VOLUME 88.NUMBER 10 PHYSICAL REVIEW LETTERS 11 MARCH 2002 Matter-Wave Interferometer for Large Molecules Bjorn Brezger,Lucia Hackermuiller,Stefan Uttenthaler,Julia Petschinka,Markus Arndt,and Anton Zeilinger* Universitit Wien,Institut fuir Experimentalphysik,Boltzmanngasse 5,A-1090 Wien,Austria (Received 20 November 2001;published 26 February 2002) We demonstrate a near-field Talbot-Lau interferometer for C7o fullerene molecules.Such interfero- meters are particularly suitable for larger masses.Using three free-standing gold gratings of I um period and a transversally incoherent but velocity-selected molecular beam,we achieve an interference fringe visibility of 40%with high count rate.Both the high visibility and its velocity dependence are in good agreement with a quantum simulation that takes into account the van der Waals interaction of the molecules with the gratings and are in striking contrast to a classical moire model. DOI:10.1103/PhysRevLett.88.100404 PACS numbers:03.75.Dg,03.65.Ta,39.20.+q Interferometry with matter waves has become a large Talbot-Lau count rates can exceed those of Mach-Zehnder field of interest throughout the past years [1,2].It repre- interferometers by several orders of magnitude. sents a powerful tool for the demonstration of basic quan- A Talbot-Lau interferometer has already been demon- tum phenomena and of matter-wave effects as well as for strated for potassium atoms [9]and-in the time applications in high precision measurements inertial forces domain-for sodium atoms in a Bose-Einstein conden- [3,4]and fundamental constants [5]. sate [10].Because the scaling properties of near-field It is therefore interesting to extend techniques from neu- devices are very favorable in comparison to far-field tron and atom interferometry to more massive,complex interferometers,the Talbot-Lau effect has actually been objects,such as large molecules,in order to quantitatively proposed for experiments using quantum objects up to the study decoherence by thermal coupling to the environment, size of a virus [6,11]. to get information about a wide range of molecular prop- The successful diffraction of fullerenes at both a me- erties and to work on novel lithographical or metrological chanical [12]and an optical [13]grating has now stimu- applications [6].In this Letter,we report the first lated a new set of experiments exploring the limits of demonstration of an interferometer for macromolecules, macromolecule interferometry.The experiment presented and in particular a Talbot-Lau interferometer for C7o. here takes advantage of the Talbot effect which is a self- The molecules are in different internal states with many imaging phenomenon that occurs when a periodic struc- excited vibrational and rotational degrees of freedom. ture is illuminated by a coherent beam [14-16].Images Nevertheless we observe a clear signature for a quantized of this grating are then reconstructed at discrete multiples center-of-mass motion. of the Talbot length Lr =d2/AdB,where d is the grating Up to now two types of interferometers with molecules constant and AdB is the de Broglie wave length of the in- have been demonstrated,namely,a Ramsey-Borde inter- cident object. ferometer using I2 [7]and a mechanical Mach-Zehnder Our interferometer consists of three identical gratings interferometer for Na2 [8].The use of such devices for which are equally spaced.The first grating acts as a comb larger molecules remains,however,a challenge:A direct of sources which have vanishing mutual coherence.Yet application of a Ramsey-Borde interferometer for com- each source is sufficiently small to prepare the required plex molecules is precluded by the lack of narrow resonant transverse coherence at the second grating.The second transitions.On the other hand,a simple extrapolation of grating is then coherently imaged onto the plane of the the Mach-Zehnder interferometer to heavy,fast molecules third grating by the Talbot effect.The images coming would require extremely fine gratings,good collimation, from all sources look identical but they are shifted by or large distances.This is because such an interferometer multiples of the grating constant and therefore overlap in operates in the Fraunhofer regime,where the characteris- spite of their lack of mutual coherence [16].The movable tic size of a diffraction pattern scales linearly both with the third grating blocks or transmits the molecular interference wavelength and with the distance between the diffracting fringes according to its position. structure and the plane of observation. The experimental setup,as sketched in Fig.1,consists A solution to this problem is the Talbot-Lau interfer- of three free-standing gold gratings(Heidenhain)with a ometer,which also consists of three successive gratings nominal period of d =(991.25 