Interaction enhanced imaging of individual atoms embedded in dense atomic gases
G. Günter, M. Robert-de-Saint-Vincent, H. Schempp, C. S. Hofmann, S. Whitlock, M. Weidemüller
aa r X i v : . [ phy s i c s . a t o m - ph ] J un Interaction enhanced imaging of individual atoms embedded in dense atomic gases
G. G¨unter, M. Robert-de-Saint-Vincent, H. Schempp, C. S. Hofmann, S. Whitlock, ∗ and M. Weidem¨uller Physikalisches Institut, Universit¨at Heidelberg, Philosophenweg 12, 69120, Heidelberg, Germany.
We propose a new all-optical method to image individual atoms within dense atomic gases. Thescheme exploits interaction induced shifts on highly polarizable excited states, which can be spatiallyresolved via an electromagnetically induced transparency resonance. We focus in particular onimaging strongly interacting many-body states of Rydberg atoms embedded in an ultracold gasof ground state atoms. Using a realistic model we show that it is possible to image individualimpurity atoms with enhanced sensitivity and high resolution despite photon shot noise and atomicdensity fluctuations. This new imaging scheme is ideally suited to equilibrium and dynamical studiesof complex many-body phenomena involving strongly interacting atoms. As an example we studyblockade effects and correlations in the distribution of Rydberg atoms optically excited from a densegas.
The ability to prepare and probe individual quantumsystems in precisely controlled environments is a drivingforce in modern atomic, molecular and optical physics.Manipulating single atoms [1], molecules [2] and ions [3],for example, is becoming a common practice. At theheart of these experiments are the powerful imaging tech-niques which have taken on great importance in diverseareas, such as chemical sensing and chemical reaction dy-namics [4], probing superconducting materials [5], and forquantum logic and quantum information processing [6].More recently, new single atom and single site sensitiveimaging techniques for optical lattices have opened thedoor to control and probe complex many-body quantumsystems in strongly correlated regimes [7].The usual approach to detect atoms is to measure thefluorescence or absorption of light by driving a strong op-tical cycling-transition. Weak or open transitions presenta difficulty since the maximum number of scattered pho-tons per atom becomes greatly limited. In the case oflong lived states of trapped ions, the technique of elec-tron shelving has been used as an amplifying mechanismin order to directly observe quantum jumps [8]. Anotherapproach involves the use of an optical cavity to enhancethe interaction of the atoms with the light field [9]. Thismakes it possible to reach single-atom sensitivity, butusually at the expense of greatly reduced spatial resolu-tion.Here we propose a new method to image individ-ual atoms embedded within a dense atomic gas. Theconcept exploits strong interactions of the atoms withhighly polarizable Rydberg states of the surrounding gas.The induced level shifts can then be transferred to astrong optical transition and to many surrounding atomswithin a critical radius, thereby providing two mecha-nisms which greatly enhance the effect of a single impu-rity on the light field. The Rydberg states could actas non-destructive probes for individual trapped ions,nearby surface charges, dipolar molecules, or other Ry-dberg atoms. In our approach, the interaction-inducedshifts are spatially resolved via an electromagnetically-induced-transparency (EIT) resonance involving a weak Rc c oup li ng probe U ΩcΩp ger
ΓcΓp
FIG. 1.
Scheme for imaging individual impurity atomswithin a dense atomic gas.
