The authors have measured by laser spectroscopy the energy of interaction between a sodium atom and its images in the walls of a micron-sized cavity. This cavity-QED study is the first direct quantitative test of the Lennard-Jones van der Waals interaction as a function of controlled atom-surface separation and mean-square electric dipole moment

Haroche, S.

Hinds, E. A.

Sandoghdar, V.

Sukenik, C. I.

English

Old Dominion University

Old Dominion UniversityODU Digital CommonsPhysics Faculty Publications Physics6-1992Direct Measurement of the Van Der WaalsInteraction Between an Atom and Its Images in aMicron-Sized CavityV. SandoghdarC. I. SukenikOld Dominion University, csukenik@odu.eduE. A. HindsS. HarocheFollow this and additional works at: https://digitalcommons.odu.edu/physics_fac_pubsPart of the Atomic, Molecular and Optical Physics CommonsThis Article is brought to you for free and open access by the Physics at ODU Digital Commons. It has been accepted for inclusion in Physics FacultyPublications by an authorized administrator of ODU Digital Commons. For more information, please contact digitalcommons@odu.edu.Repository CitationSandoghdar, V.; Sukenik, C. I.; Hinds, E. A.; and Haroche, S., "Direct Measurement of the Van Der Waals Interaction Between anAtom and Its Images in a Micron-Sized Cavity" (1992). Physics Faculty Publications. 6.https://digitalcommons.odu.edu/physics_fac_pubs/6Original Publication CitationSandoghdar, V., Sukenik, C.I., Hinds, E.A., & Haroche, S. (1992). Direct measurement of the van der Waals interaction between anatom and its images in a micron-sized cavity. Physical Review Letters, 68(23), 3432-3435. doi: 10.1103/PhysRevLett.68.3432VOLUME 68, NUMBER 23 P H YS ICAL R EV I EW LETTERS 8 JUNE 1992Direct Measurement of the van der Waals Interaction between an Atom andIts Images in a Micron-Sized CavityV. Sandoghdar, C. I. Sukenik, and E. A. HindsPhysics Department, Yale University, New Haven, Connecticut 06520Serge HarocheEcole Normale Superieure, Paris, France and Physics Department, Yale University, 1Vew Haven, Connecticut 06520(Received 27 March 1992)We have measured by laser spectroscopy the energy of interaction between a sodium atom and its im-ages in the walls of a micron-sized cavity. This cavity-QED study is the first direct quantitative test ofthe Lennard-Jones van der Waals interaction as a function of controlled atom-surface separation andmean-square electric dipole moment.PACS numbers: 42.50.%'m, 32.70.3zThe van der Waals interaction [1] between extendedneutral bodies is a ubiquitous feature of nature in which acooperative electrical polarization of the bodies lowers theenergy of the system. The most familiar case is that of agas in which the fluctuating dipole of each molecule po-larizes its neighbors [2], resulting in an attractive poten-tial proportional to R, ~here R is the distance betweentwo molecules. This interaction is responsible for afamous correction to the ideal-gas equation, causes shiftsand broadening of spectral lines in vapors, and plays anessential role in the latent heat and surface tension ofliquids. Another important example is the Lennard-Jones[3] dipole attraction between an atom and its image in aplane conductor, for which the potential varies as zwhere z is the distance of the atom from the mirror. Thisatom-surface coupling is involved in a variety of phys-icochemical processes, such as adsorption, in which atomscome close to surfaces. Atomic-force microscopes arealso based on the van der Waals interaction [4].Although several direct, quantitative measurementshave been made of the collective van der Waals forces be-tween macroscopic bodies [5], experimental studies of thevan der Waals interaction between a free atom or mole-cule and a macroscopic surface are scarce. Some experi-ments have been based on the observation of atomic tra-jectories near metallic surfaces [6] where a small atomicdeflection, related to the cumulative effect of van derWaals interactions at various distances, yielded qualita-tive information on the interaction itself. A small fre-quency shift, recently observed in the spectrum of cesiumvapor near a dielectric surface [7], has also been relatedto the van der Waals atom-surface interaction integratedover a range of separations. In this Letter we report thefirst direct and precise measurement of the van der Waalscoupling between an atom and a metallic surface, inwhich the interaction is quantitatively studied as a func-tion of controlled separation and of the electronic state ofthe atom.Figure 1(a) shows the main features of the experiment.An