Optical transmission through a dipolar layer
James Keaveney, Armen Sargsyan, Ulrich Krohn, Julia Gontcharov, Ifan G. Hughes, David Sarkisyan, Charles S. Adams
OOptical transmission through a dipolar layer
James Keaveney, Ulrich Krohn, Julia Gontcharov, Ifan G. Hughes, and Charles S. Adams
Department of Physics, Rochester Building, Durham University,South Road, Durham DH1 3LE, United Kingdom
Armen Sargsyan and David Sarkisyan
Institute for Physical Research, National Academy of Sciences - Ashtarak 2, 0203, Armenia (Dated: October 30, 2018)
The interaction between light and matter isfundamental to developments in quantum opticsand information. Over recent years enormousprogress has been made in controlling the in-terface between light and single emitters includ-ing ions [1], atoms [2, 3], molecules [4], quan-tum dots [5], and ensembles [6]. For many sys-tems, inter-particle interactions are typically neg-ligible. However, if the emitters are separatedby less than the emission wavelength, λ , resonantdipole–dipole interactions modify the radiativedecay rate [7] and induce a splitting or shift of theresonance [8]. Here we map out the transition be-tween individual dipoles and a strongly interact-ing ensemble by increasing the density of atomsconfined in a layer with thickness much less than λ . We find two surprising results: whereas fora non-interacting ensemble the opacity increaseslinearly with atomic density, for an interactingensemble the opacity saturates, i.e., a thin dipo-lar layer never becomes opaque regardless of howmany scatterers are added. Secondly, the relativephase of the dipoles produces an abrupt changein the optical transmission around the thickness λ/ . The requirement for strong dipole–dipole interactions— that the atoms must separated by less than λ — placesstringent demands on the spatial extent of the ensemble.For atomic gases considerable progress has been madein the ability to confine atoms in thin layers [9, 10], op-tical fibers [11, 12] and waveguides [13]. Alternatively,it has been recognised that the spatial constraints canbe relaxed by exploiting the long wavelength dipoles as-sociated with Rydberg transitions [14, 15] where strongdipole–dipole interactions between Rydberg atoms giverise to a large optical non–linearity [16, 17]. However,micron–scale spatial confinement is still required [18]. Asin the two–level case, in Rydberg systems dephasing canalso play a role in the emission process [19, 20].The underlying mechanism of light scattering is theinterference between the incident field and the fieldproduced by induced oscillatory dipoles. For indepen-dent dipoles, on resonance, the destructive interferencein the forward direction is equivalent to removing across-sectional area of the incident beam equivalent to σ = 6 πλ per atom, where λ = λ/ (2 π ) is the reducedwavelength of the dipolar radiation, corresponding to theeffective length scale of the light-matter interaction. Ifthe angular frequency ω of the incident field is detunedfrom the resonance frequency ω by ∆ = ω − ω , theoptical cross section becomes σ = σ / Γ) , (1)where Γ is the full width at half-maximum of the res-onance. The frequency dependence of the opacity of asingle dipole is illustrated in Fig. 1b. For a dilute dipolarlayer of thickness δz and N dipoles per unit volume, theopacity (optical depth) is given byOD = N σδz . (2)The optical depth increases linearly with N , as shown inFig. 1b. However, when the density reaches a value wherethe dipole spacing N − / ∼ λ , the resonance is broad-ened and the efficiency of each dipole is reduced due todipole–dipole interactions (Fig. 1c-d). This effect is anal-ogous to dipole blockade in Rydberg gases [14] but for athermal gas of two–level atoms it appears as an effectiveincrease in the linewidth Γ = Γ + Γ dd , where Γ dd = βN is typically referred to as self-broadening [21, 22] and β isthe self-broadening coefficient. Self–broadening results ina decrease in the resonant cross-section σ ∼ / ( N λ ) (seeSupplementary Information). In the high density limitthe increased scattering due to adding more dipoles isexactly cancelled by the higher damping rate due to res-onant dipole–dipole interactions, leading to the surpris-ing result of a constant broadband opacity (see Fig. 1c-d)given by (see Supplementary Information),OD ∼ δzλ . (3)As an interesting comparison, if we take a monolayerof atoms (the thickness δz is of