Impurity atoms (crosses) areembedded within a dense two-dimensional atomic gas of back-ground atoms. The background atoms interact with two lightfields (coupling and probe) via a two-photon resonance withan excited state | r i . This coupling produces an EIT reso-nance on the ground-state probe transition. However, stronginteractions with an impurity atom lead to a frequency shift U of the resonance within a critical radius R c . The changein absorption properties of many surrounding atoms makesit possible to map the impurity atom distribution to the ab-sorption profile of a probe laser for analysis. probe and a strong coupling laser in a ladder configu-ration [10]. Even though the Rydberg state is barelypopulated, the EIT resonance is extremely sensitive toits properties [11, 12], thereby providing the means toobtain a strong absorption signal and great sensitivitycombined with high spatial resolution for detecting indi-vidual atoms.We exemplify our imaging scheme for the specific caseof probing many-body states of strongly-interacting Ry-dberg atoms in a quasi-two-dimensional atomic gas (de-picted in Fig. 1). Rydberg atoms are of great inter-est because their typical interaction ranges are compa-rable to, or larger than, the typical interatomic separa-tions in trapped quantum gases. Traditionally Rydbergatoms are field ionized and the resulting ions are sub-sequently detected, which provides rather limited spa-tial resolution. As a result, much of the work done sofar, such as the scaling laws for excitation [13], excita-tion statistics [14] and light-matter interactions [11], hasbeen restricted to the study of cloud averaged proper-ties. M¨uller et al. proposed to use a single Rydbergatom to conditionally transfer an ensemble of atoms be-tween two states [15]. Our method exploits the strongRydberg interactions with a background gas of atoms torealize non-destructive single-shot optical images of Ry-dberg atoms with high resolution and enhanced sensitiv-ity. We anticipate this technique will complement thenew optical lattice imaging techniques [7], but with thecapability to directly image many-body systems of Ryd-berg atoms. We show in particular that this will provideimmediate experimental access to spatial correlations inrecently predicted crystalline states of highly excited Ry-dberg atoms [16].To quantitatively describe the absorption of probe lightby a background gas of atoms surrounding a Rydbergatom we follow an approach based on the optical Blochequations [10]. The Hamiltonian describing the atom-light coupling is H = ~ (cid:0) Ω p | e ih g | + Ω c | r ih e | +∆ p | e ih e | + (∆ p + ∆ c ) | r ih r | + h.c. (cid:1) . (1)For resonant driving ∆ p = ∆ c = 0 a dark-state is formed | dark i ≈ Ω c | g i − Ω p | r i , which no longer couples to thelight field. Consequently, the complex susceptibility χ of the probe transition vanishes and the atoms becometransparent.The presence of a nearby Rydberg atom, however,causes an additional energy shift U = ~ C / | d | for thestate | r i , where d is the distance to the Rydberg atomand the interaction coefficient C reflects the sign andstrength of interactions on the | r i state. One should alsoaccount for interactions between atoms in state | r i , butthese can be neglected for Ω p ≪ Ω c when the populationin | r i becomes small. We also include spontaneous decayfrom the states | e i and | r i with rates Γ p , and Γ c respec-tively. From the master equation for the density matrix ρ we calculate the steady-state absorption and solve forthe complex susceptibility of the probe transition numer-ically.In the weak probe limit (Ω p ≪ Ω c , Γ p ) we assume thepopulation stays mostly in the ground state ( ρ gg ≈ χ = i Γ p (Γ p − i ∆ p ) + Ω c (Γ c − i ∆) − , (2)where ∆ = ∆ p + ∆ c + C / | d | .Fig. 2 shows the probe absorption proportional to theimaginary part of χ for different laser parameters andfor different distances to a Rydberg atom. Far from the - - - D p (cid:144) G p I m @ Χ D H a L d (cid:144) R c I m @ Χ D H b L FIG. 2.
Probe absorption given by the imaginary partof the susceptibility. (a) Im[ χ ] as a function of probe de-tuning for Ω c = 1, Γ c = 0 .
05 and ∆ c = 0 (in units of Γ p )for various distances from the Rydberg atom. The solid lineis for d → ∞ , dashed corresponds to d = R c and the dottedline is for d = R c /
2. (b) Dependence of Im[ χ ] as a functionof distance from the Rydberg atom with ∆ p = 0. influence of the Rydberg atom ( d → ∞ ), the suscep-tibility takes on a characteristic shape with vanishingabsorption on resonance. For shorter distances, inter-actions tend to shift the transparency window and theon-resonant susceptibility increases. At a critical dis-tance d = R c , Im[ χ ] = 1 /
2. For d < R c , the excitedstates | r i become far detuned and the background atomseffectively act as two-level systems. In this case the usualLorentzian lineshape is recovered with maximum absorp-tion on resonance. Hence, in the presence of a Rydbergatom, N = n D πR c ≫ ∼ λ , can scatter many photons toproduce a dark disk with radius R c . From Eq. 2 andtaking ∆ p = 0 and Γ c ≈ R c = (2 C Γ p / Ω c ) / , and the distance dependent sus-ceptibility Im[ χ ] ≈ (cid:0) d/R c ) (cid:1) − . Since for typicalparameters N ≈
50, and R c ≈ µ m comparable to theoptical resolution, the spatially resolved probe absorp-tion provides an excellent signature for the presence of aRydberg atom within a dense gas.To apply this scheme to realistic situations one also hasto analyze the influence of noise. We identify two majorsources, one can be attributed to photon shot noise ofthe probe while a second contribution is associated withintrinsic atomic density fluctuations. Both noise sourcescan be accurately described as Poissonian processes. Thissuggests that the signal-to-noise ratio can be made arbi-trarily large for large intensity and large density. How-ever, to neglect interactions between background atomswe require the probability to find more than one atom inthe | r i state within the range of background-backgroundinteractions R ′ c to be . R ′ c ≈ R c ). This constraint imposes a relationship be-tween the maximum density of background atoms andthe maximum probe intensity n D . Ω c /πR ′ c Ω p . Abovethis critical density the contrast of the image would de-crease due to the blockade effect, where only one atom µ m (a) µ m (b) FIG. 3.
