Stochastic dynamics of a few sodium atoms in a cold potassium cloud
Rohit Prasad Bhatt, Jan Kilinc, Lilo Höcker, Fred Jendrzejewski
SStochastic dynamics of a few sodium atoms in a cold potassium cloud
Rohit Prasad Bhatt, Jan Kilinc, Lilo H¨ocker, and Fred Jendrzejewski
Universit¨at Heidelberg, Kirchhoff-Institut f¨ur Physik,Im Neuenheimer Feld 227, 69120 Heidelberg, Germany (Dated: January 5, 2021)We report on the stochastic dynamics of a few sodium atoms immersed in a cold potassiumcloud. The studies are realized in a dual-species magneto-optical trap by continuously monitoringthe emitted fluorescence of the two atomic species. We investigate the time evolution of sodiumand potassium atoms in a unified statistical language and study the detection limits. We resolvethe sodium atom dynamics accurately, which provides a fit free analysis. This work paves the pathtowards precise statistical studies of the dynamical properties of few atoms immersed in complexquantum environments.
I. INTRODUCTION
The random evolution of a small system in a largebath can only be described by its statistical properties.Such stochastic dynamics occur in a wide range of set-tings including financial markets [1], biological systems[2], impurity physics [3] and quantum heat engines [4].Their evolution is hard to describe from microscopic prin-ciples, stimulating strong efforts to realize highly con-trolled model systems in optomechanics [5], cavity QED[6], superconducting circuits [7], trapped ions [8] and coldatoms [9]. For cold atoms, the high control is comple-mented by the access to a number of powerful statisti-cal approaches, like the precise analysis of higher-ordercorrelation functions of a many-body system [10] or theextraction of entanglement through fluctuations [11, 12].Cold atomic mixtures offer a natural mapping ofphysical phenomena involving system-bath interactions,wherein one species realizes the bath, while the otherspecies represents the system. If a mesoscopic cloud ofthe first species is immersed in a Bose-Einstein conden-sate formed by the second species, it implements the Bosepolaron problem [13–16]. In recent quantum simulatorsof lattice gauge theories, the small clouds of one speciesemulate the matter field, which is properly coupled to thegauge field realized by the second atomic species [17–19].Proposed technologies even go towards quantum errorcorrection, where the logical qubits are implemented inone atomic species and the second atomic species medi-ates entanglement between them [20]. The feasibility ofimmersing a few atoms into a large cloud was demon-strated in a dual-species magneto-optical trap (MOT) ofrubidium and cesium [21]. This was extended towardsthe study of position- and spin-resolved dynamics with asingle tracer atom acting as a probe [22, 23]. However,combining such experiments with a statistical descriptionof system and bath remains an open challenge.In this work, we investigate the stochastic dynamics offew sodium atoms and a large cloud of potassium atomsin a dual-species MOT. It builds upon atom counting ex-periments with a single atomic species [24–27] and atomicmixtures of Rb and Cs [21, 28, 29]. The mixture of Naand K, as employed in our experiment, has shown excel-
FIG. 1. Experimental platform for atom counting. The atomsare trapped and laser cooled in a dual-species MOT inside thescience chamber. The emitted fluorescence is collected by ahigh-resolution imaging system onto the cameras. We observethe stochastic dynamics of single sodium atoms (orange), im-mersed in a large cloud of potassium atoms (blue). lent scattering properties in the degenerate regime [30].It further provides the option to replace the bosonic Kby the fermionic K isotope through minimal changesin the laser cooling scheme [31]. The two atomic speciesare cooled through standard laser cooling techniques in adual-species MOT, as shown in figure 1. In a MOT, cool-ing and trapping is achieved through a combination ofmagnetic field gradients with continuous absorption andemission of resonant laser light. We collect the resultingfluorescence on a dedicated camera for each species andtrace their spatially integrated dynamics. We presenta statistical analysis for the dynamics of both species,which separates the fluctuations induced by the statisti-cal loading process from those caused by technical lim-itations. Furthermore, we achieve single atom countingfor sodium, which we employ to study its full countingstatistics.The paper is structured as follows. In section II, weprovide a detailed discussion of the experimental appa- a r X i v : . [ phy s i c s . a t o m - ph ] J a n ratus, and how it is designed to fulfill the requirementsof modern quantum simulators. In section III, we studythe dynamics of the observed fluorescence signal for bothatomic species. The analysis of their mean and varianceafter an ensemble average is then employed to statisti-cally investigate the origin of different fluctuations. Insection IV, we leverage the single atom counting resolu-tion of sodium to extract the full counting statistics ofatom load and loss events. In section V, we end with anoutlook of the next steps for the experimental platform. II. EXPERIMENTAL APPARATUS
In this section, we describe the different elements of ournew experimental apparatus. In the course of designingthis machine, effort was taken to optimize the versatilityand stability of the system. To achieve this, the experi-mental setup was designed for a continuous developmentof the vacuum and the laser system.
