Manipulation of an elongated internal Josephson junction of bosonic atoms
A. Farolfi, A. Zenesini, D. Trypogeorgos, A. Recati, G. Lamporesi, G. Ferrari
MManipulation of an elongated internal Josephson junction of bosonic atoms
A. Farolfi, ∗ A. Zenesini, † D. Trypogeorgos, ‡ A. Recati, G. Lamporesi, and G. Ferrari
INO-CNR BEC Center, Dipartimento di Fisica, Universit`a di Trento and TIFPA-INFN, 38123 Povo, Italy (Dated: February 1, 2021)We report on the experimental characterization of a spatially extended Josephson junction realizedwith a coherently-coupled two-spin-component superfluid, trapped in an elongated potential. Westudy how dimensionality and inhomogeneity affect the control of the local magnetization andidentify a protocol for the preparation of the whole system in the ground state.
I. INTRODUCTION
Superfluid mixtures have recently attracted a renewedinterest thanks to the ability of cooling and manipulatingmulti-component atomic gases. In particular, the pres-ence of population transfer deeply modifies the physicsof the mixtures and have been widely studied and ex-perimentally realized both in double-well potentials andin coherently-coupled two spin states. The richness ofthese systems resides in the possibility to study phe-nomena spanning from nonlinear dynamics [1–4] to non-classical states [5–9]. In most of the cases, the mixturesare treated and behave as point-like entities, with frozenspatial degrees of freedom [10, 11].Spatially extended environments require more sophis-ticated theoretical approaches which incorporate, amongmany, dephasing–rephasing dynamics effects [12, 13] andlocal squeezing [14]. Experimentally, more stringent con-ditions are required to have full control and precise de-tection over an extended system. Up-to-now this has sofar not been fully investigated. Pioneering works havebeen reported on one-dimensional (1D) systems study-ing the dynamics in the large-coupling regime [15, 16],the phase transition dynamics [17] and in the case whereinteractions and coupling compete, creating inhomoge-neous magnetic-like heterostructures [18].While it is not an issue for single-mode systems, thecreation of a spatially-uniform coherent initial conditionfor an extended inhomogeneous sample is a difficult taskbecause of inhomogeneity-induced nonlinear effects. Dis-tant parts of the system can react differently to the exter-nal driving, depending on local properties of the systemas, for example, atomic density or external field inhomo-geneities. A strong Rabi coupling could overcome issuesrelated to the inhomogeneity, but requires very strongfields that cannot always be experimentally realized.Here we present a characterization of the techniquesused to initialize an effective spin 1 / ∗ arturo.farolfi@unitn.it † [email protected] ‡ Current address: CNR Nanotec, Institute of Nanotechnology, viaMonteroni, 73100, Lecce, Italy for an effectively 1D system. In Sec. V, we study howdensity effects prevents to produce a homogeneous state.We report on an adiabatic method that optimizes theachievement of a uniform state in Sec. VI. Finally, wepresent characterization measurements on plasma oscil-lations (Sec. VII) and on the loss of coherence of thesample (Sec. VIII).
