gap andthe theoretical value is . µs . From Fig. 3(a), we can see that the amplitude, or the contrast C ( t OL ) , decreases with the increase of t OL , where p D ( t OL ) = [1 + C ( t OL ) cos( ωt OL + φ )] / . (4)We defined a characteristic time τ , which corresponds to the time when the C ( t OL ) de-creases to /e . Temperature can affect the length of τ . To improve RI’s coherent time and performance, we should analyze the mechanisms that causeRI signal attenuation. By solving the Gross-Pitaevskii equation(GPE), which considers themechanism that may lead to decay, we can get the process of contrast decay in theory. InFig. 3(b), The following mechanisms are introduced in turn: the effect of the imperfection ofthe π/ pulse (brown dashed line), inhomogeneity of laser wavefront (blue dotted line), thetransverse expansion caused by the many-body interaction (blue dashed line), laser intensityfluctuation (the dash-dotted line), and the thermal fluctuations (the orange solid line). Fig. 3(b)illustrates that the theoretical (the orange solid line) and experimental (black dots) curves of thefinal result are very consistent. In order to extend the coherence time τ , we proposed a quantum echo method. The echo processrefers to a designed π pulse ( ˆ L ( π ) ) that flips the atomic populations of the two bands. So the12able 1: The effects for the contrast decay.Decay factor Beam inhomogeneity Echo recovery Dephasing
Momentum dispersion Yes
Collision
Unbalance of population Yes
Decoherence fluctuation Noevolution operator of Echo-RI is ˆ L ( π/ Q ( t OL / n ) ˆ L ( π ) ˆ Q ( t OL / n )] n ˆ L ( π/ , where n is thenumber of the π pulse inserted between the two π/ pulses.Fig. 3(c) illustrates the characteristic time τ for different n and temperatures. And the effectsfor the contrast decay are listed in Table. 1. It can be seen from Fig. 3(c) that the interferometerwith the longest characteristic time (14.5 ms) was obtained when n ≥ and T = 50 nK. As an essential indicator, the resolution evaluates the performance of interferometers. The res-olution is theoretically restricted to shot-noise limit, or sub-shot noise limit (
55, 56 ), however,it will decay easily due to other experimental noises, with those upper limits beyond reach.Therefore, we have made several efforts to increase the resolution in practice. Improvementof the phase resolution was accomplished by a noise-resilient multi-component interferomet-ric scheme. With the relative phase of different components remaining stable, increasing thenumber of paths could sharpen the peaks in the interference fringes, which leads to a resolutionnearly twice compared with that of a conventional double-path two-mode interferometer withhardly any attenuation in visibility. Moreover, improvement of the momentum resolution isfulfilled with optical lattice pulses. Under the condition of 10 E R OL depth, atomic momentumpatterns with interval less than the double recoil momentum can be achieved, exhibiting 10 mainpeaks, respectively, where the minimum one we have given was 0.6 (cid:126) k L . The demonstration ofthese techniques is shown in the next four subsections.13 .1 Time evolution of two-component Bose-Einstein condensates with acoupling drive For the multicomponent interferometer, it is necessary to study the interference characteristicsof multi-component ultra-cold atoms. Here we introduced a basic method to deal with thisproblem, which simulates the time evolution of the relative phase in two-component Bose-Einstein condensates with a coupling drive ( ).We considered a two-component Bose-Einstein condensate system with weak nonlinear in-teratomic interactions and coupling drive. In the formalism of the second quantization, theHamiltonian of such a system can be written as ˆ H = ˆ H + ˆ H + ˆ H int + ˆ H driv , (5) ˆ H i = (cid:90) dx Ψ † i ( x )[ − (cid:126) m ∇ + V i ( x ) + U i ( x )Ψ † i ( x )Ψ i ( x )]Ψ i ( x ) , (6) ˆ H int = U (cid:90) dx Ψ † ( x )Ψ † ( x )Ψ ( x )Ψ ( x ) , (7) ˆ H driv = (cid:90) dx [Ψ † ( x )Ψ ( x ) e iω rf t + Ψ ( x )Ψ † ( x ) e − iω rf t ] , (8)where i = 1 and .Then the interference between two BEC’s is I ( t ) = 12 N + 12 ( N − N ) cos ω rf t + 12 e − A ( t ) sin ω rf t R ( t ) . (9)Previous analysis can be used to simulate the time evolution of the relative phase in two-component Bose-Einstein condensates with a coupling drive, as well as to study the interferenceof multi-component ultra-cold atoms. This simulation would help to construct a multimodeinterferometer of a spinor BEC (see Subsecs. 3.3).14