A. Vacuum system
Ultracold atom experiments require an ultra-high vac-uum (UHV) at pressures below 10 − mbar, in order toisolate the cold atoms from the surrounding environment.Our group operates another machine for quantum simula-tion experiments with atomic mixtures [13, 14, 19], whichconsists of a dual-species oven and a Zeeman slower con-nected to a science chamber [32]. In this apparatus thefirst cooling stages of the two species are highly coupled,which renders the optimization of the system very com-plex.In the new vacuum system, we decoupled the precool-ing stages of sodium and potassium up to the sciencechamber, as sketched in figure 2. The compact vac-uum system contains two independent two-dimensionalmagneto-optical trap (2D-MOT) chambers for sodiumand potassium and a dual-species science chamber, whereexperiments are performed [33, 34]. The two 2D-MOTchambers are connected to the science chamber from thesame side under a 12 . ◦ angle. The entire apparatus ismounted on a 600 mm x 700 mm aluminium breadboard,which is fixed to a linear translation stage [35]. There-fore, we are able to move the science chamber out of thecontraption of magnetic field coils and optics. This al-lows for independent improvement of the vacuum systemand in-situ characterization of the magnetic field at theposition of the atoms. The design of the 2D-MOT setup is inspiredby ref. [34]. The chamber body is manufactured from ti-tanium (fabricated by
SAES Getters ), where optical ac-cess is ensured by standard CF40 fused silica viewportswith broadband anti-reflection coating (BBR coating).The 2D-MOT region has an oven containing a 1 g atomicingot ampoule. The oven is heated to 160 ◦ C (70 ◦ C) forsodium (potassium), thereby increasing the pressure to
FIG. 2. Vacuum system. The separated 2D-MOT chambersare connected from the same side to the dual-species sciencechamber. The vacuum pumps are shown in red. The wholevacuum system is mounted on a translation stage, such thatthe science chamber can be moved out of the region of the3D-MOT coils and optics. − mbar in this region. To maintain an UHV in thescience chamber, a differential pumping stage separatesthe two vacuum regions from each other. Two gate valvesensure full decoupling of the two atomic species by iso-lating different chambers. Each region is pumped withits separate ion getter pump (from SAES Getters ) [36].We employed four stacks of nine (four) neodymium barmagnets to generate the required magnetic quadrupolefield inside the sodium (potassium) 2D-MOT chamber.
The rectangular titanium science chamberis designed such that the two atomic beams from the 2D-MOT chambers intersect in the center. Optical access forvarious laser beams and a high-resolution imaging systemis maximized by four elongated oval viewports (fused sil-ica, BBR coating), which are sealed using indium wire.The quadrupole magnetic field required for the 3D-MOT is produced by the MOT coils, which are placedon the sides of the science chamber. Applying a cur-rent of 20 A to the coils results in a magnetic fieldgradient of 17 G/cm. The fast control of the currentin the coils, required during an experimental sequence,is achieved through an insulated-gate bipolar transistor(IGBT) switching circuit. In order to cancel stray fieldsin the vicinity of the atomic clouds, we use three inde-pendent pairs of Helmholtz coils carrying small currents( < B. Laser cooling
In order to cool and trap the atoms, the laser light isamplified and frequency-stabilized on a dedicated opticaltable for each atomic species. The light is transferred tothe main experiment table via optical fibers. The layoutof the laser systems for both species is shown in figure 3.
Sodium.
Laser cooling and trapping of sodium atomsis achieved using the D -line at 589 nm, which is ob- FIG. 3. Sketch of the optical setup for laser cooling sodium and potassium atoms. The laser light is split into different paths,enabling the individual control of laser power and frequency for the 2D-MOT, the 3D-MOT, and the push beam. The frequencyand intensity of these beams is controlled with the help of acousto-optic modulators (AOMs) in double-pass configuration. Therf-frequencies for AOMs and EOMs are given in MHz.
