New J. Phys. 17 (2015) 053046 doi:10.1088/1367-2630/17/5/053046 PAPER Imaging single Rydberg electrons in a Bose–Einstein condensate TomaszKarpiuk1,2,MirosławBrewczyk1,3, Kazimierz Rzążewski3,4, AnitaGaj4, JonathanBBalewski4, Alexander TKrupp4,Michael Schlagmüller4, Robert Löw4, SebastianHofferberth4 andTilmanPfau4 1 Wydział Fizyki, Uniwersytet wBiałymstoku, ul. Lipowa 41, 15-424 Białystok, Poland 2 Centre forQuantumTechnologies, National University of Singapore, 3 ScienceDrive 2, Singapore 117543, Singapore 3 Center for Theoretical Physics PAN,Al. Lotników 32/46, 02-668Warsaw, Poland 4 5. Physikalisches Institut andCenter for IntegratedQuantumScience andTechnology IQST,Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany E-mail:m.brewczyk@uwb.edu.pl Keywords:Rydberg atoms, Bose–Einstein condensation, imaging of electron orbitals Supplementarymaterial for this article is available online Abstract The quantummechanical states of electrons in atoms andmolecules are distinct orbitals, which are fundamental for our understanding of atoms,molecules and solids. Electronic orbitals determine a wide range of basic atomic properties, allowing also for the explanation ofmany chemical processes. Here, we propose a novel technique to optically image the shape of electron orbitals of neutral atoms using electron–phonon coupling in a Bose–Einstein condensate. To validate ourmodel we carefully analyze the impact of a single Rydberg electron onto a condensate and compare the results to experimental data. Our scheme requires onlywell-established experimental techniques that are readily available and allows for the direct capture of textbook-like spatial images of single electronic orbitals in a single shot experiment. Thewavefunction is a fundamental concept of quantummechanics. Our current understanding of atoms, molecules and solids is based on the fact that the probability density of an electron is the absolute square of the electronicwavefunction. The theoretical description of electron orbitals was founded at the beginning of the last century. Yet, to date the spatial structure ofmost orbitals has not been observed directly.Most current techniques to study thewavefunction of electrons in atoms andmolecules rely on tomographic reconstruction. Based on this technique thewavefunction of the energetically highest orbital of variousmolecules has been obtained using high harmonics generated in the interaction of intense femtosecond laser pulses with molecules [1] and photoemission spectroscopy [2]. Anothermethod, which has been until now applied to larger polymers, is based on scanning tunnelingmicroscopy [3–5]. Furthermore, an image of the electron wavefunction in single hydrogen atoms has been reconstructed only recently [6]. There, electronic states in a strong static electric field have been observed via photoionization and subsequent electron detection using a magnifying electrostatic lens. Here, we propose amethod to optically image the orbitals of electrons excited to a Rydberg state. These orbitals are larger in size than optical wavelengths.Moreover, due to the different sizable quantumdefects and dipolar selection rules, they can be prepared in awell-defined quantum state providing clean s, p, d and f series. The proposedmethod is based on the interaction of the Rydberg electronwith a dense ultracold gas [7]. Due to this interaction the probability density of a single Rydberg electron can be imprinted on the density of surrounding Bose–Einstein condensate (BEC) atoms. Thus, textbook-like optical images of hydrogenic states can be obtained using alreadywell-established imaging techniques for cold atoms. In order tomodel the Rydberg excitation dynamics and the phase imprint onto afinite-size BECwe develop a numericalmodel describing the probabilistic Rydberg excitation process and the subsequent interactionwith thefinite-size BEC.Our approach agrees well with available experimental data onRydberg excitations in a BEC [7] and confirms electron–phonon coupling as the underlyingmechanism, which has been studied previously in the framework of Bogoliubov approximation.We discuss the experimental requirements and challenges to OPEN ACCESS RECEIVED 19December 2014 REVISED 2April 2015 ACCEPTED FOR PUBLICATION 27April 2015 PUBLISHED 27May 2015 Content from this work may be used under the terms of theCreative CommonsAttribution 3.0 licence. Any further distribution of this workmustmaintain attribution to the author(s) and the title of thework, journal