0.25)nm,an open frac- but operates in the near-field or Fresnel regime,where the tion (ratio of slit width to period)of f =0.48+0.02,a characteristic size of a diffraction pattern scales with the thickness of b =500 nm,and a huge usable area with a square root of both the wavelength and the distance.A diameter of 16 mm.The distance between the gratings is Talbot-Lau interferometer can accept a spatially incoher- set to LI =L2 =0.22 m.The whole setup is placed in a ent beam,which implies that no collimation is needed, vacuum chamber at a pressure of 3 x 10-8 mbar.A beam and it works with a spatially extended detector.Therefore of C7o fullerene molecules is generated by sublimation 100404-1 0031-9007/02/88(10)/100404(4)$20.00 2002 The American Physical Society 100404-1
VOLUME 88, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 11 MARCH 2002 Matter-Wave Interferometer for Large Molecules Björn Brezger, Lucia Hackermüller, Stefan Uttenthaler, Julia Petschinka, Markus Arndt, and Anton Zeilinger* Universität Wien, Institut für Experimentalphysik, Boltzmanngasse 5, A-1090 Wien, Austria (Received 20 November 2001; published 26 February 2002) We demonstrate a near-field Talbot-Lau interferometer for C70 fullerene molecules. Such interferometers are particularly suitable for larger masses. Using three free-standing gold gratings of 1 mm period and a transversally incoherent but velocity-selected molecular beam, we achieve an interference fringe visibility of 40% with high count rate. Both the high visibility and its velocity dependence are in good agreement with a quantum simulation that takes into account the van der Waals interaction of the molecules with the gratings and are in striking contrast to a classical moiré model. DOI: 10.1103/PhysRevLett.88.100404 PACS numbers: 03.75.Dg, 03.65.Ta, 39.20.+q Interferometry with matter waves has become a large field of interest throughout the past years [1,2]. It represents a powerful tool for the demonstration of basic quantum phenomena and of matter-wave effects as well as for applications in high precision measurements inertial forces [3,4] and fundamental constants [5]. It is therefore interesting to extend techniques from neutron and atom interferometry to more massive, complex objects, such as large molecules, in order to quantitatively study decoherence by thermal coupling to the environment, to get information about a wide range of molecular properties and to work on novel lithographical or metrological applications [6]. In this Letter, we report the first demonstration of an interferometer for macromolecules, and in particular a Talbot-Lau interferometer for C70. The molecules are in different internal states with many excited vibrational and rotational degrees of freedom. Nevertheless we observe a clear signature for a quantized center-of-mass motion. Up to now two types of interferometers with molecules have been demonstrated, namely, a Ramsey-Bordé interferometer using I2 [7] and a mechanical Mach-Zehnder interferometer for Na2 [8]. The use of such devices for larger molecules remains, however, a challenge: A direct application of a Ramsey-Bordé interferometer for complex molecules is precluded by the lack of narrow resonant transitions. On the other hand, a simple extrapolation of the Mach-Zehnder interferometer to heavy, fast molecules would require extremely fine gratings, good collimation, or large distances. This is because such an interferometer operates in the Fraunhofer regime, where the characteristic size of a diffraction pattern scales linearly both with the wavelength and with the distance between the diffracting structure and the plane of observation. A solution to this problem is the Talbot-Lau interferometer, which also consists of three successive gratings but operates in the near-field or Fresnel regime, where the characteristic size of a diffraction pattern scales with the square root of both the wavelength and the distance. A Talbot-Lau interferometer can accept a spatially incoherent beam, which implies that no collimation is needed, and it works with a spatially extended detector. Therefore Talbot-Lau count rates can exceed those of Mach-Zehnder interferometers by several orders of magnitude. A Talbot-Lau interferometer has already been demonstrated for potassium atoms [9] and —in the time domain — for sodium atoms in a Bose-Einstein condensate [10]. Because the scaling properties of near-field devices are very favorable in comparison to far-field interferometers, the Talbot-Lau effect has actually been proposed for experiments using quantum objects up to the size of a virus [6,11]. The successful diffraction of fullerenes at both a mechanical [12] and an optical [13] grating has now stimulated a new set of experiments exploring the limits of macromolecule interferometry. The experiment