II. THE EXPERIMENTAL SYSTEM
In our apparatus [19–21] we produce Bose-Einsteincondensates (BECs) of Na in an elongated far-off-resonance optical dipole trap, which presents a cylin-drical symmetry with tunable aspect ratio [Fig. 1(a)].Typical experimental conditions consist in radial andlongitudinal trap frequencies ω ρ / π = 500 Hz-2 kHz and ω x / π ≈
10 Hz, and a variable atom number N up to3 × .Atoms are initially prepared in the | F, m F (cid:105) = | , − (cid:105) state (later referred as |↓(cid:105) ), where F is the total atomicangular momentum and m F its projection on the quan-tization axis set by a uniform and highly stable magneticfield of 1 . µ G over tens of minutesof continuous experimental cycling.A two-photon Raman microwave transition to the | , +1 (cid:105) (later referred as |↑(cid:105) ) is suddenly introduced [seeFig.1(b)]. The two microwave frequencies are detuned FIG. 1. (a) The trapped cloud presents an elongated andcylindrically symmetric shape. The cloud profile in the x direction follows the Thomas-Fermi inverted parabola. (b)Level scheme and microwave radiations used to couple thetwo states | , ± (cid:105) . δ represents the detuning between thetwo-photon coupling and the | , ± (cid:105) energy difference. ∆ isthe detuning from the virtual state | , (cid:105) . a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n by ∆ from the state | , (cid:105) . The effective Rabi coupling Ωbetween |↓(cid:105) and |↑(cid:105) is inversely proportional to ∆ and weuse the latter to tune Ω while keeping the single-photonRabi frequencies fixed. The two-photon process can bedetuned from the | , ± (cid:105) transition by δ , that we tune byvarying the magnetic field. An additional microwave ra-diation introduces a quadratic Zeeman shift on the | , (cid:105) to suppress spin-changing collisions.After applying the coherent coupling for a given time t , the population in each state is independently imagedafter a short time of flight.States |↑(cid:105) and |↓(cid:105) have equal intrastate coupling con-stants g − = g +1 = g and interstate coupling constant g − , +1 , with a positive difference δg = g − g − , +1 . Thisleads to a full miscibility of the spin mixture [23–25]. III. DIMENSIONALITY REDUCTION
The dynamics of the density and of the pseudo-spinin an elongated two-component Bose-Einstein conden-sate can be either effectively one- or three-dimensional(3D) depending on the characteristic lengths of spin anddensity excitations in comparison to the radial size ofthe condensate. In an equally populated uniform samplewith total density n D , the density and spin excitationsare characterized by the healing length ξ = (cid:126) / (cid:112) mn D g and by the spin healing length ξ s = (cid:126) / (cid:112) mn D δg , re-spectively. In a trapped condensate, the density followsthe Thomas-Fermi distribution n D = n D (1 − ρ R ρ − x R x )and the radial and axial sizes are given by the Thomas-Fermi radii R ρ and R x . The ratios R ρ /ξ and R ρ /ξ s , eval-uated in the center of the sample, depend on the choice ofthe trap parameters and the peak density n D as follows R ρ ξ = 2 n D g (cid:126) ω ρ (1) R ρ ξ s = 2 n D g (cid:126) ω ρ (cid:115) δgg . (2)In our case, R x is always much larger than ξ and ξ s .Since (cid:112) δg/g ≈ / ξ s ∼ R ρ ), while the to-tal density of the sample is still well described by theThomas-Fermi approximation ( ξ (cid:28) R ρ ) and the relevantquantity characterizing the radial size is simply the 3DThomas-Fermi radius R ρ .In order to verify which value of R ρ /ξ s discriminatesclouds with a 1D spin dynamics from clouds with a 3Done, we search for radial spin dynamics applying the fol-lowing protocol for different values of trap frequenciesand atom number. We apply a resonant coherent cou-pling to the initial condensate with all atoms in |↓(cid:105) for atime t = π/ Ω, i.e., a Rabi π -pulse. We set Ω ≈ . n D δg , FIG. 2. Spatial distribution of the two components ( |↑(cid:105) inblue and |↓(cid:105) in red) for an effectively 1D (a) and 3D (b) sam-ple. R ρ /ξ s are 1.6 and 4.9, respectively. (c) Magnetizationalong y at x = 0, integrated along z , after a π -pulse for dif-ferent values of R ρ /ξ s (values are reported above the plots).Confidence interval of one standard deviation is indicated asshaded region. in order to guarantee a region at the center of the cloud,where the magnetization remains self-trapped close to |↓(cid:105) [26][Chap. 21].Since R ρ is comparable to our imaging resolution, welet the system expand for a short time, prior to imag-ing, in order to magnify the radial distribution of thepopulation. After releasing the atoms from the trap, welet them freely expand for 2 ms (3 ms) before the state |↓(cid:105) ( |↑(cid:105) ) is imaged. Different expansion times are usedto separately image the two states. Due to the differ-ent expansion time, the observed clouds have differentradial dimension. We rescale the second image along y by considering that the radial size expands accordingto R ρ ( t ) = R ρ (0) (cid:113) ω ρ t [27]. While this relation isstrictly correct only for an expanding single-componentcondensate, we observe that this is a good approxima-tion also for the total density of a two-component sys-tem, even in the presence of magnetic excitations. In-deed, the large energy difference between density- andspin-excitations allows the former one to dominate theexpansion of the condensate. Moreover, since the expan-sion times are much shorter than 1 /ω x , the radial expan-sion is ensured with negligible axial motion, allowing fordirect imaging of the radial distribution of population.Figure 2(a) and Fig. 2(b) highlight the differences be-tween the 1D and 3D regime. In an effective 1D system,radial features in the magnetization are absent and thepopulation in the center of the cloud remains in |↓(cid:105) , asshown in Fig. 2(a), for which R ρ /ξ s = 1 .