Na:
The repumping light for the 2D- and 3D-MOT is generated byelectro-optic modulators (EOMs). K: In the 2D- and 3D-MOT paths the green AOM controls the K cooling frequency andthe blue AOM is responsible for the creation of the K repumping light. The repumping light for K is generated by EOMs. tained from a high-power, frequency-doubled diode laser(
TA-SHG pro , from
Toptica Photonics ). The laser lightis stabilized to excited-state crossover transition of theD -line using saturated absorption spectroscopy (SAS)and Zeeman modulation locking [37]. The modulatedSAS signal is fed into a digital lock-in amplifier and PI-controller, which are programmed on a STEMLab 125-14 board from
Red Pitaya using the
Pyrpl module [38].
Potassium.
Laser cooling and trapping of potassiumatoms is achieved using the D -line at 767 nm. The lightis obtained from a master-slave laser configuration (both DL pro , from
Toptica Photonics ). The master laser fre-quency is locked to the ground-state crossover transitionof the D -line of K with a scheme similar to sodium.The slave laser is frequency-stabilized through an off-set beat lock (405 MHz) and its output is amplified toa power of 800 mW, using a home-built tapered amplifier(TA) module. This light is used to supply all the cool-ing and trapping beams. The offset locking scheme alsofacilitates switching between the two isotopes, K and K. To cool the fermionic K, the slave laser frequencyis increased by approximately 810 MHz via the offset lockand the blue acousto-optic modulators (see figure 3) areturned off. . On the experiment table, the light from theoptical fibers is distributed into three independent pathsfor the operation of the dual-species MOT in a retro- reflected configuration. For both species the number ofatoms loaded into the 3D-MOT can be tuned in a con-trolled way by adjusting the 2D-MOT beam power andthe oven temperature. The pre-cooled atoms in the 2D-MOT region are transported to the 3D-MOT with a pushbeam. For accurate atom counting of sodium, we use1 . ◦ C, whichincreases the lifetime of atoms in the 3D-MOT due tobetter vacuum.
C. Fluorescence imaging
The cold atoms are characterized by collecting theirfluorescence through an imaging system with a high nu-merical aperture (NA) onto a camera (fig. 1). The imag-ing setup comprises an apochromatic high-resolution ob-jective, which features an NA of 0.5 and chromatic focalcorrection in the wavelength range 589 −
767 nm (fabri-cated by
Special Optics ). The fluorescence of sodium andpotassium is separated by a dichroic mirror, built into acage system, which is mounted on stages for x-, y- andz-translation along with tip-tilt adjustment.
FIG. 4. Experimental sequence. A: A series of images (black)is taken. While the MOT beams (red) are always on, the mag-netic field (green) is switched off for reference images markedin grey. B: Typical time trace from the series of images ofsodium (orange) and potassium (blue).
Both imaging paths contain a secondary lens and anadditional relay telescope. This allows us to do spatial fil-tering with an iris in the intermediate image plane of thesecondary lens and achieve a magnification of 0.75 (0.25)for sodium (potassium). For imaging the sodium atomswe use an sCMOS camera (
Andor ZYLA 5.5 ) [39, 40],while for the potassium atoms we use an EMCCD camera(
NuVu H-512 ). In total, we estimate the conversion effi-ciency from photons to camera counts to be 0 .
2% (0 . III. ATOM DYNAMICS
Our experimental sequence to investigate the atom dy-namics is shown in figure 4 A. We start the atom dy-namics by switching on the MOT magnetic field (with agradient of 21 G/cm) and then monitor the fluorescencein N img = 200 images. Each image has an integrationtime τ = 75 ms, such that the camera counts overcomethe background noise. Since the motion of the atomsduring the integration time washes out any spatial in-formation, we sum up the counts over the entire MOTregion for each image. This results in a time trace ofcamera counts N c , as shown in figure 4 B. Each experi-mental run is preceded and succeeded by a series of 100reference images to quantify the background noise ∆ bg ,induced by the fluctuations in the stray light from theMOT beams.The camera counts for sodium exhibit random jumps,corresponding to single atom load and loss events. The stochastic nature of the observed signal and large relativefluctuations require a statistical analysis of the dynamicsin terms of expectation values. The single atom resolu-tion provides additional access to the full counting statis-tics, which is discussed in section IV. The few sodiumatoms are immersed in a cloud of potassium atoms, whichwe pre-load for 5 s to ensure large atom numbers. In con-trast to sodium, we do not observe discrete jumps, butrather a continuous loading curve with higher counts andsmaller relative fluctuations. These are typical featuresof a bath, which can be characterized by its mean andvariance.To extract expectation values through an ensemble av-erage, we perform 100 repetitions of the previously de-scribed experimental sequence for sodium and potassiumindependently [41]. To further access the small atomregime for potassium in this analysis, we reduce the 2D-MOT power and do not perform pre-loading. The ob-served dynamics are shown in figure 5 A. We calculatethe mean N c and standard