presented here takes advantage of the Talbot effect which is a selfimaging phenomenon that occurs when a periodic structure is illuminated by a coherent beam [14–16]. Images of this grating are then reconstructed at discrete multiples of the Talbot length LT d2ldB, where d is the grating constant and ldB is the de Broglie wave length of the incident object. Our interferometer consists of three identical gratings which are equally spaced. The first grating acts as a comb of sources which have vanishing mutual coherence. Yet each source is sufficiently small to prepare the required transverse coherence at the second grating. The second grating is then coherently imaged onto the plane of the third grating by the Talbot effect. The images coming from all sources look identical but they are shifted by multiples of the grating constant and therefore overlap in spite of their lack of mutual coherence [16]. The movable third grating blocks or transmits the molecular interference fringes according to its position. The experimental setup, as sketched in Fig. 1, consists of three free-standing gold gratings (Heidenhain) with a nominal period of d 991.25 6 0.25 nm, an open fraction (ratio of slit width to period) of f 0.48 6 0.02, a thickness of b 500 nm, and a huge usable area with a diameter of 16 mm. The distance between the gratings is set to L1 L2 0.22 m. The whole setup is placed in a vacuum chamber at a pressure of 3 3 1028 mbar. A beam of C70 fullerene molecules is generated by sublimation 100404-1 0031-90070288(10)100404(4)$20.00 © 2002 The American Physical Society 100404-1
VOLUME 88.NUMBER 10 PHYSICAL REVIEW LETTERS 11 MARCH 2002 Figure 2 shows the detected signal from a single scan at nearly optimal settings for maximum contrast.Apart ion detector from some noise it corresponds well to a sine,as expected 1 laser beam from theory for our open fractions near 0.5,which suppress (onisation) grating 3 higher Fourier components.Fast-Fourier transformation grating 2 (FFT)gives a sharp single peak corresponding to the lattice period.A signal-to-noise ratio of 50:I was achieved in grating 1 lateral limiter gold just 150 s of total measuring time.It was not necessary to correct the extracted visibilities for dark counts because the source height limiter (v-selection) gratings 1 um period dark count rate was about 0.2 s-compared to count rates (oven) in the range from 50 to 450 s-l at central velocities of 80 and 160 m/s,respectively [17].The phase of the peak FFT FIG.1.Setup of the Talbot-Lau interferometer inside the vac- component gives the spatial position of the fringe pattern. uum chamber:It consists of three gratings which are inco- herently illuminated by the molecular beam.The interference Comparison of subsequent scans gives the lateral drift of fringes are detected by transversal scanning of the third grating the three-grating setup,which is of the order of 2 nm/min. and integral detection using an ionizing laser beam.The tilt of Tilting the three gratings with respect to each other di- each grating and the longitudinal position of the second grating minishes the visibility.Each grating was prealigned to are critical and may be adjusted with actuators.The height limi- several milliradians with respect to gravity by observing ter ensures that only the trajectories within a narrow velocity range,tunable by varying the vertical position of the oven,pass a laser diffraction pattern outside the vacuum chamber. from the oven to the detector. Fine adjustment of the grating parallelism-better than 2 mrad-was achieved by maximizing the fullerene fringe visibility.A much smaller variation of the visibility was in an oven at a temperature of 650C.Its velocity distri- observed when all three gratings were tilted by the same bution is close to that of an effusive source.with a most amount.This is because the calculated interferometer probable velocity of about 200 m/s.A laser beam with phase shift from gravitation is A-0.2 rad per mrad a power of 26 W traverses the apparatus behind the third of tilt (see below)and only its nonuniformity due to the grating in the horizontal direction.It is generated by a finite velocity distribution diminishes the visibility. multiline visible argon ion laser focused to a beam waist Longitudinal displacement of one grating also affects of 8 um(1/e2 radius).It ionizes the arriving molecules the visibility.Again,the requirement LI=L2 was pre- regardless of their transversal position.The interferome- aligned before evacuating the chamber and fine-aligned on ter signal is obtained by detecting the resulting ions and the 100 um level by maximizing the contrast.Vibration scanning the third grating transversally,using an actively isolation of the optical table by pneumatic feet proved to stabilized piezotranslation stage (Piezosystem Jena). be