200 400 600 800 1000100200 z / ! y / ! (a) |2> |1> |0> |-1> |-2> (b1) (b2) (b3) (b4)(b5) (c) Figure 4: (a) One typical interference picture. These spatial interference fringes come from thefive sub-magnetic states of | F = 2 (cid:105) hyperfine level. (b1-5) Density distributions correspondingto different sub-magnetic components respectively, where the points are the experimental dataand the curves are fitting results according to the empirical expression ( ). (c) Averageof 15 consecutive experimental shots with a visibility reduction to zero for the chosen state | m F = − (cid:105) . Reproduced with permission from Ref. ( ).15 a ) (a!)(a") (a Figure 5: (a1)-(a4) Histograms of relative phases distributions ( φ − φ , φ − − φ − , φ − φ − , and φ − φ − ) respectively. These relative phases show good reproducibility, for the firsttwo are concentrated at about o , while the latter two are concentrated at about o in 61consecutive experimental shots; (b) Relative phase distributions of 41 consecutive experimentalshots with t = 3 . ms. Distributions of relative phases φ − φ and φ − φ − are shown in (b1)and (b2); (c)When t = 3 . ms, distributions of relative phases φ − φ and φ − φ − are shownin (c1) and (c2). The polar plots of relative phase vs visibility (shown as angle vs radius) areshown as these insets, respectively, where the value of visibility is an average of the visibilityinvolved in calculation. Reproduced with permission from Ref. ( ).16 .2 Parallel multicomponent interferometer with a spinor Bose-Einsteincondensate Revealing the wave-particle duality, Young’s double-slit interference experiment plays a criti-cal role in the foundation of modern physics. Other than quantum mechanical particles suchas photons or electrons which had been proved in this stunning achievement, ultra-cold atomswith long coherent time have got the potential of precision measurements when utilizing thisinterferometric structure. Here we have demonstrated a parallel multi-state interferometer struc-ture ( ) in a higher spin atom system ( ), which was achieved by using our spin-2 BECof Rb atoms.The experimental scheme is described as following. After the manufacture of Bose-Einsteincondensates in an optical-magnetic dipole trap, we switched off the optical harmonic trap andpopulate the condensates from | F = 2 , m F = 2 (cid:105) state to | m F = 2 (cid:105) and | m F = 1 (cid:105) sub-magneticlevel equally. After the evolution in a gradient magnetic field for time t , these two wave packetswere converted again into multiple m F states ( m F = ± , ± , ) as our spin states, leading tothe so-called parallel path. All these states were allowed to evolve for another period time t , then the time-of-flight (TOF) stage t for absorption imaging. Spatial interference fringeshad been observed in all the spin channels. Here, we used the technique of spin projectionwith Majorana transition ( ) by switching off the magnetic field pulses nonadiabaticallyto translate the atoms into different Zeeman sublevels. The spatial separation of atom cloudin different Zeeman states was reached by Stern-Gerlach momentum splitting in the gradientmagnetic field.A typical picture after 26 ms TOF is shown in Fig. 4(a). Fig. 4(b1-b5) are the densitydistributions for each interference fringes. To reach the maximal visibility, we studied the cor-relation between the interference fringes’ visibility and the time interval applying Stern-Gerlachprocess. Though separated partially, the interfering wave pockets must overlap in a sort of way.17he optimal visibility was about 0.6, corresponding to t = 210 µ s and t = 1300 µ s . We alsomeasured the fringe frequencies of different components, which exhibited a weak dependenceon m F .Special attention is required in Fig. 4(c). After an average of 15 consecutive CCD shots in re-peated experiments, the interference fringe almost disappeared for the chosen state | m F = − (cid:105) .This result manifested the phase difference between the two copies of each component in ev-ery experimental run is evenly distributed. The poor phase repeatability could be attributed touncontrollable phase accumulation in Majorana transitions.However, the relative phase across the spin components remained the same after more than60 continuous experiments, just as Fig. 5(a) illustrates. Furthermore, evidence has been spottedthat the relative phase can be controlled by changing the time t before the first Majorana tran-sition, as shown in Fig. 5(b)(c), paving a way towards noise-resilient multicomponent parallelinterferometer or multi-pointer interferometric clocks ( ). The experiment described above was achieved by Stern-Gerlach momentum splitting, sepa-rating the wave pockets in different spin states or Zeeman sub-magnetic states in space. Theconclusion that relative phases across the spin components remain stable gives us an inspirationto carry on the double-path multimode matter wave interferometer scheme. With the number ofpaths increased, it will suppress the noise and improve the resolution (