the order of the Bohrradius, a ) and take the transition wavelength as λ ∼ a /α , where α is the fine structure constant, we obtainan opacity OD ∼ α (see Supplementary Information),very similar to the result of πα recently demonstratedfor graphene [23].To confirm the saturation of opacity we measure theoptical transmission of a single dipolar layer, where the a r X i v : . [ phy s i c s . a t o m - ph ] S e p Detuning ( ω − ω ) / Γ D e n s i t y N -1000 0 1000 -1000 0 1000 -6 -3 0 3 6Detuning (GHz) 0.010.11 T r a n s m i ss i o n a b c d Figure 1. Evolution of opacity with density. Dipoles are represented in ( a ) by semitransparent discs, whose radii are thedipole–dipole interaction range λ . At low density the discs do not overlap on average, and hence the interactions are negligible,whereas at high density the interactions dominate. ( b ) With no dipole–dipole interactions the lineshape is a Lorentzian withconstant width and the sample quickly becomes optically thick (black). ( c ) When dipole–dipole interactions are considered,the line broadens linearly with density, eventually leading to a constant broadband value at high density. ( d ) Experimentaldata for cell thickness δz = 390 nm showing transmission over the Rb D line, where there are now four separate absorptionlines owing to the hyperfine splitting of the ground states of Rb and Rb. Opacity is represented by the colour map, whichgoes from completely transparent (white) through to completely opaque (black).Figure 2. Experimental setup and cell. ( a ) Schematic ofoptical setup. A laser is scanned across the D atomic reso-nance line in Rb (wavelength 780 nm). The light is filteredto low power to avoid optical pumping effects and focussed toa 30 µ m spot size inside the cell, leading to a local thicknessvariation that is limited by the surface flatness of the win-dows. The transmission is then recorded on a photodiode.( c ) The ‘Newton’s Rings’ interference pattern is observed asthe gap between the two sapphire windows changes thickness( b ), due to the curvature of one of the windows, with a ra-dius R >
100 m. At the centre of these rings the cell has athickness δz ∼
30 nm. layer thickness must be less than or of the order of λ . Inour experiment we confine a gas of Rb vapour within ananocell formed by two super–polished sapphire surfaceswith a separation between 30 nm and 2 µ m, similar tothose in refs. [9, 10]. To achieve such small separationsthe plates have a surface flatness of less than 3 nm and a radius of curvature R >
100 m. After several iterations ofcell design it has been possible to fabricate a nanocell ofexceptional quality that allows the scaling of atom–lightinteractions in a dipolar layer to be elucidated.The experimental set up and the nanocell are shown inFig. 2. A narrow band laser with a linewidth of less than1 MHz is frequency tuned in the vicinity of the D res-onance lines in Rb ( λ = 780 nm), and the transmissionthrough the cell is recorded at different temperatures. Bychanging the cell temperature between 20 ◦ C and 350 ◦ Cwe can vary the number density between
N λ (cid:28) N λ >
100 . In doing so we tune the energy of dipole–dipole interactions ( V dd ∝ N ) over 6 orders of magni-tude. The measured transmission as a function of laserdetuning is shown in Fig. 1d. The similarity with theinteracting case in the high density limit (Fig. 1c) is im-mediately apparent. At low temperature the spectrumconsists of four main Dicke narrowed resonance lines cor-responding to the two isotopes, Rb and Rb, that areboth split by the ground state hyperfine interaction (theexcited state hyperfine splitting is also resolved in thecase of Rb). As the number density is increased thisstructure is completely lost and one observes a broadbandabsorption that saturates at a finite non–zero value.To make a more quantative comparision, in Fig.3 we plot transmission spectra for a cell length of λ/ T r a n s m i ss i o n T r a n s m i ss i o n ab Figure 3. Transmission spectra - experiment and theory. Ex-perimental ( a ) and theoretical ( b ) transmission spectra atthickness δz = 390 nm for measured resevoir temperatures of130 ◦ C (black), 160 ◦ C (purple), 220 ◦ C (blue), 250 ◦ C (green),280 ◦ C (yellow), 310 ◦ C (orange), and 330 ◦ C (red). Theoryspectra are based on fitting the experimental data, using aMarquardt–Levenberg method (see [30]). Zero on the detun-ing axis represents the weighted line centre of the D2 line. of Dicke narrowing [25, 26], cavity effects between thecell walls [27] and dipole–dipole interactions (see [28] andreferences therein). Although this is thicker than a sin-gle layer, the saturation of opacity can still be observed.Additional broadening and shifts due to van der Waals(vdW) atom–surface interactions are also present for δz < λ [29]. We restrict our investigation to δz (cid:38)