atomic beam of sodium effuses from an oven atDetector/4ion RateLaser BeamsX, and h.3SAtomic Beam(a j!!SPVXXPEXEEPXE8!!nSIOD!X -425nm2A~(= 589nm! ~8(b)F lG. l. (a) Sodium ground-state atomic beam enters theparallel-plate cavity and is excited by laser beams with wave-lengths A, ] =589 nm and A, 2 —425 nm to an upper level nS(n =10-13) or lODg2. The wavelength k2 is scanned to obtainan excitation spectrum. Only those atoms excited near thecenter of the cavity reach the detector. (b) Relevant energylevels of sodium.200'C into a 10 -torr vacuum and enters the space be-tween two plane, gold-coated mirrors (flatness -5 nm),which is 8 mm long in the beam direction and 3 cm high.Approximately half a millimeter before the cavity exit,the atoms are excited by two superimposed laser beams,one yellow (A, =589 nm) and the other blue (X-425 nm),either to the 10Dq~2 state or to one of the states nS, withn =10-13. The relevant energy levels are shown in Fig.1(b). The laser beams are focused to a 300 pm waist andare carefully aligned to be normal to the mirrors, andtherefore to the atomic beam, in order to avoid any first-order Doppler effect. Excited atoms leave the cavity andenter a detector where they are field ionized and countedusing a channel electron multiplier. An excitation spec-trum is obtained by scanning the wavelength X2 while3432 1992 The American Physical SocietyVOLUME 68, NUMBER 23 PHYSICAL REVIEW LETTERS 8 JUNE 1992where g(3) —= 1.20 is the Riemann zeta function, d is theelectric dipole operator, and L is the width of the gap.We have computed the expectation values of d using theknown oscillator strengths between nS and n'P states[10] and find that the result is given to better than1% accuracy by the hydrogenic formula (nS lid )nS)=e aon (5n* + I )/2, where e is the electron charge, aois the Bohr radius, and n =n —1.349 is the principalquantum number corrected for the quantum defect of theS states.At distances of the order of atomic dimensions,higher-multipole interactions and electron exchangeeffects become important and the atom-surface interac-tion no longer has the van der Waals form. For atom-mirror separations greater than (2x) ' times the charac-teristic atomic absorption or emission wavelengths, thepropagation time of the field exceeds the autocorrelationtime of the atomic dipole, and the van der Waals interac-tion vanishes. It is replaced in ground states by theCasimir-Polder interaction [11],which varies as z andis a purely quantum electrodynamic effect. The long-range behavior for excited states, which is an oscillatoryfunction of the separation, is dominated instead by theresonant radiative coupling between the atom and its re-Nx —100—I2 —200"Vl~ —300-~—400 ~W-500—0.5I—0.3I I I—0. 1 0. 1Position (pm)0.3 0.5FIG. 2. van der Waals shift of the sodium I3S state vs posi-tion in a I -pm-wide, parallel-plate cavity.recording the rate of detected ions. The essence of theexperiment is to study the shifts of the nS excitation spec-tra at various mirror spacings, the 10D5g2 spectra beingused to monitor stray electric fields as discussed below.Figure 2 shows the van der Waals potential for the 13Sstate of sodium in a parallel-plate cavity 1 pm wide dueto the infinite series of images resulting from successivefield reflections. The interaction energy is negative anddiverges at each mirror surface. At the center of the gap,this potential reaches a maximum having the value [8,9]&&vdw = 3 &nS Ild InS&,—7g(3)24xepLtarded reflections [8]. The nS states of this study have anelectronic extension of order 10 nm and their characteris-tic emission and absorption wavelengths (nS n'P) arein the 50-150 pm range. Thus the van der Waals formu-la (I) holds for L between —100 and —10000 nm and isvalid for the full range of gap sizes we have explored.Moreover, our spectra are sensitive only to the shifts ofthe excited nS levels since the shifts of the 3S and 3Pstates are completely negligible in comparison.At first sight, the potential shown in Fig. 2 suggestsvery asymmetric spectral lines with a high-frequencyedge due to atoms at the center of the gap and a long tailon the low-frequency side from atoms passing closer to ei-ther mirror. In fact, the lines are typically only 20-30MHz wide, indicating that the atoms we detect are excit-ed only near the center of the gap over a region of width0.2L-0.3L. This is due to a combination of severaleffects. First, before laser excitation, atoms far from thecenter are deflected by the atom-cavity interaction ontothe mirrors where they stick and are therefore removedfrom the beam. Second, the excited atoms