Simulated absorption images of atom distribu-tions including photon shot noise and atomic densityfluctuations.
In (a), without the coupling beam, a regularabsorption image of the background atoms is obtained. Thecolor code indicates absorption. With the coupling on (b)the background atoms are rendered transparent, except forthose in the vicinity of a Rydberg atom. Parameters of thesimulation can be found in the text. can contribute to the EIT signal, while the remainderact as two-level atoms and couple resonantly to the probelaser[11, 17]. In general, the maximum signal-to-noise ra-tios (see supplementary material) are achieved for largecoupling strengths Ω c and long exposure times τ , butin practice these will be limited by the available laserpower and by the required time resolution, which shouldbe compared to the typical lifetime of a Rydberg atom( ∼ µ s).To show the potential of this imaging scheme we havecarried out numerical calculations of the EIT imagingprocess on simulated distributions of Rydberg atoms ex-cited from a quasi-2D ideal gas. This situation can berealized with an optical dipole trap made using cylin-drically focused Gaussian beams. Of particular in-terest for current experiments is the possibility to di-rectly observe strong spatial correlations between Ryd-berg atoms induced by interactions in an otherwise dis-ordered gas [14, 16, 18].We use a simple semi-classical model to simulate theexcitation of Rydberg atoms during a chirped Rydbergexcitation pulse. The model is closely related to thoseused to describe optical control of cold collisions [19, 20].We consider a thermally distributed gas of 25 000 back-ground atoms with a peak density n D = 40 atoms /µ m and a cloud radius of σ = 10 µm . Each atom can bein either the electronic ground state or a Rydberg state.We take the strength of the Rydberg-Rydberg interac-tions as 2 π ×
50 GHz µm , typical of the 55 S state [21].The detuning of the coupling field is swept from 0 to+200 MHz within 6 µ s with an effective Rabi frequencyof Ω = 2 π × . . µ m) in the plane of the atomsand we assume a numerical aperture of 0 .
25 and an ex-posure time of 10 µ s. For the EIT ladder system, we takethe Rb states, | S / , F = 2 , m F = 2 i for the groundstate, | P / , F = 3 , m F = 3 i for the intermediate state,and | r = 28 S i for the excited state. The decay ratesare Γ p = 2 π × . c ≈ π ×
10 kHz, and forthe coupling laser we assume Ω c = 2 π ×
50 MHz. Laserline widths of 2 π × | S i and | S i states were calculated to obtain a van der Waalscoefficient of C (28 S − S ) = − π × . µ m giv-ing R c = 0 . µ m. Interactions between backgroundatoms are taken as C (28 S − S ) = 2 π × . µ m ( R ′ c = 0 . µ m). For these parameters the optimal signal-to-noise ratio is obtained for n D ≈
40 atoms/ µ m . Theprobe intensity (Ω p = 2 π × . | r i state density remains below 1 per πR ′ c .Such parameters are readily achieved in current experi-ments with quasi-2D atomic gases [23].In the background region of the image the signal isdominated by photon-shot noise, while at the centeratom-shot noise dominates. With the coupling laser onthe ground state atoms are rendered mostly transpar-ent, except for regions of high absorption around eachRydberg atom (Fig. 3b). The locations of the individ-ual Rydberg atoms are clearly resolved in the image asbright (absorbing) spots with a spatial extent of 2 . µ mFWHM comparable to the assumed optical resolution.One can easily envisage higher resolutions using state-of-the-art imaging systems [7], with the fundamental limitgiven by the density of background atoms surroundingthe impurities. The signal-to-noise ratio of our imagesis sufficiently high that we can fit the position of eachRydberg atom with subpixel precision.Despite the simplicity of the excitation model, the finaldistribution of Rydberg atoms appears highly-correlated,reproducing some of the features of a full quantum me-chanical treatment [16]. To characterize the translationalorder of the simulated Rydberg distributions, we calcu-late the pair distribution function g ( r ) from 15 simulated g (r) - r (μm) dx (μm) d y ( μ m ) FIG. 4.
