High signal to noise absorption imaging of alkali atoms at moderate magnetic fields
Maurus Hans, Finn Schmutte, Celia Viermann, Nikolas Liebster, Marius Sparn, Markus K. Oberthaler, Helmut Strobel
HHigh signal to noise absorption imaging of alkali atoms at moderatemagnetic fields
Maurus Hans, a) Finn Schmutte, Celia Viermann, Nikolas Liebster, Marius Sparn, Markus K. Oberthaler, and Helmut Strobel Kirchhoff-Institut für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg,Germany. (Dated: 18 February 2021)
We present an improved scheme for absorption imaging of alkali atoms at moderate magnetic fields, where the excitedstate is well in the Paschen-Back regime but the ground state hyperfine manifold is not. It utilizes four atomic levels toobtain an approximately closed optical cycle. With the resulting absorption of the corresponding two laser frequencieswe extract the atomic column density of a K Bose-Einstein condensate. The scheme can be readily applied to allother alkali-like species.
I. INTRODUCTION
Absorption imaging is a standard technique for observa-tions in quantum gas experiments which relies on resonantatom light interaction in ideally closed optical cycle schemes. At very high magnetic fields in the Paschen-Back regime thesecan be found for every ground state. At moderate fields, whereonly the excited state is well in the Paschen-Back regime,this is only possible for the atom’s stretched states with max-imal or minimal magnetic quantum number. Efficient opti-cal pumping schemes to reach these stretched states arenot available for arbitrary initial states. However, when usingFeshbach resonances to tune the atomic interaction strength, the choice of atomic states is fixed. Recently, a scheme forfluorescence imaging has been developed that improves thesingle-atom detection in these states. It makes use of twoatomic transitions in order to obtain an approximately closedfour-level optical cycle. Here, we adapt this scheme to ab-sorption imaging of dense atomic clouds.
II. EXPERIMENTAL SETTING
We exemplify the technique with a Bose-Einstein conden-sate (BEC) of K in the state which corresponds to | F , m F (cid:105) = | , − (cid:105) at low magnetic fields. For our experiments we workat 550 G, close to a broad Feshbach resonance. Figure 1 com-pares the absorption signal obtained with the improved ab-sorption scheme (blue points) to the signal using only onelaser frequency (red points). The latter results in a vanish-ing scattering after ∼ µ s. With the addition of the secondfrequency, a drastic enhancement is achieved. In the experi-mental setup, each laser frequency is generated by a dedicatedexternal cavity diode laser that is offset-locked to the coolinglaser stabilized on the D2 line of K. After double-pass AOMpaths for pulsing, both laser frequencies are coupled into thesame single-mode optical fiber with orthogonal polarizationsand pass through the same quarter-wave plate after the fiber.A CCD camera detects the total absorption signal (see Fig. 1, a) Electronic mail: highfi[email protected]
150 5 10 20 E n e r g y S c a tt e r e dph o t o n s Imaging time0 5002041068
FIG. 1.
Absorption imaging of a BEC of K. The number of scat-tered photons N scatt levels off within ∼ µ s when imaging the atomswith a single laser frequency ( σ − , red points). By adding a secondfrequency ( σ + ), the signal can be enhanced drastically (blue points).The difference is clearly visible in the absorption images of the atomcloud after 20 µ s (same color scale used for both images; in the darkblue regions no photons are scattered). The upper inset shows theenergy eigenstates of the ground state S / and the excited state P / hyperfine manifold as a function of the magnetic field B . The twoimaging transitions are indicated with arrows. The lower inset showsa simplified schematic of the experimental setup. From left to right:laser light impinges on the atomic cloud, which is imaged via an ob-jective and a secondary lens onto a CCD camera. lower inset). The number of scattered photons is estimated by N scatt = − G ( C f − C i ) , where C f and C i are the number of in-tegrated counts on the CCD camera with and without atoms,respectively. The factor G includes the camera gain as wellas a correction factor for the solid angle of the objective, re-flection loss along the imaging beam path, and the quantumefficiency of the camera. a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b FIG. 2.
Four-level scheme for imaging.