crucial for obtaining high visibility. The fullerene beam travels 2.38 m from the oven to the The appearance of a periodic signal as a function of the detection laser.In the horizontal direction,the beam is position of the third grating is not necessarily a sign of restricted only by the rectangular oven orifice of 1.2 mm quantum interference-it could also result from classical length and a 500 um wide slit that determines the used moire fringes,i.e.,shadow patterns which result from the segment of the first grating.This means that in comparison to the diffraction angles of about I urad,the illumination is transversally incoherent. In the vertical direction,the oven orifice of 200 um 5000 height and the laser beam fix two regions where the trajectories of the detected molecules are tightly confined 4000 in comparison to a typical free-fall distance during their flight.Halfway in between,the vertical positions of the detected molecules are therefore correlated with their .s3000 longitudinal velocities.We introduce an adjustable slit (Piezosystem Jena)that limits the beam height to 150 um 82000 or less at a distance of 1.38 m from the oven,0.12 m before the first grating.It selects a narrow velocity max-min 1000 visibility =38.5% distribution,which has been measured using a mechanical max+min chopper in front of the oven orifice.Its center may be varied between 80 and 215 m/s by adjusting the 55.5 5656.557 57.5 58 vertical oven position.This corresponds to de Broglie position third grating [um] wavelengths of 5.9 and 2.2 pm,respectively.The FWHM FIG.2. Interference fringes (raw data)resulting from a typical of the wavelength distribution goes up from 8%of the single scan of the third grating.A central velocity of 115 m/s mean to 35%with decreasing center wavelength. was selected. 100404-2 100404-2
VOLUME 88, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 11 MARCH 2002 source (oven) height limiter (v-selection) grating 1 grating 2 laser beam (ionisation) grating 3 gold gratings 1 m period } µ L1 L2 g z x lateral limiter ion detector FIG. 1. Setup of the Talbot-Lau interferometer inside the vacuum chamber: It consists of three gratings which are incoherently illuminated by the molecular beam. The interference fringes are detected by transversal scanning of the third grating and integral detection using an ionizing laser beam. The tilt of each grating and the longitudinal position of the second grating are critical and may be adjusted with actuators. The height limiter ensures that only the trajectories within a narrow velocity range, tunable by varying the vertical position of the oven, pass from the oven to the detector. in an oven at a temperature of 650 ±C. Its velocity distribution is close to that of an effusive source, with a most probable velocity of about 200 ms. A laser beam with a power of 26 W traverses the apparatus behind the third grating in the horizontal direction. It is generated by a multiline visible argon ion laser focused to a beam waist of 8 mm (1e2 radius). It ionizes the arriving molecules regardless of their transversal position. The interferometer signal is obtained by detecting the resulting ions and scanning the third grating transversally, using an actively stabilized piezotranslation stage (Piezosystem Jena). The fullerene beam travels 2.38 m from the oven to the detection laser. In the horizontal direction, the beam is restricted only by the rectangular oven orifice of 1.2 mm length and a 500 mm wide slit that determines the used segment of the first grating. This means that in comparison to the diffraction angles of about 1 mrad, the illumination is transversally incoherent. In the vertical direction, the oven orifice of 200 mm height and the laser beam fix two regions where the trajectories of the detected molecules are tightly confined in comparison to a typical free-fall distance during their flight. Halfway in between, the vertical positions of the detected molecules are therefore correlated with their longitudinal velocities. We introduce an adjustable slit (Piezosystem Jena) that limits the beam height to 150 mm or less at a distance of 1.38 m from the oven, 0.12 m before the first grating. It selects a narrow velocity distribution, which has been measured using a mechanical chopper in front of the oven orifice. Its center may be varied between 80 and 215 ms by adjusting the vertical oven position. This corresponds to de Broglie wavelengths of 5.9 and 2.2 pm, respectively. The FWHM of the wavelength distribution goes up from 8% of the mean to 35% with decreasing center wavelength. Figure 2 shows the detected signal from a single scan at nearly optimal settings for maximum contrast. Apart from some noise it corresponds well to a sine, as expected from theory for our open fractions near 0.5, which suppress higher Fourier