experience amuch larger attraction to the walls and cannot escapefrom the cavity to be detected unless they are sufficientlyclose to the center at the point of laser excitation. Final-ly, the detected atoms contribute to the spectrum in pro-portion to the inverse of the slope of the potential at theirposition, which again favors those excited at the center.We now give some more details about the experimentalprocedure. The mirror surfaces are made by thermalevaporation of chromium (- I nm) followed by gold (40nm) onto a substrate of fused silica. This results in mod-est reflectivities at the laser wavelengths (67% at 589 nmand 32% at 425 nm) and allows us to couple light into thecavity even though it is not resonant at these wavelengths.On the other hand, the reflectivity of the gold film isessentially unity at the much longer wavelengths (50-150pm) relevant for the van der Waals interaction of the nSstates. The mirrors are pressed against each other toform a wedge whose width is zero at the bottom, whilethe width at the top can be varied by pushing on the mir-rors with screws in order to compress a 3-pm nickel foilspacer acting as a spring. We set the width of the gap byilluminating the cavity with monochromatic light (mercu-ry green, lj. =546 nm) and adjusting the screws until oneof the interference fringes coincides with the laser beams.Excitation spectra were observed for four nS states(n =10-13) for each of the seven cavity widths L corre-sponding to the third through ninth fringes of the greenlight. In Fig. 3, graphs (a)-(e) show some of the 13Sspectra in successively narrower cavities. The peak onthe right of each trace is a reference line obtained on un-perturbed atoms in an auxiliary atomic beam. Looking atthe peak on the left, the van der Waals shift is clearlyseen.Three main systematic effects must be taken into ac-count when analyzing these spectra. First, in spite of thestrong spatial selection of the atomic beam, the peaks of3433VOLUME 68, NUMBER 23 P H YSICAL R EV I EW LETTERS 8 JUXE 1992—30-N"0N~~E-0z(b)(e)—60-—90-0U—120-SU-1800.4 0.8I I1.2 1.6 2 2.4Plate Separation (tim)f. lG. 4. vanv der Waals shifts vs plate separation for an atomat cavity center. Series a-d correspond to states 135-105, re-spectively, with crosses indicating experimental data and linescorresponding to the calculated shifts.-300 -200 -100 0 100Frequency Shift (MHz)3 Sh'~ Shifted excitation spectra of the sodium l3S state ina cavity. Traces (a)-(e) were measured in cavities of widths2100, 1554, 1281, 1007, and 734 nm ( ~ 2l nm), respectively.the spectral lines lie at a slightly lower frequency thanthey would if all the atoms were exactly at the center ofthe cavity. A simple numerical model shows that thisoffset B,ass, t grows with linewidth and reaches a maximumo — ~ 3 MHz when the observed linewidth is larger than.5I, I being the free space linewidth which is 11 MHzin our experiment.A second small correction is required because of theStark shift 6 .sta, k of the nS states due to an unavoidablestray electric field in the cavity. This electric field pro-duces much larger Stark shifts of the very polarizable10D~/2„,, -5/2 sublevel, which we measure to determinethe required correction. In order to find the nS Starkshift 6'sta, k, we subtract the calculated van der Waals shiftof the 10DO ~/2„„=5/2 level from the measured shift of thatlevel and divide by the ratio of electric polarizabilities,a 1 OD5/ jJ5jg)/a(nS)Thisr. atio is calculated to varyfrom 24 for n =13 to 150 for n =10, a result that wechecked by recording Stark spectra in known electricfields outside the cavity. We note that due to the largevalues of these ratios, the Stark correction is actually veryinsensitive to the assumed value of the 10D5/2, m& =5/2 Vander Waals shift. The largest nS Stark shift is found inthe 135 state and in the smallest gap, where 6= —178 MHz wh 1ereile usta„k= —6 MHz, corresponding toa stra field f' y o 44 V& cm. I n every ease the correction isless than 10% of the van der Waals shifts observed. Inorder to reach this low level of stray electric field, we findit important to bake the mirrors at 120 C for severalhours after placing them in the vacuum system. Theelectric field is then found to vary roughly as L '-, whichis consistent with a patch effect due to the crystallinestructure of the evaporated gold surface.The last systematic correction concerns our measure-ment of the cavity width using the green light fringes. Ifthe phase of the reflection coefficient is written as n+4,the width of the cavity at the mth fringe is (m —p/ir)k/2.Whenen p is estimated