25, 26, 69–72 ) comparedwith the conventional double-path single-mode structure. The results show that resolution ofthe phase measurements is increased nearly twice in time domain interferometric fringes ( ).The experimental procedure is similar to the previous one. The major difference lies in thesplitting stage, during the optical harmonic trap participating in the preparation of the conden-18 a ) (a!) (a")(b ) (b!) (b")(c) Figure 6: (a1-a3) Single-shot spatial interference pattern with five interference modes afterTOF = 26 ms. Fringes of each mode are (a1)in phase (a2)partially in phase (a3)complementaryin space. (b1-b3) Black points are the experimental data by integrating the image in panels(a1-a3) along the z direction. Red solid lines are fitted by Thomas-Fermi Distribution ( ).Visibilities are 0.55, 0.24, and 0.05, respectively. (c) Schematic of the spatial interferenceimage. ∆ φ ( T d , T N ) is the relative phase between adjacent mode fringes. The fringe in eachcolor represents the interference between the two wave packets of a single mode. Reproducedwith permission from Ref. ( ). 19 ! " ! (a) ! " (b) ! " ! " { % } = {&,&, &, &} { % } = {&, &,’,’} { % } = {&. (’, &.)’, &. *’, ’}{ % } = {&, &, (cid:133) &} V N = V N ∆ϕ Figure 7: Dependence of the visibility on the number of modes N and initial relative phase φ m F of the same mode in two paths. (a) Dependence on N in a situation that φ m F are all zero.FWHM of the N-mode fringe is 2/N times that of the two-mode fringe. (b) Dependence on φ m F using N = 4 as an example. The green dashed line, red solid line, and purple dottedline show the fringes with ( φ , φ , φ , φ ) = (0 , , , , (0 , , π, π ) , and (0 . π, . π, . π, π ) ,respectively. Reproduced with permission from Ref. ( ).20ates is not going to switch off until the TOF stage, thus the Stern-Gerlach process in the gradi-ent magnetic field mentioned above cannot significantly split the wave packets. With differentmomentum atomic clouds are spatially separated only for tens of nanometers, approximately1% of the BEC size, thus well overlapped ( ). As a result, multi-modes from two paths willinterfere in one region instead of five. Another difference lies in the second spin projection withnon-adiabatic Majorana transition. Here we replace it with a radio frequency pulse for its higherefficiency as a 1 to 5 beam splitter, although that we still use it to transfer the initial condensatesinto | m F = 2 (cid:105) and | m F = 1 (cid:105) sub-magnetic levels. The performance of Majorana transition isbetter than RF pulse as a 1 to 2 beam splitter. There are also some changes with experimentalparameters that count a little and we would not discuss them here.Hence the global view of our interferometer is as follows: The magnetic sublevels are con-sidered as modes in the interferometer, each has its own different phase evolution rates in gra-dient magnetic field. The double path configuration is made up of Majorana transition as wellas the evolution of the first two m F superposed states during time T d , makes up (path I, pathII). RF pulse leading to the multiple m F superposed states together with their evolution in time T N forms the multi-modes configuration. During the TOF stage, atomic clouds expand and in-terfere with each other. Owing to the different state-dependent phase evolution rate ω ( I,II ) m F , theabsorption image shows something more than spatial interference fringes, which is a periodicdependence of the visibility on phase evolution time as the function V N ( T d , T N ) . We refer to itas the time domain interference.Fig. 6(a) shows a group of absorption images with various combinations of T d , T N . Theobserved fringe is a superposition of the interference fringes of different modes. Consequently,the visibility depends on the relative phase ∆ φ ( T d , T N ) between the interference fringes of eachmode [Fig. 6(c)] and can also be modulated.By carefully analyzing with expression V N ( T d , T N ) = (cid:104) Ψ ( I ) | Ψ ( II ) (cid:105) [34], we can acquire the21xpression of the relative phase between two adjacent components: ∆ φ ( T d , T N ) = (∆ φ m F − ∆ θ m F − )= ∆ ωT N + ∆ θ (10)where ∆ ω is the relative phase evolution rate between the two paths, ∆ θ is the relative initialphase introduced through the double path stage T d . Yet we have already demonstrated thatthe visibility V N is modulated with the period π/ ∆ ω along with how the time domain fringeemerges theoretically.A remarkable feature of the multi-modes interferometer is the enhancement of resolution,which is defined as (fringe period)/(full width at half maximum). We have investigated