components. Fast-Fourier transformation (FFT) gives a sharp single peak corresponding to the lattice period. A signal-to-noise ratio of 50:1 was achieved in just 150 s of total measuring time. It was not necessary to correct the extracted visibilities for dark counts because the dark count rate was about 0.2 s21 compared to count rates in the range from 50 to 450 s21 at central velocities of 80 and 160 ms, respectively [17]. The phase of the peak FFT component gives the spatial position of the fringe pattern. Comparison of subsequent scans gives the lateral drift of the three-grating setup, which is of the order of 2 nmmin. Tilting the three gratings with respect to each other diminishes the visibility. Each grating was prealigned to several milliradians with respect to gravity by observing a laser diffraction pattern outside the vacuum chamber. Fine adjustment of the grating parallelism—better than 2 mrad—was achieved by maximizing the fullerene fringe visibility. A much smaller variation of the visibility was observed when all three gratings were tilted by the same amount. This is because the calculated interferometer phase shift from gravitation is Dwg 0.2 rad per mrad of tilt (see below) and only its nonuniformity due to the finite velocity distribution diminishes the visibility. Longitudinal displacement of one grating also affects the visibility. Again, the requirement L1 L2 was prealigned before evacuating the chamber and fine-aligned on the 100 mm level by maximizing the contrast. Vibration isolation of the optical table by pneumatic feet proved to be crucial for obtaining high visibility. The appearance of a periodic signal as a function of the position of the third grating is not necessarily a sign of quantum interference—it could also result from classical moiré fringes, i.e., shadow patterns which result from the max min visibility 38.5 % max min - = = + FIG. 2. Interference fringes (raw data) resulting from a typical single scan of the third grating. A central velocity of 115 ms was selected. 100404-2 100404-2
VOLUME 88.NUMBER 10 PHYSICAL REVIEW LETTERS 11 MARCH 2002 geometry of straight rays passing the gratings or being The most likely reason for this discrepancy is that the blocked at one of them [18].These patterns do not depend ansatz of a purely absorptive grating transmission func- on the particle velocity,in sharp contrast to the behavior tion is not adequate:The outcome of far-field diffrac- of quantum de Broglie waves:Here the velocity gives the tion experiments with rare-gas atoms [19]is described wavelength,hence the Talbot length,and the ratio L/LT correctly only by a theory that includes the phase shift of the grating separation to the Talbot length is a crucial by the van der Waals interaction between the particles parameter in the theory of wave propagation. and the grating.To estimate the magnitude of this ef- The quantitative evaluation of the expected velocity fect for our experimental situation,we assumed a potential dependence involved numerical simulations of the three- V(r)=-C3r-3 for a molecule at a distance r 1 nm grating setup in a quantum de Broglie wave model,in from a gold surface.The constant C3 is known for various comparison with a classical point-particle calculation.The noble gases [20]near gold surfaces and depends roughly quantum model relies on Fresnel integrals and calculates linearly on their atomic dc polarizability.We extrapolated them efficiently by a Fourier transform method under to the known dc polarizability ade =97 A3 x 4TEo of the assumption of incoherent illumination.Assuming the fullerene molecule C7o [21]and obtained the rough es- binary transmission functions corresponding to our grat- timate C0.09 eVnm For the passage through ing geometry,the classical point-particle model yields a grating slit of width fd (=475 nm)centered atx =0, a velocity-independent fringe visibility of 5%,whereas we obtain the quantum model gives the dashed curve in Fig.3 with two maxima slightly below 30%in the experimentally V[x.()dr =-Csl(fd/2 -x) accessible velocity range.A relative minimum in be- V. tween corresponds to the velocity vr =107 m/s where +(fd/2+x)-3].(1) Lr =LI.In this simulation and the following ones, our measured velocity distributions have been taken into In the quantum model the potential of Eq.