citation andDOI. © 2015 IOPPublishing Ltd andDeutsche PhysikalischeGesellschaft http://dx.doi.org/10.1088/1367-2630/17/5/053046 mailto:m.brewczyk@uwb.edu.pl http://dx.doi.org/10.1088/1367-2630/17/5/053046 http://crossmark.crossref.org/dialog/?doi=10.1088/1367-2630/17/5/053046&domain=pdf&date_stamp=2015-05-27 http://crossmark.crossref.org/dialog/?doi=10.1088/1367-2630/17/5/053046&domain=pdf&date_stamp=2015-05-27 http://creativecommons.org/licenses/by/3.0 http://creativecommons.org/licenses/by/3.0 http://creativecommons.org/licenses/by/3.0 implement our proposal, including finite imaging resolution aswell as the role of atomic and photonic shot noise for the expected images of Rydberg orbitals. The r1 4 interaction between the single electron of the Rydberg atom and polarizable ground state atoms can bewell described by a pseudopotential [8, 9] resulting in an effective potential acting on the ground state atoms of the form π Ψ⃗ = ⃗ V r a m r( ) 2 ( ) , (1) e Ryd 2 Ryd 2 whereΨ ⃗r( )Ryd is the Rydberg electronwavefunction, a denotes the electron–atom s-wave triplet scattering length, = −a 16.1a.u. for 87Rb [10], andme is the electronmass. This well-knownmodel has previously been used tomake quantitative statements about the binding energy and excitation spectra of ultralong-range Rydbergmolecules [11–14]. Higher partial waves are irrelevant for principal quantumnumber >n 100. The interaction between the ionic core and the BEC is∼300 times smaller and can be safely neglected [15]. Thus, the interaction between the Rydberg atomand ground state atoms creates a potentialVRyd around the excited atom with a structure defined by theRydberg electron orbital. Tomodel the effect of a single Rydberg electron on the BECwe introduce the pseudopotential termVRyd as a meanfield component in theGross–Pitaevskii equation (GPE). GPE describes the dynamics of the bosonic atomicfield.We adopt a classicalfield approximation (CFA), where a long-wavelength atomic field is replaced by a classical complex functionΨ ⃗r t( , ) satisfying the time-dependent GPE Ψ Ψ Ψ ∂ ∂ ⃗ = − + ⃗ + ⃗ + ⃗ − ⃗ ⃗   t r t m V r g r t f t V r R r t i ( , ) 2 ( ) ( , ) ( ) ( ) ( , ), (2) 2 2 trap 2 Ryd ⎡ ⎣⎢ ⎤ ⎦⎥ where f (t) is 1 for thefinite timewhen theRydberg atom is present in the BEC and 0 otherwise. On the right-hand side thefirst three terms are related to the kinetic energy, the trapping potential and the contact interactionwith coupling constant g. CFA is a valid treatment for describing Bogoliubov–Popov excita- tions [16]. Before we turn to the investigation of the imaging of electron orbitals, we use our approach tomodel our recent experiment, where about a few hundredRydberg atomswere excited successively at randompositions inside a BEC. In the experiment [7] a Rydberg atom in an s-state with principal quantumnumbers n ranging from110 to 202was created in a condensate of 87Rb atoms.We used a 1 μs light pulse, duringwhich theRydberg atomgot excitedwith a certain probability. μ10 s after the excitation pulse, we sent a μ2 s ionization pulse, which extracted theRydberg atomunless it has not been lost before. Although the Rydberg blockademechanism [17] ensures that at anymoment therewas notmore than a single excitationwithin the BEC,we studied only the cumulative effect ofmany successive excitations on the BEC. In the finite-size BEC the resonance frequency is modified by the spatially varying energy shift δ ⃗E R t( , ) [18] due to local density.Wemodel this complicated many-body excitation process by a stochasticmodel. Each appearance and disappearance of the potentialVRyd are examples of quantum jumps.We check if a randomly chosen atomon a grid representing the density distribution ρ ⃗r( ) is in a Rydberg state according to the excitation probability. The probability tofind an atom at position ⃗R in the Rydberg state, for sufficiently short times t and low single atomRabi frequencies ΩR, is given as Ω Ω Ω⃗ = ⃗ ⃗p R t R t R t t( , ) ( , ) sin ( , ) 2 , (3)R 2 2 2 ⎡⎣ ⎤⎦ where Ω Ω Δ⃗ = + ⃗R t R t( , ) ( , )R 2 2 is the effective Rabi frequency, which accounts for a non-zero local detuning Δ ⃗R t( , ). This spatially varying detuning Δ ⃗R t( , ) is given by a frequency difference of the detuning ΔωL of the excitation laser from the Rydberg transition in an unperturbed atom and an additionalmeanfield shift δ ⃗E R t( , ). Since the condensate density changes due to the appearance of successive Rydberg atoms, the detuningΔ and thus the excitation probability (3) depend also on time. Following