deviation ∆ c of counts at eachimage index [42, 43]. For the case of sodium, the dy-namics is extremely slow and never reaches a stationaryregime. Furthermore, the amplitude of the fluctuations iscomparable to the average camera counts throughout theentire observation. For potassium, the stationary situa-tion is achieved on average after a few seconds of loading.Once again, we observe a strong dependence of the stan-dard deviation on the average atom counts.To study this dependence quantitatively, we trace thevariance ∆ as a function of the average counts N c infigure 5 B. For sodium the variance shows a linear de-pendence on the average counts with an intercept. Thisbehavior can be understood by considering two indepen-dent noise sources. The first one is a background noise∆ bg , which is independent of the atom number and addsa constant offset to the variance. It originates from thereadout noise of the camera and intensity-varying straylight. The second noise source is the atom shot noise,which describes the random variations due to the count-ing of atoms loaded until a given image index in the timetrace. Its variance is equal to the average atom num-ber. The recorded camera signal is directly proportionalto the atom number N c = C N at , leading through errorpropagation to a variance of C N c . The two independentnoise sources add up in their variances∆ = C N c + ∆ . (1)This theoretical prediction agrees well with the exper-imental observations. The calibration constant C Na =1 . × and the background noise ∆ bg,Na = 2201(2)were independently extracted from a histogram plot,as described in section IV. This validates our assump-tion that background and shot noise are the dominat-ing noise sources for sodium. Converting the cameracounts back into atom numbers, we obtain a resolutionof 0 . FIG. 5. Characterization of atom number fluctuations forsodium (left) and potassium (right). A: Hundred time tracesof sodium and potassium with mean and error band (shown asthick lines with shaded region around them). B: Dependenceof variance on mean camera counts. For sodium (left) theinset shows the background noise level. of the variance. In the regime of few counts the vari-ance is again dominated by the background noise andthe atom shot noise. With the noise model (1), vali-dated for sodium, we perform a fit to extract the cali-bration factor C K = 560(140) and the background noise∆ bg,K = 2450(140). The resulting atom resolution of4 . .
1) atoms is similar to that achieved in precision ex-periments with Bose-Einstein condensates [11, 45].For higher atom numbers, we observe a non-linear de-pendence, which we attribute to technical fluctuations ofthe MOT. The MOT properties can be parameterized bythe loading rate Γ load and loss rate Γ loss . Consideringsingle atom load and loss only, they are connected to theatom dynamics through N at ( t ) = Γ load Γ loss (cid:2) − exp( − Γ loss t ) (cid:3) . (2)We fit each time trace with this solution and, hence, ex-tract the distribution of Γ load and Γ loss across differentruns. The variance in the atom dynamics, resulting fromthese fluctuations, is traced as the dash-dotted curve infigure 5 B. In the high atom number regime it agreeswell with our experimental observation. We expect tosubstantially reduce these fluctuations in the future byimproving the stability of intensity, frequency, and mag-netic field. IV. FULL COUNTING STATISTICS OFSODIUM
Going one step beyond the statistical analysis of en-semble averages, we use the single atom resolution ofsodium to extract its full counting statistics [46, 47]. Thisrequires the digitization of camera counts into discreteatom numbers [24], as presented in figure 6. For this,we aggregate the camera counts of 100 runs into one his-togram, which shows distinct atom number peaks. Thecalibration from camera counts to atom counts is accom-plished through Gaussian fits to individual single atompeaks. The distance between consecutive peaks corre-sponds to the calibration factor C Na = 1 . × .The width of the zero atom signal sets the backgroundnoise limit ∆ bg,Na = 2201(2) [48]. From the overlap ofthe peaks, we estimate the detection fidelity of atoms to96(3)%. With this calibration, we convert the time tracesof camera counts into digitized atom count dynamics, asshown in figure 6 B.Each change in atom counts corresponds to a load orloss event with one or more atoms, as shown in figure7 A. We observe that the dynamics are dominated by sin-gle atom events, as only 3% involve two or more atoms.Therefore, we neglect them in the following. We countthe number of single atom events in each time trace andsummarize them in a histogram, shown in figure 7 B. Onaverage we observe N load = 2 . N img = 200 taken per time trace. Given thatthe atoms come from a large reservoir, namely the ovenregion, the loading rate is independent of the number ofloaded atoms. From these observations, we describe theloading process statistically as a series of independentBernoulli trials with a success probability p load . There- FIG. 6. Accurate atom counting of sodium. A: Histogram ofrecorded camera counts. The calibration from camera countsto atom number is accomplished through Gaussian fits to dis-tinct single atom peaks. Insets show average images of zeroand one atom. B: Example time trace before and after digi-tization.