(1)leads to account. a phase variation of the de Broglie waves passing at dif- The experimental visibility curve(diamonds in Fig.3) ferent positions x through an individual slit of the second exhibits a pronounced velocity dependence and a maxi- grating.In the classical model we expect the potential mum visibility of 35%in the vicinity of the one- gradient to lead to a position-dependent force which ulti- Talbot-length criterion.On other days,up to 39%was mately makes an individual slit appear as a minilens with a obtained with the same settings,and 41%with a narrower velocity-dependent focal length.By a simple geometrical height limiter and therefore a narrower velocity distri- argument,one expects maximal visibility for f-L1, bution.This is totally incompatible with classical moire for our parameters at velocities about 1000 m/s.Indeed, fringes but corresponds also quite poorly to the described our classical simulation shows a single broad maximum quantum simulation result. of 18%height in this range.In the experimentally acces- sible velocity range,it predicts much lower visibilities than our experimental data(dash-dotted line in Fig.3).On the other hand,the measured functional dependence of fringe 50 visibility on velocity corresponds reasonably well to the exberiment quant.w.vdw numerical quantum model with van der Waals interaction % quant.w/o vdw included (solid line in Fig.3):The position of the single -·class.w.vdW maximum coincides perfectly,and the maximum experi- class.w/o vdw mental contrast attains nearly what is expected from theory. 30 Both the theoretical and the experimental curves show a single maximum at 115 m/s and are therefore asymmetri- cal with respect to the criterion LI Lr-a consequence of the complex transmission function of the combined ab- sorptive and phase grating.The remaining discrepancy is 10 not astonishing,having in mind the rather coarse approxi- mations underlying Eq.(1),such as the neglect of edge ef- fects,and the inevitable experimental imperfections. 80 90 107 120 140 180 240 Talbot-Lau interferometers allow various interferomet- v[m/s] ric measurements,especially those of inertial forces or of FIG.3.Dependence of the interference fringe visibility on the decoherence effects,which do not require spatially sepa- mean velocity of the molecular beam.Numerical simulation re- rated beams.As a first example,we have investigated the sults are plotted for four models without free parameters:clas- gravitational phase shift in our interferometer by tilting the sical or quantum behavior,with or without consideration of the van der Waals (vdW)interaction of the molecules with the sec- optical table with the whole experimental setup by a few ond grating.The quantum result including the van der Waals milliradians.As expected [18],the interference fringes are effect is clearly the only adequate one. shifted by a phase of Ag=2TLiga/(dv2)=0.2 rad 100404-3 100404-3
VOLUME 88, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 11 MARCH 2002 geometry of straight rays passing the gratings or being blocked at one of them [18]. These patterns do not depend on the particle velocity, in sharp contrast to the behavior of quantum de Broglie waves: Here the velocity gives the wavelength, hence the Talbot length, and the ratio L1LT of the grating separation to the Talbot length is a crucial parameter in the theory of wave propagation. The quantitative evaluation of the expected velocity dependence involved numerical simulations of the threegrating setup in a quantum de Broglie wave model, in comparison with a classical point-particle calculation. The quantum model relies on Fresnel integrals and calculates them efficiently by a Fourier transform method under the assumption of incoherent illumination. Assuming binary transmission functions corresponding to our grating geometry, the classical point-particle model yields a velocity-independent fringe visibility of 5%, whereas the quantum model gives the dashed curve in Fig. 3 with two maxima slightly below 30% in the experimentally accessible velocity range. A relative minimum in between corresponds to the velocity yT 107 ms where LT L1. In this simulation and the following ones, our measured velocity distributions have been taken into account. The experimental visibility curve (diamonds in Fig. 3) exhibits a pronounced velocity dependence and a maximum visibility of 35% in the vicinity of the oneTalbot-length criterion. On other days, up to 39% was obtained with the same settings, and 41% with a narrower height limiter and therefore a narrower velocity distribution. This is totally incompatible with classical moiré fringes but corresponds also quite poorly to the described quantum simulation result. 