using 0.32+2.22i for the complexrefractive index of gold at 546 nm, we find that the cavityshould be narrower than mA/2 by 72 nm. Direct mea-surements on a similar cavity using an electron micro-scope show that the actual correction is 85+ 15 nm.shifts, 6 . - —6T e crosses in Fig. 4 show the observed van d W 1er aa sti ts, meas ~onset stark, versus gap width for each fhe four nS levels. The width of the cross (+'21 nm)0re ects the total uncertainty in the mirror spacing, whilethe heigh& indicates a + 2 MHz uncertainty in frequencyshift due to residual Doppler shift and measurement er-rors. The lines correspond to Eq. (1), without any freeparameters.This experiment provides the first direct and precisecheck of the Lennard-Jones formula. The result clearlyverifies both the ." ' law and the proportionality to themean-square atomic dipole. We see that the boundaryaround the atom affects not only the natural widths of thelevels, as was demonstrated in earlier cavit -~ED t d], ut also shifts the energy levels by an amount which,in t e distance range of our experiment, is given preciselyby the Lennard-Jones theory. This shift results from thenear-field instantaneous coupling of the atomic dipolewith its images in the mirrors. In the case of larg. r mir-ror spacings, one expects the excited state to exhibit in-stead the oscillatory shift due to the far-field retarded ra-diative coupling of the atom with its images [g], which isproportional to I/n* L, rather than n* /L'. This workmay open the way to observing the far-field shift in;~3434VOLUME 68, NUMBER 23 PH YSICAL REVI EW LETTERS 8 JUNE 1992low-Q cavity, where its magnitude is typically a fractionof the natural width of the state. This radiative shift hasbeen observed in high-Q cavities resonant near an atomictransition frequency, where it is strongly enhanced [12].The retarded Casimir shift of the ground state has so faronly been investigated by the deflection or reflection of anatomic beam [13]. Since any transition to an excited lev-el is aA'ected by the much larger shift and linewidth ofthat level, a spectroscopic study of the Casimir shiftseems inadequate, but methods based on atom inter-ferometry oA'er an interesting alternative.We have shown that it is practical to perform spectros-copy on an atomic beam inside a cavity less than 1 p mwide, opening the way to direct studies of other effects in-cluding the patch potentials of a surface and the excited-state atom-surface coupling. From another point of view,this experiment shows that the van der Waals energyshift can be used to determine the position of an atomwithin a microscopic cavity with a resolution of a few tensof nanometers, a new tool which also promises to be valu-able in future cavity-QED studies.We thank Donghyun Cho for his work on the data-acquisition system. This work was supported by the Na-tional Science Foundation.[I] See, for example, D. Langbein, Springer Tracts inModern Physics, Vol. 72 (Springer-Verlag, New York,1972).[2] F. London, Z. Phys. 63, 245 (1930); Trans. Faraday Soc.33N, 8 (1937).[3] J. E. Lennard-Jones, Trans. Faraday Soc. 2$, 333 (1932).[4] G. Binnig, C. F. Quate, and Ch. Gerber, Phys. Rev. Lett.56, 930 (1986).[5] See, for example, W. Arnold, S. Hunklinger, and K.Dransfeld, Phys. Rev. B 19, 6049 (1979), and referencestherein.[6] D. Raskin and P. Kusch, Phys. Rev. 179, 712 (1969); A.Shih, D. Raskin, and P. Kusch, Phys. Rev. A 9, 652(1974); J. Bardon and M. Audiffren, J. Phys. (Paris),Lett. 44, 883 (1983); A. Anderson et al. , Phys. Rev. A 37,3594 (1988).[7] M. Oria et al. , Europhys. Lett. 14, 527 (1991).[8] G. Barton, Proc. R. Soc. London A 410, 175 (1987); E.A. Hinds and V. Sandoghdar, Phys. Rev. A 43, 398(1991).[9] E. A. Hinds, Adv. At. Mol. Opt. Phys. 2$, 237 (1991).[10] E. M. Anderson and V. A. Zilitis, Opt. Spectrosc.(USSR) 16, 99 (1964).[I I] H. B. Casimir and D. Polder, Phys. Rev. 73, 360 (1948).[12] D. J. Heinzen and M. S. Feld, Phys. Rev. Lett. 59, 2623(1987); Yifu Zhu et al. , Phys. Rev. Lett. 64, 2499(1990); F. Bernardot et al. , Europhys. Lett. 17, 33(1992); R. J. Thompson et al. , Phys. Rev. Lett. 6$, 1132(1992).[13] E. A. Hinds et al. , in Atomic Physics, Vol. I2, edited byJ. Zorn and R. Lewis (AIP, New York, 1991), p. 283;Mark Kasevich et al. , ibid. , p. 47.3435

Direct Measurement of the Van Der Waals Interaction Between an Atom and Its Images in a Micron-Sized Cavity

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Direct Measurement of the Van Der Waals Interaction Between an Atom and Its Images in a Micron-Sized Cavity

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