6. When thesample is more 3D, radial excitations lead to nonuniformradial distribution, as can be seen in Fig. 2(b), where R ρ /ξ s = 4 .
9. In Fig.2(c) we average the density along the x -axis for the central 100 µ m region for different valuesof the ratio R ρ /ξ s . Note that integration along one of theradial directions happens naturally through the absorp-tion imaging technique. We observe that the transitionbetween radially uniform and inhomogeneuos takes placeat R ρ /ξ s ≈
3. This indicates that, in the regime used inRef. [18], the system can be considered effectively 1D.For comparison, single component condensates in elon-gated traps, admit stable topological structures in thetransverse direction for R ρ /ξ ≈ R ρ /ξ s = 2 .
5, therefore, in the following, we con-sider only the 1D axial dynamics.
IV. THEORETICAL MODEL
We are mainly interested in studying the dynamics ofthe magnetization of the atomic cloud, and its coherencein a non-uniform trap. In this respect, it is convenientto describe the BEC in terms of its (position-dependent)total density n D and its spin-density s . In particular, s z describes the population difference in the |↑(cid:105) and |↓(cid:105) states and | s | = n D . Neglecting both density and spincurrents, the total density is constant and the spin dy-namics is described by the nonlinear precession equation[18, 30, 31] ˙ s ( r ) = H ( s ) × s ( r ) . (3)The effective magnetic field H ( s ) = Ωˆ x + ( δ + δg (cid:126) s z )ˆ z isdue to the presence of several SU (2) symmetry breakingterms: the homogeneous transverse microwave Rabi cou-pling Ω, the linear detuning δ and the nonlinear detuning δg (cid:126) s z arising from the difference between the intra- andinterspecies interaction constants δg . The nonlinear termin H is referred to as magnetic anisotropy in the contextof ferromagnetism and as capacitive term in the contextof Bose-Josephson dynamics (see also below).As we explained in the previous Section, the densityof our cloud is well described by a cylindrically symmet-ric Thomas-Fermi profile and in our geometry the spindynamics occurs only in the axial direction. By integrat-ing in the radial plane, we can describe the dynamicsof the spin along the axial direction introducing the 1Dspin-density s ( x ), such that | s ( x ) | = n ( x ) = n (1 − x /R x ) . (4)The spin-density obeys the following 1D version ofEq.(3): ˙ s ( x ) = [Ω , , δ + κs z ( x )] × s ( x ), where the non-linear coupling strength is κ = 56 (cid:126) δgπR ρ , (5) and is related to the 3D density through κn = 23 (cid:126) n D δg. (6)We recall here that, locally, Eq. (3) is equivalentto the so-called Bose-Josephson junction equations [32],which are written in terms of the normalized magneti-zation Z ( x ) = s z /n and of the relative phase φ ( x ) =arctan( s y /s x ). In such a context, it has been realizedthat Eq. (3) has different dynamical regimes. In the par-ticular case of δ = 0, for Ω > | κn | the dynamics resemblesRabi oscillations for any initial state. For Ω < | κn | , in-stead, a self-trapped regime appears around the points Z = ± (cid:112) − Ω / ( κn ) , φ = π , where the magnetizationnever changes sign. For Ω < | κn | /
2, the initial states Z = ± V. DENSITY-DEPENDENT SHIFT
In dense atomic clouds, transitions between energy lev-els are modified by the presence of interactions, whoseeffects can be introduced by means of mean-field correc-tions. These are commonly known as collisional shiftsand have great importance in metrology [33]. In aJosephson system, collisional shifts are dominant whenthe nonlinear mean field contributions are of the sameorder of magnitude of (or larger than) the linear couplingstrength.Starting from a fully polarized sample in |↓(cid:105) , a π -pulsewith Rabi coupling of Ω = 2 π ×