theresolution of the time domain fringe experimentally and theoretically. It can be influenced byparameters like modes number and initial phase, which is R ( N, φ m F ) . φ m F refers to the initialrelative phase of m F states accumulated in double paths T d .Fig. 7 is the numerical results considering an arbitrary number of modes. Fig. 7(a) is underthe condition that the phases φ m F are all the same for any modes. In that case, if we denote ∆ ωT N = 2 nπ/N , then the visibility achieves V N = 1 when n is the multiple of N and a majorpeak is observed in this case. A remarkable feature of our interferometer is the enhancementof resolution by N/ times without any changes in visibility nor periodical time. It is theharmonics that cause the peak width to decrease with the number of modes increasing in thiscase ( ). Fig. 7(b) indicates φ m F varies from mode to mode for comparison. Neither themaximum visibility V N = 1 nor the minimum could be reached. Meanwhile, the time domainfringe shows more than one main peak in one period. Therefore, the initial phase φ m F needs tobe well controlled to achieve the highest possible visibility and clear interference fringe in thetime domain.We also experimentally study the time domain fringes. The experimental data (not depictedhere) coincides with the numerical results of Fig. 7(b) red line, testifying its superiority to22 andmapping q/k L -1 -0.5 0 0.5 1 E / E r ω ( )q ω ( )q Superposed state
TheoryExperiment P k/k L TheoryExperiment P k/k L xy -3 -1 1 3-3 -1 1 3 t t (a )(a!) (b)(c) (d) Figure 8: (a) Shortcut method for loading atoms: (a1) after the first two pulses and the msholding time in the OL and the harmonic trap, the state becomes the superposition of the Blochstates in S-band with quasi-momenta taking the values throughout the FBZ, and is denoted by | ψ (0) (cid:105) . Then ms band mapping is added. (a2) The single pulse acted on the superposed state | ψ (0) (cid:105) . (b) the superposed Bloch states of S-band spreading in the FBZ (black circles). Thetop Patterns in (c) and (d) are the TOF images in experiments. The lower part of (c) and (d)depicts the atomic distributions in experiments (red circles) and theoretical simulations (bluesolid lines). There are seven peaks in (c) and ten peaks in (d) with q = ± (cid:126) k L . Reproducedwith permission from Ref. ( ).the resolution of the phase measurement. Moreover, the relative phase evolution rate ∆ ω canbe controlled by adjusting the difference between the two paths accumulated in T d stage ( ). With enhanced resolution, the sensitivity of interferometric measurements of physicalobservables can also be improved by properly assigning measurable quantities to the relativephase between two paths, as long as the modes do not interact with each other (
60, 75, 76 ).23 .4 Atomic momentum patterns with narrower interval
For ultra-cold atoms used in precise measurement, improving the precision of momentum ma-nipulation is also conducive to improving the measurement resolution. The method to getatomic momentum patterns with narrower interval has been proposed and verified by experi-ments ( ). Here we applied the shortcut pulse to realize the atomic momentum distributionwith high resolutions for a superposed Bloch states spreading in the ground band of an OL.While difficult to prepare this superposition of Bloch states, it can be overcome by theshortcut method. First, the atoms are loaded in the superposition of S- and D-bands ( | S, q =0 (cid:105) + | D, q = 0 (cid:105) ) / √ , where q is the quasi-momentum. Fig. 8(a1) depicts the loading sequence.The atoms in S and D bands Collision between atoms in S and D bands will cause the atoms togradually transfer to S band with non-zero quasi-momentum. After ms, as shown in Fig. 8(b),atoms cover the entire ground band from q = − (cid:126) k L to (cid:126) k L .The momentum distribution of the initial state is a Gaussian-like shape. After an OLstanding-wave pulse, which is similar to that in the shortcut process, the different patterns withthe narrower interval can be obtained. The standing-wave pulse sequence is shown in Fig. 8(a2).Fig. 8(c) and (d) show the different designs for patterns of multi modes with various numbersof peaks under OL depth 10 E r , where the top figures are the absorption images after the pulseand a ms TOF. The red circles are the experimental results of the atomic distribution alongthe x-axis from the TOF images. These results are very close to the numerical simulation result(blue lines). For the numerical simulation, we can get the initial superposition of states by fit-ting the experimental distributions (Fig. 8(b)). Fig. 8(c) and (d) depict the atomic momentumdistribution with resolutions of . (cid:126) k L interval (seven peaks within q = ± (cid:126) k L ) and . (cid:126) k L (ten peaks within q = ± (cid:126) k L ) interval, respectively.The superposed states with different quasi-momenta in the ground band cause the narrowinterval (far less than double recoil momentum) between peaks, which is useful to improve the24esolution of atom interferometer ( ). Noise identification as well as removal is crucial when extracting useful information in ultra-cold atoms absorption imaging. In general concept, systematic noises of cold atom experimentsoriginate from two sources, one is the process of detection, such as optical absorption imaging;the other is the procedure of experiments, such as the instability of experimental parameters.Here we provided an OFRA scheme, reducing the noise to a level near the theoretical limitas / √ of the photo-shot noise. When applying the PCA, we found that the noise origins,which mainly come from the fluctuations of atom number and spatial positions, much fewerthan the data dimensions of TOF absorption images. These images belong to BECs in onedimensional optical lattice, where the data dimension is actually the number of image pixels.If the raw TOF data can be preprocessed with normalization and adaptive region extractionmethods, these noises can be remarkably attenuated or even wiped out. PCA of the preprocesseddata exhibits a more subtle noises structure. When we compare the practical results with thenumerical simulations, the few dominant noise components reveal a strong correlation with theexperimental parameters. These encouraging results prove that the OFRA as well as PCA canbe a promising tool for analysis in interferometry with higher precision (
77, 78 ). Optical absorption imaging is an important detection technique to obtain information from mat-ter waves experiments. By comparing the recorded detection light field with the light field inthe presence of absorption, we can easily attain the atoms’ spatial distribution. However, dueto the inevitable differences between two recorded light field distributions, detection noises areunavoidable. 25 a ) (b )(a!) (b!)(c)
Figure 9: Comparison between ordinary method and OFRA method. The integral of the atomicdistribution in the red box in (a1) and (b1) correspond to (a2) and (b2). The atomic distribution(blue dots) is fitted by a bi-modal function to extract the temperature of atoms, which is shownin (c). Reproduced with permission from Ref. ( ).26herefore, we have demonstrated an OFRA scheme to generate an ideal reference light field.With the algorithm, noise generated by the light field difference could be eliminated, leadingto a noise close to the theoretical limit ( ). The OFRA scheme is based on the PCA, we con-firmed its validity by experiments of triangular optical lattices. The experimental configurationhas been described in our prior work (
28, 80 ). When the experiment was in process, the depthof the lattice was adiabatically raised to a final value, followed by a hold time of 20 ms tokeep the atoms in the lattice potential before the optical absorption imaging. There are severalparameters to characterize the triangle lattice system ( ), among which the visibility, the con-densate fraction, and the temperature matter. Fig. 9 (a2) and Fig. 9 (b2) are bimodal fitting tothe scattering peaks by summing up the atomic distribution within the red box in the directionperpendicular to the center. Here Fig. 9 (a) stands for the common way of calculation and 9(b)for the OFRA. The bi-modal curve is composed of two parts: a Gaussian distribution for thethermal component and an inverse parabola curve for the condensed atoms. For each part, thecolumn densities along the imaging axis can be written as n th ( x ) = n th (0) g (1) g [exp( − ( x − x ) /σ T )] , (11) n c ( x ) = n c (0) max[1 − ( x − x ) χ ] . In the formula there are 5 parameters accounting for the bi-mode fitting, the amplitude oftwo components n th (0) and n c (0) , the width of two components σ T , χ and the center position x of the atomic cloud. The Bose function is defined as g j ( z ) = (cid:80) i z i /i j . In practice, weperformed the least-squares fitting of n th ( x ) + n c ( x ) to the real distribution obtained from theimaging. From the fit, we can get the atom number and width of the two components separately.Note that the measurement in Fig. 9 (9c) is performed at different lattice depths. For each latticedepth, 30 experiments have been performed to acquire the statistical results. The temperature27s given as T = 1 / M σ T /t T OF /k B , where M described the atom mass and t T OF width of thethermal part ( ).For the number of condensed atoms, the fitting outcome is less affected by the fringe shownin Fig. 9 (a). Whereas the