Pair distribution function computed for simu-lated Rydberg images from 15 realisations.
The insetshows the averaged two-dimensional autocorrelation functionas computed from the absorption images. Taking the radialaverage gives g ( r ) − g ( r ) − . µ m bin size asobtained from the simulated Rydberg atom coordinates. Theclear shell structure which reflects translational order betweennearest and next-nearest neighbours is preserved by the im-ages. images (Fig. 4). To account for the inhomogeneous den-sity distribution we also normalize by the autocorrelationof the mean image (see supplementary material). For arandom distribution of atoms g ( r ) ≈
1. Larger correla-tion values indicate an enhanced probability to find twoRydberg atoms at a given separation, while lower val-ues indicate the absence of pairs. Since there is no pre-ferred orientation in our system g ( r ) takes on cylindricalsymmetry. We clearly observe a shell with g ( r ) ≈ ∼ µ m which reflects the strong blockadeof excitation due to Rydberg-Rydberg interactions. Atlarger distances, we observe two positive-correlated shells(around 4 and 8 µ m), which indicate translational cor-relations between nearest- and next-nearest neighbours.The observed shell structure decays rapidly indicatingthe absence of true long-range order. We note very simi-lar behaviour of g ( r ) for the raw atom positions (shadedbars). From this we conclude that the information re-garding density-density correlations can be reliably ex-tracted from the images, even under realistic imagingconditions.We have also computed the angular correlation func-tion Φ( θ ) at the radius of the first shell (see supplemen-tary material). This gives the probability, starting froman atom to find two neighbours forming an angle θ . Weobserve the presence of two peaks at ∼ ∼ π/ π/
3, reflecting the 6-fold symmetry present among most nearest-neighbours.Even more information could be obtained from these im-ages by studying higher-order correlation functions, orby first estimating the Rydberg atom positions to fullycharacterize the many-body state. Our new imaging method provides the means to opti-cally image individual atoms within a dense atomic gasusing Rydberg state electromagnetically induced trans-parency. The conditions we find to optimise the signalclosely match those of current cold atom experiments.As an example of the full potential of this new imagingscheme we have carried out numerical simulations of Ry-dberg atoms excited from a quasi-2D gas. Remarkably,this simple model already gives rise to the appearance ofstrong spatial correlations between Rydberg atoms. TheEIT imaging scheme provides the means for single-shot,non-destructive and time resolved images of such many-body states. We can foresee numerous other applicationsof the EIT imaging method. For example, Rydberg stateEIT has already been used to measure spatially inho-mogeneous electric fields near a surface [12]. Anotherexciting prospect would be to directly image single ionswithin an atomic gas which could be directly observedin current experiments [24]. Closely related ideas couldbe used to realise a single atom optical transistor [25] orto impose non-classical spatial correlations onto the lightfield [26].We would like to thank J. Evers, B. Olmos and I.Lesanovsky for valuable discussions. This work is sup-ported in part by the Heidelberg Center for QuantumDynamics and the Deutsche Forschungsgemeinschaft un-der WE2661/10.1. SW acknowledges support from theEU Marie-Curie program (grant number PERG08-GA-2010-277017).