126 Hz is applied totransfer part of the population to |↑(cid:105) . Depending onthe (global) detuning and on the (local) nonlinear con-tribution, the final magnetization will locally change [seeFig. 3(d)]. The measurement is repeated for differentvalues of the detuning δ of the coupling from the transi-tion frequency and the final magnetization is plotted inFig. 3(a).On the thermal tails of the cloud the density is lowand the system is well approximated by a pure two-levelsystem (deep Rabi regime), where the amount of trans-ferred population depends on the detuning δ accordingto the commonly known sinc-like spectroscopic curve [or-ange data and curve in Fig. 3(b)]. When the nonlinearterm is no longer negligible as compared to Ω, the dy-namics follows the Josephson equations [blue data andcurve Fig. 3(b)]. The spectroscopic curve becomes asym-metric with a shifted peak. The direction and magnitudeof the shift depend on the sign and magnitude of κn , re-spectively.We fit the data at different x [Fig. 3(b)] with the nu-merical solution of the Josephson equation by having κn as the only free parameter. The fitted local value of κn ( x ) FIG. 3. (a) Local magnetization after a pulse of duration t = π/ Ω for different values of the detuning δ . The centerof the cloud is at x = 0 and R x = 200 µ m. (b) Verticalcuts for high- (blue, x = − µ m to 20 µ m) and low-density(orange, | x | = 180 µ m to 200 µ m) regions from data in (a).Lines correspond to the solution of Eq. 3 with density as fit-ting parameter. Significant asymmetry of the resonance peakis observed in the high-density region. (c) Local nonlinearparameter at different positions (points). The data are in ex-cellent agreement with the Thomas-Fermi profile derived fromthe fit to the total density (line). Error bars coming from thefit in (b) are smaller than symbol size. (d) Spin precessionleads to different final magnetization, depending on δ and κn .For δ = 0, the orange arrow reaches the south pole, while theblue one ends on a different point. is in good agreement with Eq. (4) and Eq. (6) and pro-vides a peak value κn / π = 297(15) Hz. VI. DENSITY-DEPENDENT ADIABATICRAPID PASSAGE
Different proposals in the field of nonlinear spin-waves[34, 35], quantum computation and squeezing requirethat the full cloud must be prepared in a state with auniform magnetization Z = 0. To this task, the proce-dure presented in the previous Section can be used onlyif the regime Ω (cid:29) κn is experimentally reachable. Inthe case of Ω ∼ κn , the magnetization of the cloud af-ter a π -pulse is not uniform, as Fig. 3 clearly shows. Adifferent approach is based on the Adiabatic Rapid Pas-sage (ARP). This has been used, for instance, to generatenumber-squeezed states [36].In the ARP, the coupling is applied to a polarized statewith an initially large detuning δ , so that the system is inthe state of minimum energy. The detuning is adiabat- ically swept to a final value close to resonance. Duringthe ramp, the local magnetization and δ are connectedthrough the following relation [36] δ = Ω Z √ − Z + κnZ, (7)while φ = 0 during the whole passage.Note that, far in the Rabi regime, the magnetizationdepends only on Ω /δ , while in the Josephson regime, anadditional density-dependent term is present. At the be-ginning of the ramp, all parts of the cloud are close tothe north pole of the Bloch sphere. Due to the inhomo-geneous nonlinear interaction, the magnetization has aposition-dependent evolution. However, if δ is adiabati-cally reduced to zero, at the end of the ARP, the wholesystem will reach Z = 0 simultaneously, independent ofthe value of the local nonlinear parameter, as sketched inFig.4(d).In our experiment, we start from a polarized samplein |↓(cid:105) , turn on a coupling with Ω = 2 π ×