90 nmto avoid vdW atom–surface interactions strongly influ-encing the resonance lines, such that in the high den-sity limit of interest to this work dipole–dipole broad-ening dominates. For a random distribution of two–leveldipoles the shift of the resonance (Lorentz shift) is smallerthan the broadening [22] and is neglected. The effect ofthe Lorentz shift is apparent in Fig. 3a, but does notchange the on resonance opacity significantly. Overall,Fig. 3 suggests that the model provides an excellent de-scription of the measured transmission.To illustrate the saturation effect, we plot in Fig. 4the scaled opacity
N σλ = Im[ χ ], where χ is the sus-ceptibility of the medium (see Supplementary Informa-tion), as a function of number density. We take theopacity at the unshifted resonance position of the Rb -1 Density, N S c a l e d o p a c i t y , N σ
150 200 250 300 350 400 450 500Temperature ( ◦ C) Figure 4. Saturation of opacity due to dipole–dipole interac-tions. Without dipole–dipole interactions (dashed line) theopacity increases linearly with density. When interactionsare included into the standard theory (upper solid line) thescaled opacity for the D line is predicted to saturate at avalue Nσλ = √
2. However for thicknesses δz < λ/
4, we ob-serve a change in the behaviour which doubles the broadeningcoefficient (lower solid line). A constant additional broad-ening of Γ ad = 2 π ×
200 MHz has been added to accountfor atom–surface effects, which does not affect the value atwhich the opacity saturates at high density. Experimentaldata: δz = 90 nm (magneta pentagons); δz = 110 nm (redsquares); δz = 140 nm (green circles); δz = 180 nm (blue tri-angles); δz = 220 nm (yellow diamonds); δz = 250 nm (cyanhexagons); δz = 390 nm (purple stars). F = 3 → F (cid:48) = 4 hyperfine transition, though in principleone can choose any detuning and achieve similar results.For layer thicknesses λ/ < δz < λ , the data match ourtheoretical model including interactions extremely well.Also shown is the theoretical prediction without interac-tions, where the opacity increases linearly with densityand does not saturate. For the D2 line, the saturat-ing value is N σλ = √ δz < λ/ β thatis twice the expected value [28]. This arises because for δz < λ/ δz = λ/ β .In summary, we have demonstrated that adding more λ/ λ/ λ/ λ Layer thickness0.00.51.01.52.02.5 D i p o l e - d i p o l e i n t e r a c t i o n s t r e n g t h ( β ) Figure 5. Dipole–dipole interaction strength. Interactionstrength β is plotted as a function of layer thickness. Valuesand errors are calculated from fitting data of the form shownin Fig. 4 using a Marquardt-Levenberg method. We calculatethe average interaction strength for δz < λ/ . ± . β , whereas for δz > λ/ . ± . β . scatterers in a thin layer does not make the mediummore opaque. Instead the opacity saturates at a well de-fined value, when dipole–dipole interactions within themedium dominate the light–matter interactions. In ad-dition, when the layer thickness is less than λ/
4, we ob-serve a dramatic change in behaviour whereby the dipo-lar interaction doubles in strength, resulting in a differ-ent opacity that saturates at a value that is half that ofthe conventional case. This effect is independent of thetransition dipole moment, and so is predicted to occur inother systems of two-level interacting dipoles. This resultcould have important consequences for the generation ofnon-classical light in confined geometries and in futurework we will investigate the photon statistics associatedwith the transition between a 2D and 3D dipolar layer.We would like to thank S. J. Clark and M. P. A. Jonesfor stimulating discussions. We acknowledge financialsupport from EPSRC and Durham University. [1] Harlander, M., Lechner, R., Brownnutt, M., Blatt, R. & H¨ansel,W. Trapped-ion antennae for the transmission of quantum infor-mation.
Nature , 200–3 (2011).[2] Raimond, J., Brune, M. & Haroche, S. Manipulating quantumentanglement with atoms and photons in a cavity.