Note added.–
During preparation of this manuscript webecame aware of related work [27]
SUPPLEMENTARY MATERIALInfluence of noise
We provide the criteria to optimize the quality of im-ages in the presence of realistic noise sources. Supposean impurity atom is situated in region ( A ) and we wishto distinguish its position from another region ( B ) bymeasuring the difference in transmission ∆ T . N ( A/B ) ph isthe number of detected photons (proportional to CCDcounts; we assume a quantum efficiency ≈
1) and N r is a reference used to normalize probe intensity varia-tions. We consider the resonant case (∆ p = ∆ c = 0)since this maximizes the contrast while keeping the den-sity of atoms in state | r i small. In region ( A ) the ab-sorption is greatly enhanced and N Aph ≈ T A N r with T A = exp ( − σ n D Im[ χ ]) and σ is the resonant absorp-tion cross-section for the probe transition. In region ( B )Im[ χ B ] ≈ T > T can bemade depends on the noise in both regions:var(∆ T ) ≈ var( N r ) h N ( A ) ph i h N r i + var( N ( A ) ph ) h N r i + 2 var( N r ) h N r i . We assume Poisson distributed noise for the inten-sity and density fluctuations, so var( N r ) = h N r i andvar( N ph ) ≈ h T A ih N r i + h N r i var( T A ). Atom shot noise isaccounted for by var( T A ) = σ Im[ χ A ] h T A i n D /a , with a the area of each region (for example the area of a pixel),var(∆ T ) = h T A i + h T A i h N r i + 2 h N r i + σ n D a Im[ χ A ] h T A i . The first two terms can be attributed to photon shotnoise while the last term is from density fluctuations.Including saturation, Im[ χ A ] = Γ p / (Γ p + 2Ω p ). This sug-gests that the signal-to-noise ratio (SNR) can be madearbitrarily high for large h N r i and large n D . How-ever, to ensure that interactions between backgroundatoms can be neglected, we require that the densityof atoms in the | r i state is kept low ( ρ rr n D πR ′ c . ρ rr ≈ Ω p / Ω c and this implies h N r i . aτ Ω c /σ n D πR ′ c Γ p , with exposure time τ . Inthe limit of strong absorption h T A i ≪
1, and substitut-ing for the maximum value of h N r i :var(∆ T ) = 2 σ Γ p n D πR ′ c a Ω c τ (3) × (cid:18) c τ σ π Γ p R ′ c Im[ χ A ] exp ( − σ n D Im[ χ A ]) (cid:19) with Im[ χ A ] = (cid:0) c / Γ p πR ′ c n D (cid:1) − .In general, the best SNR is obtained for large couplingstrengths Ω c and long exposure times τ , but in practicethese will be limited by the available laser power and bythe required time resolution. To find the optimal val-ues for n D and Ω p given fixed values of τ and Ω c wenumerically maximize the SNR using Eq. (3). The finalparameters used in the paper include the additional ef-fect of finite laser linewidths which tends to increase ρ rr slightly for the same Ω p . This shifts the optimum densityto slightly lower values. Assuming Ω c = 2 π ×
50 MHz and τ = 10 µ s we find n opt D = 40 µ m − (neglecting linewidth n opt D ≈ µ m − ). Rydberg excitation model
To simulate the excitation of Rydberg atoms by achirped laser pulse we consider a randomly (thermally)distributed ensemble of atoms. Each atom is treated asa point-like classical particle which can be in either theelectronic ground state or in a Rydberg state. As thecoupling field is swept from low to high detuning, eachatom can undergo a transition. The transition probabil-ity is estimated using the Landau-Zener (LZ) formula for a sweep through an avoided crossing [22]. The effect ofRydberg-Rydberg interactions causes level shifts for thenearby atoms which subsequently alters their probabil-ity to be excited by the laser pulse, giving rise to strongspatial correlations.The simulation starts with zero detuning for the exci-tation laser and one atom is chosen at random to startin the Rydberg state. In the next time step the laserfrequency is varied according to a fixed sweep rate, andwe calculate all level shifts due to Rydberg-Rydberg in-teractions. From the atoms which crossed the resonancecondition in the previous timestep we randomly selectnewly excited atoms based on their LZ probabilities. Anysuccessful excitation immediately influences all other sur-rounding atoms, and thus the simulation also reproducesthe excitation blockade effect. For each time step we alsosolve the Newtonian equations of motion of the Rydbergatoms to account for the interparticle mechanical forces.We do not consider the motion of the ground state atomsfor the simulation (frozen gas regime). The simulation re-turns a list of the final coordinates of all the ground-stateand Rydberg atoms within the gas after the laser sweep.These coordinates are then used as inputs to calculatethe corresponding absorption image.
Correlation analysis
To characterize the translational order of the simulatedRydberg distributions, we define a pair distribution func-tion from the absorption images n ( ~r ): G [ n ]( ~r ) = R d r n ( ~r ) n ( ~r + ~r ) (cid:0)R d r n ( ~r ) (cid:1) . (4)To account for the inhomogeneous density and finite sizeof the system we define the following rescaled pair distri-bution function : g ( ~r ) = h G [ n ] i G [ h n i ] (5)where the brackets reflect averages over independent re-alisations. For a random distribution of atoms g ( r ) ≈ θ ) ∝ *R d r n ( ~r ) R dφ n ( ~r + R nn ~e φ ) n ( ~r + R nn ~e φ + θ ) (cid:0)R d r n ( ~r ) (cid:1) + (6)where ~e φ is defined as the unit vector with angle φ withrespect to a reference axis ~e x , and R nn is the radius of thefirst positive shell of the pair distribution function. This Φ ( θ ) θ - rad FIG. 5.
Angular correlation function computed from15 simulated images.
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