a coherent evolution at all possible grid points a localized Rydberg atom is potentially generated in our simulation on the time scale of the decoherence rate due to elastic scatteringwith a ground state atom. Thereforewe choose the time step for our coherent evolution to be 200 ns, whichwe then interrogate for the presence of a localized Rydberg atom.We repeat checking the atoms every 200 ns until the end of the excitation pulse if noRydberg atomwas found in the previous iteration.Once a Rydberg atom is createdwe propagate the GPEwith theVRyd term included to calculate the evolution of the perturbed BECwhile the Rydberg atom is present in the BEC. The interaction of a single Rydberg electronwith surounding BEC atoms decays 2 New J. Phys. 17 (2015) 053046 TKarpiuk et al exponentially in timewith a time constant of∼10 μs5. After 13 μs thewhole procedure starts again, however, the density distribution of the BEC is changed by the previous cycle. A cycle consisting of the excitation of the Rydberg atom followed by afinite interaction timewith the condensate atoms is repeated 300 or 500 times as it was done in the experiment. The energy of the system increases by the time-dependent potentialVRyd. Thus the condensate fraction is reduced. Some of the ground state atoms are promoted from the condensate to the thermal cloud.Within the CFA the two components of the bosonic gas—the condensate and the thermal cloud are identified by accounting for the coarse graining as an unavoidable element of themeasurement process [19], [20]6.Here, we define a coarse grained one-particle densitymatrix as resulting from the column integration [21] ∫ρ Ψ Ψ′ ′ = ′ ′x y x y t N z x y z t x y z t¯ ( , , , ; ) 1 d ( , , , ) *( , , , ), (4) where the x-axis is the condensate symmetry axis and the imaging is performed along the radial direction. The resulting densitymatrix, upon spectral decomposition [22], determines the fraction of the condensed atoms as a dominant eigenvalue. We calculate the total condensate losses at the end of the excitation sequence, divide themby the number of excitation cycles and study the dependence of this quantity as a function of the laser detuning ΔωL and the principal quantumnumber n of the Rydberg state (figure 1). On the blue side of the resonance theRydberg atom is created almost in every shot but losses are small because Rydberg atoms are excited in regions of low density, far from the center (seemiddle panel in figure 1, right frame). Towards the center of the linemore Rydberg atoms are excited around the center of the trapwhere the density of the condensate is high. This leads to the increase of losses reaching amaximumapproximately at the point where ΔωL is equal to δ ⃗ E R t( , ) calculated Figure 1.Theoreticallymodeled losses of atoms from the condensate per laser excitation pulse versus the detuning from the non- interacting Rydberg level (top frame). Solid lines areGaussian fits.Middle panel: a single realization real-space distribution of Rydberg atoms for n=110 and Δω = −13 MHzL (left frame),−9 MHz (middle frame), and 1 MHz (right frame). Bottompanel: excitation probability, equation (3), along the trapping symmetry axis averaged over 500 cycles (taken at themoments of timewhenRydberg atoms are created). 5 See supplementarymaterial avaliable at stacks.iop.org/njp/17/053046/mmedia. 6 A close analogy exists with the classical,Maxwell electrodynamics. At amicroscopic level at each point in space and time an electric field has awell defined value even for themost complicated field.While a product of electricfields at two points in space–time is awell defined but usually useless number, a coarse graining caused by the detectorsmakes such a quantity useful, enabling, for instance, a proper definition of coherence length/time. 3 New J. Phys. 17 (2015) 053046 TKarpiuk et al http://stacks.iop.org/njp/17/053046/mmedia at the center of the trap.On the red side of the resonance stillmanyRydberg atoms are excited in the center of the trap (compare left and central frames of themiddle panel infigure 1), however, not in every excitation cycle and thus the overall losses decrease. In the case of n=110 and Δω = − ∼12 MHzL two third of the excitation pulses creates a Rydberg excitationwhile only every seventh trial is successful at Δω = −16L MHz. The asymmetry of the process with respect to the center of the line stems from the detuningwhich is a function of the local density. As in [7] our BEC exhibits quadrupole oscillations after the