influence of the fringe on the fitting of temperature is much moreevident. The temperature is proportional to the width of the Gaussian distribution σ T as men-tioned above. Fig. 9(c) shows the temperature extracted from the TOF absorption images withand without the OFRA separately, namely Fig. 9(a1) and 9(b1). Fig. 9(a2) and 9(b2) are thecorresponding integrated one-dimensional atomic distributions for each method. Fig. 9 depictsthat the temperature we get with the common way of calculation has a large error of 400 nK,extraordinary higher than the initial BEC temperature of 90 nK. The turning on the procedureof lattice potential would indeed lead to a limited heating effect, nevertheless the proportion ofcondensed atom should be reduced significantly considering our system has been heated up by4 times. This is still not consistent with the observation. However, the temperature is measuredwith much small variance at a much reasonable value if we dive into the OFRA scheme. Forexample, the measured temperature is 123.5 nK for a lattice depth of V = 4 E r , with 183.9 nKfor V = 9 E r . Comparison between these two results illustrates that only by using the fringeremoval algorithm we can get a reliable result, especially in the case of small atom numberswhen fitting physical quantities such as the temperature.In conclusion, with this algorithm, we can measure parameters with higher contradictionto the conventional methods. The OFRA scheme is easy to implement in absorption imaging-based matter-wave experiments as well. There is no need to do any changes to the experimentalsystem, only some algorithmic modifications matter.28 .2 Extraction and identification of noise patterns for ultracold atoms inan optical lattice Furthermore, on the basis of the absorption images after preprocessing by OFRA, the PCAmethod is used to identify the external noise fluctuation of the system caused by the imperfec-tion of the experimental system. The noise can be reduced or even eliminated by the corre-sponding data processing program. It makes the task more difficult that these external system-atic noises are often coupled, covered by nonlinear effects and a large number of pixels. PCAprovides a good method to solve this problem (
77, 83–86 ).PCA can decompose the fluctuations in the experimental data into eigenmodes and providean opportunity to separate the noises from different sources. For BEC in a one-dimensional OL,it was proved that PCA could be applied to the TOF images, where it successfully separated andrecognizing noises from different main contribution sources, and reduced or even eliminatednoises by data processing programs ( ).The purpose of PCA is to use the smallest set of orthogonal vectors, called principal compo-nents (PCs) to approximate the variations of data while preserving the information of datasetsas much as possible. The PCs correspond to the fluctuations of the experimental system, whichcan help to distinguish the main features of fluctuations. In the experimental system of BECs,the data are usually TOF images. A specific TOF image A i can be represented by the sum ofthe average value of the images and its fluctuation: A i = ¯ A + (cid:88) ε ij P j , (12)Here P j is the different eigenmodes of fluctuation, ε ij is the weight of the eigenmodes P j .Taking the BEC experiment (
27, 28, 43, 87 ) in an OL as an example, we demonstrated theprotocol of PCA for extraction noise. A TOF image for ultra-cold atoms in experiments canbe represented as a h × w matrix. The PCA progress, shown in Fig. 10, will be applied to the29 a) (b) (c) (d)(e) Figure 10: Process diagram of PCA method. (a) Transform the h × w matrix (raw images) intoa × hw vector, denoted by A i . And stack these vectors together. (b) Calculate the mean vector ¯ A = n (cid:80) ni =1 A i and the fluctuations δ i = A i − ¯ A . Leaving only the fluctuation term in thematrix. (c) Stack δ i together to form a matrix X = [ δ , δ , · · · , δ n ] . Then the covariance matrix S is obtained by S = n − X · X T . (d) Decompose covariance matrix so that V − SV = D where D is a diagonal matrix. (e) Transform eigenvectors of interest back to a new TOF image.Reproduced with permission from Ref. ( ).images: (1) Transform the h × w matrix into a × hw vector, denoted by A i . (2) Express A i as the sum of the average value ¯ A and the fluctuation δ i , where ¯ A = n (cid:80) ni =1 A i and δ i = A i − ¯ A . (3) Stack δ i together to form a matrix X = [ δ , δ , · · · , δ n ] . Then the covariance matrix S isobtained by S = n − X · X T . (4) Decompose covariance matrix with V − SV = D , where V is the matrix of eigenvectors.Fig. 11 shows the PCA results of the TOF images. Fig. 11(b)-(f) correspond to the first tofifth PCs, respectively.The