The imaging transition σ − transfers atoms from the initial state | g − (cid:105) = √ p |− / , − / (cid:105) + √ − p | / , − / (cid:105) (black dot) to the excited state | e − (cid:105) (cid:39)|− / , − / (cid:105) . Here, the states | m J , m I (cid:105) are the basis states of theelectron’s total angular momentum J and the nuclear spin I . p = . | g + (cid:105) = √ p | / , − / (cid:105) + √ − p |− / , − / (cid:105) (dotted arrow). Asecond laser frequency drives the transition σ + , which couples thestate | g + (cid:105) to the excited state | e + (cid:105) (cid:39) | / , − / (cid:105) from where theatoms can decay back into | g + (cid:105) and | g − (cid:105) only. This results in aclosed optical cycle when the excited states are sufficiently pure inquantum numbers m J , m I . III. PRINCIPLE OF THE METHOD
The upper inset of Fig. 1 shows the Breit-Rabi diagram ofthe S / ground state and P / excited state manifold, withthe employed transitions depicted as arrows. The relevantfour-level scheme is depicted in Fig. 2. The atoms are ini-tially prepared in | g − (cid:105) ∼ | m J , m I (cid:105) = |− / , − / (cid:105) with m J and m I denoting the magnetic quantum numbers of the elec-tron’s total angular momentum J and the nuclear spin I , re-spectively. Imaging at a single frequency involves a σ − tran-sition to the state | e − (cid:105) (cid:39) |− / , − / (cid:105) in the P / excitedstate manifold. The nearby states ( <
15 MHz) with the same m J are not addressed, since the nuclear spin quantum num-ber m I is not changed by electric dipole transitions and theatomic eigenstates are pure up to 10 − in the | m J , m I (cid:105) states.The excited state | e − (cid:105) has a small leakage into a dark state | g + (cid:105) ∼ | / , − / (cid:105) , which causes the quick saturation of thesignal in Fig. 1. Specifically, the two ground states read | g − (cid:105) = √ p |− / , − / (cid:105) + (cid:112) − p | / , − / (cid:105) , | g + (cid:105) = √ p | / , − / (cid:105) + (cid:112) − p |− / , − / (cid:105) , (1)with p (cid:39) .
98 at a field of 550 G. As both ground stateshave an admixture of |− / , − / (cid:105) , the excited state | e − (cid:105) (cid:39)|− / , − / (cid:105) can decay into both. The 2 % admixture is con-sistent with the observed time scale of 2 . µ s, after which halfof the atoms are transferred into the dark state.To enhance the signal we address the state | g + (cid:105) with thesecond laser frequency. This σ + light couples | g + (cid:105) to the ex-cited state | e + (cid:105) (cid:39) | / , − / (cid:105) . It closes the optical cycle togood approximation and results in the effective four-level sys- Intensity ratio S c a tt e r e dph o t o n s FIG. 3.
Optimization of the intensity ratio.
The number of scat-tered photons is measured at different ratios r = I − / I tot and differenttotal intensities I tot = I − + I + . The imaging pulse length is 10 µ s,and the data points correspond to total intensities of 23 mW / cm (tri-angle), 42mW / cm (diamond), 60mW / cm (square), 79mW / cm (circle). For the highest intensities, the largest signal is found at r (cid:39) .
5. For decreasing light intensities this optimum slightly shiftsto larger ratios as it takes longer to reach the steady state. For thehighest intensities we can compare the data to numerical solutionsof the optical Bloch equations for the four-level system, scaled by aglobal factor (solid curves). tem shown in Fig. 2. During a typical 10 µ s imaging pulsewe expect to lose only 2 % of the atoms into the ground states ∼ |− / , / (cid:105) and ∼ | / , − / (cid:105) , which are not addressed bythe imaging light. This results from the limited purity of theexcited states in the | m J , m I (cid:105) basis. Off-resonant coupling toother excited states is negligible for typical imaging intensitiessince the closest transitions are detuned by at least 350 MHz. IV. OPTIMAL INTENSITY RATIO
We optimize the absorption signal by varying the ratio r = I − / I tot between the intensities I − and I + on the two imag-ing transitions σ − and σ + , respectively. Here, the total imag-ing beam intensity I tot = I − + I + is kept constant. Figure 3shows the number of scattered photons N scatt for different con-figurations and compares the results to numerical solutions ofthe optical Bloch equations for the four-level system (scaledby a constant factor). In the case without σ + light ( r =
1) thetotal signal is limited by the decay into the dark state. Imag-ing without σ − light ( r =
0) results in no signal, as the initialstate of the atoms is not addressed by this light. For the highestimaging intensities the maximum number of scattered photonsis obtained at r (cid:39) .
5, as expected from the steady state solu-tion. For smaller intensities, the optimum is at higher ratios r . This results from the initial pumping dynamics starting in | g − (cid:105) that are still relevant for the short imaging duration of10 µ s. FIG. 4.