excitation sequence isfinished.However, the losses do not continue. The absolute values ofmaximal losses determined from theGaussianfits to our numerical data (figure 1) are compared to experimental data and Bogoliubov calculations from [7] infigure 2 (top panel).We extract also the position of the resonance (middle panel) and thewidth of the line (bottompanel). The numerical results agree remarkablywell with the experimental data considering the fact that only estimated values for the Rabi frequencies from themeasurement and no additional free parameters were used. While the overall atom loss was already quantitatively predictedwithin a Bogoliubov approach in [7], our method presented here provides additional insights and describes time evolution of the BECduring the experimental sequence in detail. This fact is of importance, since for every excitation but thefirst one, the condensate is already distorted due to the influence of the previous Rydberg atoms.Ourmethod is nonperturbative and goes beyondfirst order approximation in phonon production.Moreover we predict the whole resonance line shape and include in themodel the inhomogeneous density of the condensate caused by the trapping potential. Having demonstrated that ourmodel reproduces the experimental data verywell, we now turn to the proposal of observing an electronic orbital by imaging the condensate density responding to the Rydberg potential (equation (1)). Our scheme relies on optical access with high numerical aperture as is readily available inmany BEC experiments. Such high resolution optics enable the tight focusing of the excitation lasers into the center of the condensate, to define the position of the Rydberg atom(s) with high precision (figure 3).Moreover, this enables high resolution absorption images of the BEC to be taken.We consider Rydberg s- and d-states, which are accessible in typical two-photon excitation schemes [23, 24]. First, we study the case where the excitation lasers are kept on continuously (figure 4(e)) to re-excite the Rydberg state as soon as the previous one has decayed.Here, the localization of the Rydberg atommust bewithin an area smaller than the expected structure size.Otherwise, the combined impact ofmanyRydberg excitations will wash out the Rydberg electron orbital imprint on the BEC. To resolve the overall structure of the exemplary Figure 2.Comparison of theoretical results obtainedwithinCFA (red open squares) and Bogoliubov approximation from [7] (blue open triangles) with experimental data (black dots). The frames depict (from top to bottom) themaximum losses of atoms from the condensate per laser excitation pulse, the position of the resonance, and the FWHMof the resonance lines. The error bars of theCFA results are the statistical errors from theGaussianfits. 4 New J. Phys. 17 (2015) 053046 TKarpiuk et al 180D state (orbital radius μ=r 3.3 m) it is sufficient to have the excitations within a diameter of μ=d 1.5 m. Sufficient sharpness and a good contrast of the image requires about 50 excitations, since the scattering potential depth is lower than the chemical potential of the condensate. To visualize the electron orbit of the Rydbergwavefunctionwith only oneRydberg excitation cycle (figure 4(d)) the parameters of the experiment have to be chosenmore carefully. A detailed study of this situation can be found in the (see footnote 5) and is summarized in the following. The principal quantum number of the Rydberg statemust not be chosen too large because although the orbital radius scales with n*2, the effective potential drops with −n* 6. The calculations show that the effective potential should be at least one order ofmagnitude deeper than the chemical potential of the atoms so that they can react during the lifetime of the Rydberg atom. This situation is reached for a principal quantumnumber around 140. Additionally, the thickness of the condensate, which the imaging light is traveling through, should not be larger than the orbital Figure 3.Rydberg atoms in 140S (blue) and 180D (orange) states are excited in the center of the condensate by a tightly focused laser beam (red). Dashed lines indicates the projection of the respective Rydberg blockade radii. All the sizes are to scale. Figure 4. (a) and (b): calculated orbitals for different Rydberg electrons convolvedwith afinite imaging resolution of μ1 m ( e1 2 width of the point spread function). This is themaximumcontrast which can be expected from the imprint on the condensate. (c): center part of the BECdensity distribution. The condensate consists of ×5 104 rubidium atoms and is confined in a harmonic trap with radial and axial frequencies ω π= ×2 200 HzR and ω π= ×2 10z Hz, respectively. (d) and (f): simulated density change caused by a single Rydberg atomwithout (d) andwith atomnumber shot noise (f). These density patterns form after a single Rydberg atom lasting for μ30 s (whichwill be the case for every twentieth shot for a lifetime of μ10 s) and an additional evolution time of μ180 s. (e): density distribution of the BEC after 50Rydberg atoms have been consecutively excited in the center region ( e1 2 width: μ1.2 m). 