Rev. Mod.Phys. , 565–582 (2001).[3] Alton, D. J. et al. Strong interactions of single atoms and photonsnear a dielectric boundary.
Nature Phys. , 159–165 (2010).[4] Celebrano, M., Kukura, P., Renn, A. & Sandoghdar, V. Single-molecule imaging by optical absorption. Nature Photon. , 95–98(2011). [5] Shields, A. J. Semiconductor quantum light sources. Nature Pho-ton. , 215–223 (2007).[6] Hammerer, K., Sorensen, A. S. & Polzik, E. S. Quantum interfacebetween light and atomic ensembles. Rev. Mod. Phys. , 1041–1093 (2010).[7] DeVoe, R. G. & Brewer, R. G. Observation of Superradiant andSubradiant Spontaneous Emission of Two Trapped Ions. Phys.Rev. Lett. , 2049–2052 (1996).[8] Hettich, C. et al. Nanometer resolution and coherent optical dipolecoupling of two individual molecules.
Science (New York, N.Y.) , 385–9 (2002).[9] Sarkisyan, D., Bloch, D., Papoyan, A. & Ducloy, M. Sub-Dopplerspectroscopy by sub-micron thin Cs vapour layer.
Opt. Comm. , 201–208 (2001).[10] Sarkisyan, D. et al.
Spectroscopy in an extremely thin vapor cell:Comparing the cell-length dependence in fluorescence and in ab-sorption techniques.
Phys. Rev. A , 065802 (2004).[11] Slepkov, A. D., Bhagwat, A. R., Venkataraman, V., Londero, P.& Gaeta, A. L. Spectroscopy of Rb atoms in hollow-core fibers. Phys. Rev. A , 053825 (2010).[12] Bajcsy, M. et al. Laser-cooled atoms inside a hollow-core photonic-crystal fiber.
Phys. Rev. A , 063830 (2011).[13] Wu, B. et al. Slow light on a chip via atomic quantum statecontrol.
Nature Photon. , 776–779 (2010).[14] Lukin, M. D., Fleischhauer, M. & Cote, R. Dipole Blockade andQuantum Information Processing in Mesoscopic Atomic Ensem-bles. Phys. Rev. Lett. , 037901 (2001).[15] Pritchard, J. D., Adams, C. S. & Mølmer, K. Correlatedphoton emission from multi–atom Rydberg dark states. (2011).arXiv:1108.5165.[16] Pritchard, J. et al. Cooperative Atom-Light Interaction in a Block-aded Rydberg Ensemble.
Phys. Rev. Lett. , 193603 (2010).[17] Gorshkov, A. V., Otterbach, J., Fleischhauer, M., Pohl, T. &Lukin, M. D. Photon-Photon Interactions via Rydberg Blockade.(2011). arXiv:1103.3700.[18] K¨ubler, H., Shaffer, J. P., Baluktsian, T., L¨ow, R. & Pfau, T.Coherent excitation of Rydberg atoms in micrometre-sized atomicvapour cells.
Nature Photon. , 112–116 (2010).[19] Honer, J., L¨ow, R., Weimer, H., Pfau, T. & B¨uchler, H. ArtificialAtoms Can Do More Than Atoms: Deterministic Single PhotonSubtraction from Arbitrary Light Fields. Phys. Rev. Lett. ,093601 (2011).[20] Bariani, F., Dudin, Y. O., Kennedy, T. A. B. & Kuzmich, A. De-phasing of multiparticle Rydberg excitations for fast entanglementgeneration. (2011). arXiv:1107.3202.[21] Lewis, E. Collisional relaxation of atomic excited states, linebroadening and interatomic interactions.
Phys. Rep. , 1–71(1980).[22] Maki, J. J., Malcuit, M. S., Sipe, J. E. & Boyd, R. W. Linear andnonlinear optical measurements of the Lorentz local field. Phys.Rev. Lett. , 972–975 (1991).[23] Nair, R. R. et al. Fine structure constant defines visual trans-parency of graphene.
Science (New York, N.Y.) , 1308 (2008).[24] Siddons, P., Adams, C. S., Ge, C. & Hughes, I. G. Absolute ab-sorption on rubidium D lines: comparison between theory andexperiment.