80 90 107 120 140 180 240 0 10 20 30 40 50 visibility [%] v [m/s] experiment quant. w. vdW quant. w/o vdW class. w. vdW class. w/o vdW FIG. 3. Dependence of the interference fringe visibility on the mean velocity of the molecular beam. Numerical simulation results are plotted for four models without free parameters: classical or quantum behavior, with or without consideration of the van der Waals (vdW) interaction of the molecules with the second grating. The quantum result including the van der Waals effect is clearly the only adequate one. The most likely reason for this discrepancy is that the ansatz of a purely absorptive grating transmission function is not adequate: The outcome of far-field diffraction experiments with rare-gas atoms [19] is described correctly only by a theory that includes the phase shift by the van der Waals interaction between the particles and the grating. To estimate the magnitude of this effect for our experimental situation, we assumed a potential Vr 2C3r23 for a molecule at a distance r * 1 nm from a gold surface. The constant C3 is known for various noble gases [20] near gold surfaces and depends roughly linearly on their atomic dc polarizability. We extrapolated to the known dc polarizability adc 97 Å3 3 4pe0 of the fullerene molecule C70 [21] and obtained the rough estimate CAu2C70 3 0.09 eV nm3. For the passage through a grating slit of width fd 475 nm centered at x 0, we obtain Z Vx, zt dt 2 b yz C3 fd2 2 x 23 1 fd2 1 x 23 . (1) In the quantum model the potential of Eq. (1) leads to a phase variation of the de Broglie waves passing at different positions x through an individual slit of the second grating. In the classical model we expect the potential gradient to lead to a position-dependent force which ultimately makes an individual slit appear as a minilens with a velocity-dependent focal length. By a simple geometrical argument, one expects maximal visibility for fy 2L1, for our parameters at velocities about 1000 ms. Indeed, our classical simulation shows a single broad maximum of 18% height in this range. In the experimentally accessible velocity range, it predicts much lower visibilities than our experimental data (dash-dotted line in Fig. 3). On the other hand, the measured functional dependence of fringe visibility on velocity corresponds reasonably well to the numerical quantum model with van der Waals interaction included (solid line in Fig. 3): The position of the single maximum coincides perfectly, and the maximum experimental contrast attains nearly what is expected from theory. Both the theoretical and the experimental curves show a single maximum at 115 ms and are therefore asymmetrical with respect to the criterion L1 LT—a consequence of the complex transmission function of the combined absorptive and phase grating. The remaining discrepancy is not astonishing, having in mind the rather coarse approximations underlying Eq. (1), such as the neglect of edge effects, and the inevitable experimental imperfections. Talbot-Lau interferometers allow various interferometric measurements, especially those of inertial forces or of decoherence effects, which do not require spatially separated beams. As a first example, we have investigated the gravitational phase shift in our interferometer by tilting the optical table with the whole experimental setup by a few milliradians. As expected [18], the interference fringes are shifted by a phase of Dwg 2pL2 1gady2 0.2 rad 100404-3 100404-3
VOLUME 88.NUMBER 10 PHYSICAL REVIEW LETTERS 11 MARCH 2002 per mrad of table inclination a.This means that the Y177,and by the Marie Curie Fellowship HPMF-CT- gravitational acceleration component in the interferome- 2000-00797 of the European Community (B.B.). ter plane,which is on the order of 10-3g,can be measured within a short integration time. In conclusion,we have demonstrated for the first time an interferometer for massive molecules that are internally *Email address:zeilinger-office@exp.univie.ac.at in a highly excited thermal state.The fact that the experi- www.quantum.univie.ac.at mental visibility is not significantly lower than the theo- [1]Special issue edited by A.Aspect and J.D.Dalibard,C.R retical expectation shows that decoherence by emission of Acad.Sci.Ser.IV 2.No.4 (2001). blackbody radiation does not play a significant role in this [2]Atom Interferometry,edited by P.R.Berman (Academic experiment.However,there is a chance to study decoher- Press,San Diego,1997). ence in future experiments involving laser heating of the [3]H.Rauch and S.A.Werner,Neutron Interferometry (Ox- ford University Press,New York,2000). fullerene molecules [4]T.L.Gustavson,P.Bouyer,and M.A.Kasevich,Phys.Rev. The successful implementation of this near-field inter- Let.78,2046(1997). ferometer paves the way towards interference of even more [5]D.S.Weiss,B.C.Young,and S.Chu,Phys.Rev.Lett.70, massive objects.Talbot-Lau interferometry has a favor- 2706(1993). able scaling behavior.If one leaves the longitudinal length [6]M.Arndt,O.Nairz.and A.Zeilinger.in Ouantum scale Lr =d2/A unchanged,a reduction of the grating /Un/Speakables,edited by R.Bertlmann and A.Zeilinger period by a factor of 4 already allows the observation of (Springer,New York,2002). quantum interference with 16 times shorter de Broglie [7]C.Borde,N.Courtier,F.D.Burck,A.Goncharov,and wavelength-that means 16 times heavier molecules,if M.Gorlicki,Phys.Lett.A 188,187(1994). one assumes