185 Hz withan initial detuning δ ≈ π × δ on the magnetic field B , the sweep of the detuning FIG. 4. Adiabatic Rapid Passage (a) Local magnetizationafter a ARP as a function of the final detuning δ f . (b)Magnetization at positions x = − µ m to 20 µ m(blue) and x = − µ m to − µ m (orange). Errorbars are standarddeviation in the integrated region. Line is a sigmoidal func-tion fitted to the data. (c) Nonlinear parameter extractedfrom the derivative of the sigmoidal fit. (d) Blue and orangearrows correspond to high and low density regions, respec-tively. The second one is rotated from z to x as a single atomwould, while the first one has an initial larger velocity and afinal smaller velocity because the nonlinear effect of interac-tions. However, the nonlinear effects compensates along thepath and they arrive simultaneously on the equator. is performed by keeping constant microwave frequenciesand by varying the strength of the magnetic field in 50 mswith a nonlinear ramp. The ramp is stopped to a variablefinal value B f corresponding to a final δ f and in Fig. 4(a)we plot the magnetization of the sample as a function ofthe coordinate x and δ f .The magnetization at δ = 0 of the ARP procedureis less sensitive to magnetic field fluctuations, since, ex-panding Eq. (7) near Z = 0, one gets ∂Z∂δ = 1Ω + κn (8)that is lowered by the nonlinear term. Figure 4(b) showshow the final value of the magnetization is sensitive tothe final detuning, with a much smaller sensitivity in thecentral part of the system (blue points) rather than atthe edges (orange).By fitting the dynamics of the magnetization for eachposition x in the vicinity of zero magnetization, we ex-tract the slope of the magnetization as a function of δ andhence κn applying Eq. (8) [see Fig. 4(c)]. Note that theefficiency of the full rotation is increased by the nonlinearterm.The determination of the single state population andthe not full adiabaticity of the process introduce system-atic errors, and the nonlinear term derived from the ARPprocedure provides a larger value for κn than the oneobtained in Sec. V. However this method allows for aclean preparation of the extended system in a uniform Z = 0 state, at the expected value δ f = 0. VII. PLASMA OSCILLATIONS
In the presence of coherent coupling and at δ = 0, theground-state of the system is uniformly Z = 0, φ = 0.For small deviations near the ground-state, the Joseph-son dynamics predicts small oscillations around Z = 0and φ = 0, which are known as plasma oscillations [seeFig. 5(a)]. Their frequency follows ω p = (cid:112) Ω(Ω + κn ) , (9)allowing to determine κn from independent measure-ments of Ω and ω p .The sample is prepared in Z = 0 , φ = 0 with the previ-ously described ARP procedure. Then, the phase of thecoupling is suddenly modified from φ = 0 to φ = 0 . π ,starting the oscillatory dynamics [see Fig. 5(b)]. We in-dependently determine Ω driving Rabi dynamics on avery dilute thermal cloud, in order to ensure Ω (cid:29) | κn | .We repeat the procedure for different Ω. By extractingthe oscillation frequency at the center of the cloud andby taking advantage of the dependence of ω p on Ω wedetermine κn [Fig. 5(c)]. As for the methods used in theprevious Section, local fits in the condensate allow to de-termine κn for different x -position according to Eq. (9)[Fig. 5(d)]. In this case we obtain κn / π = 276(3) Hz. FIG. 5. Plasma oscillations. (a) The plasma oscillation areprecession around the x -axis, where the vector Ω lays. (b)Evolution of the magnetization for the initial state Z = 0 and φ = 0 . π , showing plasma oscillations. (c) Plasma oscillationfrequency at x = 0 for different values of Ω. The line is afit according to Eq. 9, hence providing the κn parameter atthe position where the plasma frequency is measured. (d)Local nonlinear parameter extracted from plasma oscillationfrequency. By means of this spectroscopic measurement of plasmafrequencies, we can determine κn with high precisionand overcome systematic uncertainties in the determina-tion of the atom number. Furthermore the results arenot sensitive to the uncertainties on the initial positionon the Bloch sphere. Together with the measurement of R x from the absorption images, we can fully and accu-rately measure κn ( x ), as we did in a previous work [18]. VIII. ROLE OF DECOHERENCE