5 New J. Phys. 17 (2015) 053046 TKarpiuk et al radius of the Rydberg electron.Otherwise the imaging light passes through an area that is not affected by the imprint of the electronicwavefunction, which results in a reduction of contrast. In such a single-shot experiment the atomnumber shot noise [25] is themain source of noise. For the proposed parameter set (figure 4(f)), a peak density ρ ∼ 1014 cm−3 and a radial size of the condensate of μ1.5 m results in a ∼6 %background noise level (see supplementarymaterial). This is well below the expected signal contrast of ∼24 %. Therefore, the Rydberg orbital imprint on the BECdensity should be observable. Note, that the impact of the shot noisemay be reduced by averaging images frommultiple runs. To summarize, we have presented and verified a theoretical,microscopicmodel of a single Rydberg electron in a BEC.Our theoreticalmodel has several simplifications: (1) we do not account for the real losses of the trapped gas due to three body recombination. (2) The electron in itsmotion is not slow in the vicinity of the ionic core.We assume the electron–atom scattering length to be velocity independent, which is a valid description beyond >n 80. 3. (3) The impact of the ionic core on the heating process is neglected. After verification of our theoreticalmodel, we have proposed a novel scheme formapping the electronic orbital onto the density of the condensate, thereby realizing amethod to directly observe various electronic orbitals. Of course, with the available resolutionwe can image only the angular probability distribution of a Rydberg orbital. Its radial structure will bewashed out and this is not only due to limited imaging resolution but also because tiny oscillating radial structure of higher Rydberg orbitals occurs on the space scale shorter than the healing length. Also exotic shapes of single electron probability densities in electric andmagnetic fields including circular Rydberg states [26], Stark states, Bohr-likewavepackets [27, 28] and one-dimensional atoms [29] could be investigated in awaywe propose. Furthermore, this approach can also be extended tomore complex systems like Rydberg atommacrodimers [30] andmulti-electron systems. Phase-sensitive images could be obtained if a structureless reference state is used in a coherent superposition state. The technical requirements with respect to resolution, both for the local excitation of Rydberg atoms and the detection of the resulting structures, aremet by state of the art experimental setups. Furthermore, various techniques like dark ground imaging [31], phase- contrast imaging [32], polarization contrast imaging [33, 34] and adapted forms of absorption imaging [35] are readily available to precisely determine the density distribution of a BEC in situ. The optical imaging of a single electron in a single shot experiment thus seems in direct reach. Acknowledgments Weare grateful toMariuszGajda andTomasz Sowiński for helpful discussions. Theworkwas supported by the National Science Center grantsNo.DEC-2011/01/B/ST2/05125 (TK) andDEC-2012/04/A/ST2/00090 (MB, KR). KR acknowledges the financial support from the project ‘Decoherence in long range interacting quantum systems and devices’ supported by contract research ‘Internationale Spitzenforschung II’ of the Baden- Württemberg Stiftung. TheCQT is a ResearchCenter of Excellence funded by theMinistry of Education and the National Research Foundation of Singapore. The experimental work is funded by theDeutsche Forschungsgemeinschaft (DFG)within the SFB/TRR21 and the project PF 381/4-2.We also acknowledge support by the ERCunder contract number 267100 and from EUMarie Curie program ITN-Coherence 265031.MS acknowledges support from theCarl Zeiss Foundation. SH is supported by theDFG through projectHO4787/1-1. 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