FIG. 7. Counting statistics of sodium with and withoutpotassium atoms present. A: Digitized example time traceof sodium with single atom load (loss) events marked with up(down) arrows. Only jumps during the MOT loading stageare taken into account. B: Histogram of the number of singleatom losses and loads per time trace. The dashed lines showPoisson distributions with mean N loss and N load (extractedfrom the counting statistics). fore, the single atom loading probability is given by p load = N load N img . (3)The large number of images and the low loading proba-bility means that the number of loading events N load con-verges towards a Poisson distribution with mean N load .This stands in full agreement with the experimental ob-servation.Once an atom is present, it can be lost from the MOTwith a probability p loss . We observe an average numberof N loss = 1 . N loss enables us to extract theloss probability p loss = N loss (cid:80) i N i . (4)The normalization factor is the sum of average numberof atoms present in each image i . Similar to the load-ing case, we observe a Poisson distribution for the lossevents with mean N loss , which can be attributed to theoccurrence of only a few loss events over a large set ofimages. TABLE I. Comparison of load and loss probabilities in afew atom sodium MOT with and without the presence of apotassium cloud. The uncertainties were obtained throughbootstrap resampling. p load [%] p loss [%]Without K 1.06(3) 2.76(23)With K 1.02(3) 2.47(24) To study the influence of the large potassium cloudon the dynamics of the few sodium atoms, we comparethe load and loss statistics of the sodium atom countswith and without potassium atoms present (see fig. 7 B).The extracted load and loss probabilities are summarizedin table I. The values corresponding to the absence andpresence of potassium are indistinguishable to roughlywithin five percent. To exclude experimental errors, werepeated the analysis for various configurations of rela-tive positions of the two clouds, magnetic field gradientsand laser detunings. All results were compatible with ourobservation of no influence of potassium on the sodiumatom dynamics. We attribute these results to the ex-tremely low density of the atomic clouds. To increasethe density of both clouds in future studies, we plan towork at higher magnetic field gradients with water-cooledcoils [49]. At higher densities, we expect to observe inter-species interaction, which should influence the loadingdynamics similar to previous studies [21, 50].
V. OUTLOOK
In this work, we presented a detailed experimentalstudy of the stochastic dynamics of a few cold Na atomsimmersed in a cloud of K atoms in a MOT. The ex-perimental setup is designed to be directly extendabletowards quantum degenerate gases through evaporativecooling [34]. Defect-free optical tweezer arrays will pro-vide high control over single atoms [51, 52] and their re-peated observation, as recently demonstrated for stron-tium atoms [53].Our study opens the path for investigating time-resolved dynamics, including transport and thermaliza-tion, in atomic mixtures over a wide range of parameters.The emergence of ergodicity will become directly observ-able as the equivalence of the ensemble averages withtime averages for individual time traces for sufficientlylong times. Optimizing the photon detection should fur-ther allow us to reduce imaging times sufficiently to reachposition-resolution [54, 55] without a pinning lattice [22].This will extend our work towards the quantum regime,in which we might continuously monitor thermalizationof impurity atoms in a Bose-Einstein condensate [28, 56].
ACKNOWLEDGEMENT
The authors are grateful for fruitful discussions and ex-perimental help from Apoorva Hegde, Andy Xia, Alexan-der Hesse, Helmut Strobel, Valentin Kasper, Lisa Rin-gena and all the members of SynQS. We thank Gia-como Lamporesi as well as Gretchen K. Campbell andher team for valuable input on the design of the experi- mental setup.This work is part of and supported by the DFG Col-laborative Research Centre “SFB 1225 (ISOQUANT)”.F. J. acknowledges the DFG support through theproject FOR 2724, the Emmy- Noether grant (Project-ID 377616843) and support by the Bundesministeriumf¨ur Wirtschaft und Energie through the project ”En-erQuant” (Project- ID 03EI1025C). [1] R. Cont and J.-P. Bouchaud, Herd behavior and aggre-gate fluctuations in financial markets, MacroeconomicDynamics , 170 (2000).[2] G. Kucsko, P. C. Maurer, N. Y. Yao, M. Kubo, H. J.Noh, P. K. Lo, H. Park, and M. D. Lukin, Nanometre-scale thermometry in a living cell, Nature , 54 (2013).[3] F. Grusdt and E. 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