Calibration of the imaging system.
For each ratio r = I − / I tot the effective saturation intensity I effsat = α I sat is chosen suchthat the resulting atomic column density n c is invariant under changesof the imaging intensity I tot . The inset shows this procedure for r = .
5. The theoretical predictions obtained from the steady statesolution and the numerical simulation of the dynamics are shown bythe dashed and the solid curves, respectively. The experimental val-ues are scaled by the mean of the three points around r = .
5, andthe theoretical curves by their respective values at r = .
5. The errorbars are estimated by bootstrap resampling.
V. CALIBRATION
To obtain an accurate estimate of the atomic density, wecalibrate the imaging system following the method presentedin Reinaudi et al. Each atom in the cloud is described as aneffective two-level system, which includes saturation effects.One solves the resulting Beer-Lambert-type differential equa-tion for resonant light and integrates along the direction of theimaging beam. This leads to the atomic column density n c = σ eff (cid:20) ln (cid:18) I i I f (cid:19) + I i − I f I effsat (cid:21) . (2)Here, the final intensity I f and the initial intensity I i are thetotal intensities measured via the signal on the CCD camerawith and without the presence of atoms, respectively. σ eff isthe effective scattering cross-section and I effsat = α I sat is the ef-fective saturation intensity. The deviation from the bare satu-ration intensity I sat of a single closed two-level optical cyclecaptures effects of polarization, detuning fluctuations of thelaser from atomic resonance, and optical pumping effects. Weestimate the effective saturation intensity by taking absorptionimages for a constant atom number with different total imag-ing intensities. I effsat is optimized such that the column density n c is invariant under changes in intensity. As shown in the in-set of Fig. 4, we find an optimum for I effsat = ( ± ) I sat , where I sat is the saturation intensity of a single transition. With thevalue of I effsat at hand an absolute atom number can be cali-brated by comparing atomic density distributions with theo-retical predictions or the detection of atomic shot noise. To predict a value for I effsat we scale the theoretical resultsfor the scattering rate versus intensity of the four-level systemto the expectation for an effective two-level system. For thesteady state the analytic solution reads I effsat ( r ) = I sat r ( − r ) . (3)We use that the coupled four-level system can be describedby two two-level systems with equal I sat which are only cou-pled to each other via the incoherent spontaneous decay oftheir excited states. Thus, no coherence is built up and thetwo subsystems can be described as being independent. In thesteady state this leads to an imbalance in population of thetwo systems for r (cid:54) = .
5. In the case of r = . I effsat mainly to instabilities of the imaging laserfrequencies.In Fig. 4 we show experimental results for the dependenceof I effsat on the ratio r and compare them to the analytic solution(dashed curve). At the largest and smallest ratios, deviationsbetween experimental and analytic behavior arise due to theinitial population dynamics of the four-level system. Thesecan be captured by a numerical simulation, as shown by thesolid curve in Fig. 4. As before, the numerical results for thescattering rate versus total intensity are scaled to those of aneffective two-level system. From r ∼ . . VI. GENERAL PERSPECTIVES
Finally, we note that the imaging procedure can be general-ized to all alkali-like atoms. The ground states can always bewritten as a superposition of maximally two | m J , m I (cid:105) states.This is a consequence of the the fact that the spin operator F z commutes with the Hamiltonian H = a h f / ¯ h J · I + µ B B z / ¯ h ( g J J z + g I I z ) (4)of the ground state hyperfine manifolds. Here, a h f is the mag-netic dipole constant and g J , g I the electron and nuclear g -factors, respectively. This means that the z-projection m F = m J + m I of F is always a good quantum number. Since J = / m J = ± /
2) thereare maximally two states with the same m F . Except for thestretched states with maximal | m F | , all states can be written inthe form of Eq. 1, and the imaging scheme can be applied. ACKNOWLEDGMENTS
We thank Selim Jochim, Martin Gärttner and BenediktErdmann for discussions. This work was supported bythe DFG Collaborative Research Center SFB1225 (ISO-QUANT), the ERC Advanced Grant Horizon 2020 Entan-gleGen (Project-ID 694561), and the Deutsche Forschungs-gemeinschaft (DFG) under Germany’s Excellence StrategyEXC-2181/1 - 390900948 (the Heidelberg STRUCTURESExcellence Cluster). M. H. acknowledges support from theLandesgraduiertenförderung Baden-Württemberg, C. V. andN. L. acknowledge support from the Heidelberg graduateschool for physics.
DATA AVAILABILITY
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