first PC is from the fluctuation in the atom number. The normalization process canbe applied to reducing the fluctuation in the atomic number. Because for a macroscopic wavefunction Ψ( r ) = √ N φ ( r ) ( ), we usually concentrate on the relative density distribution,instead of the √ N . After the normalization, the impact of this PC becomes very small.30 μ/xmμ/x x/μmmμ/xmμ/xrow sum (a") (b")(a Figure 11: PCA results of the TOF images. (a) Example of a raw TOF image. (b)-(f) corre-spond to the fluctuation in atom number (b), atom position (c)(d), peak width (e) and normalphase fraction (f), respectively. (a ), (b ), (d ), (e ), and (f ) are the integrated results of atomdistributions along x direction. (c ) is for the atom distribution along z direction. The blue linesare the experimental results, and the orange lines are the simulation results. Reproduced withpermission from Ref. ( ). 31he second and third PCs correspond to the position fluctuations along the z - and x -directions,respectively. The fluctuation in spatial position of the TOF images originates from the vibrationof the system structure, such as the OL potential, trapping potential, and imaging system. Weused a dynamic extraction method to eliminate the fluctuation in spatial position. We chosea region whose center is also the center of the density distribution. We first set a criterion todetermine the center of the density distribution in the extraction area, and then used this centeras the center of the new area to extract the new one. We repeated this process until the regionto be extracted becomes stable.The fourth PC is from the fluctuation in the width of the Bragg peaks in the TOF images.The final PC shown in Fig. 11(f1) comes from the normal phase fraction fluctuation.By studying the first five feature images, we have identified the physical origins of severalPCs leading to the main contributions. We numerically simulated this understanding usingthe GPE with external fluctuation terms, and got very consistent results ( ). It is helpful tounderstand the physical origins of PCs in designing a pretreatment to reduce or even eliminatefluctuations in atom number, spatial position and other sources. Even in the absence of anyknowledge of the system, the PCA method is very effective to analyze the noise, so that it canbe applied to interferometers with higher precision (
77, 78 ). Inertia measurements ( ), especially those for gravity acceleration g , have always drawnlots of attention. Until now, the performance of atom interferometry has reached a sensitivityof × − at 1 second (
90, 91 ), pushing forward the determination of the Newtonian gravita-tional constant G (
92, 93 ) or the verification of equivalence principle (
94, 95 ). Yet the bulky sizeof these quantum sensors strictly restricts their application for on-site measurements. There-fore, based on our previous study of ultra-cold atoms in precision measurements, we intend to32igure 12: Schematic of the experimental protocol. Rb atoms are evaporation cooled as aBose-Einstein Condensate in the | F = 2 , m F = 0 (cid:105) initial state. However, they are transferredto | F = 1 , m F = 0 (cid:105) as the two arms of interferometry, followed by P sequence of accelerateoptical lattice pulse to maintain the atoms against gravity: When atoms fall to a velocity of q × v Recoil , they acquire a velocity of q × v Recoil upwards. The delay T Bloch is chosen as T Bloch = 2 qv Recoil /g to eliminate the fall caused by gravity. Still, the probe beam should consista laser light resonant with the F = 1 ground state to upper levels, thus we can take absorptionphotos of F = 1 population for analysis.precisely measure the local gravity acceleration with our Rb Bose-Einstein Condensates in asmall displacement. Note that this conception bases on previous research of Perrier Clad´e ( ).Fig. 12 illustrates the protocol of this BEC gravimeter. Instead of the Mach-Zender method ( π/ − π − π/ widely used in free-falling or atomic fountain gravimeters, we utilize theRamsey-Bord´e approach by two pairs of π/ Raman pulses to get a small volume. Applicationof the Doppler-sensitive Raman beam rather than the 6.8 GHz microwave field provides a farmore efficient way to realize larger momentum splitting, which will significantly boost theinterference resolution. Raman light pulse can also attain the effects of velocity distribution.Consequently, the first pair of π/ pulses selects the initial velocity while the second pair canmeasure the final distribution. It should be noticed that right after the velocity selection step, acleaning light pulse resonant with the D line will shine on the condensate to clear away atomsremaining in F = 2 state, leading to the two arms in