In the previous sections, we treated the system asa fully coherent condensate. In experimental condi-tions, however, external sources of decoherence are al-ways present and can limit the investigation of the con-densate dynamics at long times.A first source of decoherence is given by the interac-tion with the incoherent normal component that is al-ways present. Decoherence effects in a finite-temperaturecloud have been first studied in Ref.[37] as a function ofthe cloud temperature and are critical for future inves-tigations in elongated clouds. In this work, we limit theeffects of the normal component by lowering the temper-ature of the condensate. For the data shown in the pre-vious Sections, the total normal component is less than20% of the total atom number. In this conditions weestimate the amount of normal component at the centerof the cloud from the Hartree-Fock model, and find it Z FIG. 6. Decay of Rabi oscillations for a quasi condensed cloud.The measured coherence time is τ = 370 ms. Error bars arestandard deviation on 5 realizations. less than 3% of the condensate component. We there-fore neglect its contribution to the decoherence of thecondensate.A second source of decoherence comes from externalnoise. To estimate this contribution, we drive Rabi dy-namics on a very dilute thermal sample, where mean-fieldeffects can be neglected. In Fig. 6(a), we show the ob-served Rabi oscillations for a fully thermal cloud. Weobserve coherence times of τ ∼
370 ms, presumably lim-ited by residual collisional effects and technical noise.For a small Rabi coupling, the measurement of such along coherence time for condensed clouds is limited notonly by interactions, but also by the rise of other phenom-ena such as spin-torque induced magnetic excitations andshock waves which mask the bare coherence of the sys-tem. The observed long coherence is very promising forfuture investigations on Rabi coupled magnetic solitonsand related phenomena.
IX. CONCLUSIONS AND OUTLOOK
We have characterized the properties of an elongatedJosephson junction based on two coherently coupledatomic spin states of Na. After finding the regime,where the dynamics is more 1D-like, we demonstrate thecapability to adiabatically manipulate the pseudo-spinstate on the Bloch sphere. We apply this to calibrate thenonlinear term of the Hamiltonian and we investigatethe coherence of the prepared state. Additionally to thepresented ARP procedure, future investigation can be fo-cused on the search for shortcuts to adiabaticity, basedon different ramps of the driving detuning and ampli-tude, in order to further decrease losses and decoherenceof the system during the state preparation [38].The full control of the quantum state of an elongatedJosephson junction represents a cornerstone to future in-vestigations in the field of nonlinear dynamics and to-wards new metrological tools. The system can be drivento points of the Bloch sphere that are far from the equilib-rium position, but present locally different evolution dueto the non-uniform nonlinearity, leading to localized andpropagating instability [4, 18]. In an elongated cloud, theinterplay between a spatially non-uniform squeezing andthe long-range entanglement requires further theoreticaland experimental investigations, with particular focus onlocal and global correlations [14].
X. ACKNOWLEDGEMENTS
We thank I. Carusotto and P. Hauke for fruitful discus-sions and R. Cominotti for experimental support duringthe final stage of the data acquisition. We acknowledgefundings from INFN through the FISH project, from theEuropean Union’s Horizon 2020 Programme through theNAQUAS project of QuantERA ERA-NET Cofund inQuantum Technologies (Grant Agreement No. 731473),from Italian MIUR under the PRIN2017 project CEn-TraL (Protocol Number 20172H2SC4) and from Provin-cia Autonoma di Trento. We thank the BEC Center inTrento, the Q@TN initiative and QuTip. [1] B. D. Josephson, Possible new effects in superconductivetunnelling, Phys. Lett. , 252 (1962).[2] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall,M. J. Holland, J. E. Williams, C. E. Wieman, and E. A.Cornell, Watching a superfluid untwist itself: Recurrenceof rabi oscillations in a bose-einstein condensate, Phys.Rev. Lett. , 3358 (1999).[3] M. Albiez, R. Gati, J. F¨olling, S. Hunsmann, M. Cris-tiani, and M. K. 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