interferometry.33he critical feature lies in the evolving stage between the two pairs of π/ pulses. By pe-riodically inverting the velocity, the two arms shall replicate parabolic trajectory in a confinedvolume, without any decreasing of interrogation time. Choosing appropriate parameters, dis-placement of ultra-cold atoms can be limited in a few centimeters, at least 1 order less than thatof a Mach-Zender gravimeter. This assumption is accomplished by a succession of Bloch os-cillation (BO) in a pulsed accelerated optical lattice which transfers many photon recoils to thecondensates ( ). Here the application of ultra-cold atoms instead of optical molasses ( )makes it more efficient when loading the atoms in the first Brillouin zone adiabatically, owingto their wave function consistency and narrow velocity distribution. This pulsed acceleratedoptical lattice should be manufactured along the direction of gravity with higher lattice depth,to minimize the effect of Landau-Zener Tunneling loss.To deduce the value of g, we may scan the evolving time between the two pairs of π/ pulses, keeping the Raman frequency of each pair of π/ pulses fixed. When the time in-terval equals P qv
Recoil /g , where P is the number of pulses and q is the number of recoilvelocities( v Recoil ) obtained by a single pulse (shown in Fig. 12), the two arms are in phase andthe value of g can be extrapolated. Here the absorption image is used to extract the interfer-ence information, due to the number of atoms being one order less than that obtained by theconventional method.We also give a qualitative analysis of this compact BEC gravimeter. Besides its enormouspotential in transportable instruments, prospective sensitivity maybe even better. This encour-aging outlook can be attributed to a longer coherent time of ultra-cold atoms where the phaseshift scales quadratically. The smaller range of movement possesses other superiorities. Sys-tematic errors stemming from the gradients of residual magnetic fields and light fields becomenegligible, especially for Gouy phase and wave-front aberrations (
91, 101, 102 ). Furthermore,the value of the gravity acceleration is averaged over a smaller height compared with the Mach-34ender ones. Finally, this vertical Bloch oscillation technique offers a remarkable ability tocoherently and efficiently transfer photon momenta ( ), though decoherence induced by theinhomogeneity of the optical lattice must be taken into consideration.In conclusion, we believe that this compact BEC gravimeter will have a sensitivity of afew tens of µ Gal at least. The falling distance will be no more than 2 centimeters. Furtherimprovement should be possible by performing atom chip-assisted BEC preparation as well asthe interaction-suppressing mechanism ( ). Gravity measurements with sub- µ Gal accuraciesin miniaturized, robust devices are sure to come in the future.
In summary, we review our recent experimental developments on the performance of inter-ferometer with ultra-cold atoms. First, we demonstrated a method for effective preparationof a BEC in different bands of an optical lattice within a few tens of microseconds, reducingthe loading time by up to three orders of magnitude as compared to adiabatic loading. Alongwith this shortcut method, a Ramsey interferometer with band echo technique is employed toatoms within an OL, enormously extend the coherence time by one order of magnitude. Ef-forts to boost the resolution with multimode scheme is made as well. Application of a noise-resilient multi-component interferometric scheme shows that increasing the number of pathscould sharpen the peaks in the time-domain interference fringes, which leads to a resolutionnearly twice compared with that of a conventional double-path two-mode interferometer. Wecan somehow boost the momentum resolution meanwhile. The patterns in the momentum spacehave got an interval far less than the double recoil momentum, where the narrowest one is givenas . (cid:126) k L . However, these advancements are inseparable with our endeavor to optimize dataanalysis based on the PCA. Extrinsic systematic noise for absorption imaging can be reduced ef-ficiently. A scheme for potential compact gravimeter with ultra-cold atoms has been proposed.35e believe it will tremendously shrink the size of a practical on-site instrument, promotinganother widely used quantum-based technique. Acknowledgment
This work is supported by the National Basic Research Program of China (Grant No. 2016YFA0301501),the National Natural Science Foundation of China (Grants No. 61727819, No. 11934002, No.91736208, and No. 11920101004), and the Project funded by China Postdoctoral Science Foun-dation.
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