a beam of equal velocity,or even 256 times [8]M.S.Chapman,C.R.Ekstrom,T.D.Hammond,R.A with effusive characteristics at the same temperature. Rubenstein,J.Schmiedmayer,S.Wehinger,and D.E. However,the present experiment teaches us that one has Pritchard,Phys.Rev.Lett.74,4783(1995). [9]J.F.Clauser and S.Li,Phys.Rev.A 49,R2213 (1994) to take into account the effect of the van der Waals interac- [10]L.Deng,E.W.Hagley,J.Denschlag,J.E.Simsarian, tion.With a decreasing grating period the influence of the M.Edwards,C.W.Clark,K.Helmerson,S.L.Rolston, r-3 potential increases strongly.In numerical simulations, and W.D.Phillips,Phys.Rev.Lett.83,5407(1999). the resulting contributions of high diffraction orders at [11]J.Clauser,in Experimental Metaphysics,edited by R.Co- the second grating restrict the self-imaging phenomenon hen et al.(Kluwer,Dordrecht,1997). and therefore the fringe visibility to extremely narrow [12]M.Arndt,O.Nairz,J.Voss-Andreae,C.Keller,G.V.der velocity ranges,e.g.,Av/v0.01 for m =16mc and Zouw,and A.Zeilinger,Nature (London)401,680(1999). d=dc/4,assuming both an unchanged velocity and an [13]O.Nairz,B.Brezger,M.Arndt,and A.Zeilinger,Phys. unchanged C3 constant for simplicity.For experimentally Rev.Lett.87,160401(2001). accessible velocity distributions.the inclusion of the [14]H.F.Talbot,Philos.Mag.9,401(1836). [15]M.S.Chapman,C.R.Ekstrom,T.D.Hammond, van der Waals interaction gives therefore a more strin- gent limit to interferometer scalability than previously J.Schmiedmayer,B.E.Tannian,S.Wehinger,and D.E.Pritchard,Phys.Rev.A 51,R14(1995). discussed by Schmiedmayer et al.[2]. [16]K.Patorski,Prog.Opt.27,1 (1989). For an experimental implementation,it seems therefore [17]These specific values refer to Fig.3. more promising to implement the second grating as an opti- [18]M.K.Oberthaler,S.Bernet,E.M.Rasel,J.Schmiedmayer, cal phase grating like the one we recently demonstrated for and A.Zeilinger,Phys.Rev.A 54,3165 (1996). fullerenes [13].Exploiting the position-dependent ioniza- [19]R.E.Grisenti,W.Schollkopf,J.P.Toennies,G.C. tion and fragmentation probability of molecules in a strong Hegerfeldt,and T.Kohler,Phys.Rev.Lett.83,1755 standing light wave,one could also replace the first and (1999). third gratings by laser light,thereby eliminating the fragile [20]G.Vidali,G.Ihm,H.Y.Kim,and M.W.Cole,Surf.Sci. free-standing microstructures [22]. Rep.12,133(1991). [21]M.S.Dresselhaus,G.Dresselhaus,and P.C.Eklund, We acknowledge help in the design of the experiment Science of Fullerenes and Carbon Nanotubes (Academic by Gerbrand van der Zouw and William Case.This work Press,San Diego,1998),2nd ed. has been supported by the European TMR network,Con- [22]Combinations of absorptive and phase gratings have been tract No.ERBFMRXCT960002,by the Austrian Science discussed for the quite different case of atoms by B.Du- Foundation(FWF),within the projects F1505 and START betsky and P.R.Berman [Phys.Rev.A 59,2269 (1999)]. 100404-4 100404-4
VOLUME 88, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 11 MARCH 2002 per mrad of table inclination a. This means that the gravitational acceleration component in the interferometer plane, which is on the order of 1023g, can be measured within a short integration time. In conclusion, we have demonstrated for the first time an interferometer for massive molecules that are internally in a highly excited thermal state. The fact that the experimental visibility is not significantly lower than the theoretical expectation shows that decoherence by emission of blackbody radiation does not play a significant role in this experiment. However, there is a chance to study decoherence in future experiments involving laser heating of the fullerene molecules. The successful implementation of this near-field interferometer paves the way towards interference of even more massive objects. Talbot-Lau interferometry has a favorable scaling behavior. If one leaves the longitudinal length scale LT d2l unchanged, a reduction of the grating period by a factor of 4 already allows the observation of quantum interference with 16 times shorter de Broglie wavelength—that means 16 times heavier molecules, if one assumes a beam of equal velocity, or even 256 times with effusive characteristics at the same temperature. However, the present experiment teaches us that one has to take into account the effect of the van der Waals interaction. With a decreasing grating period the influence of the r23 potential increases strongly. In numerical simulations, the resulting contributions of high diffraction orders at the second grating restrict the self-imaging phenomenon and therefore the fringe visibility to extremely narrow velocity ranges, e.g., Dyy 0.01 for m 16mC70 and d dC70 4, assuming both an unchanged velocity and an unchanged C3 constant for simplicity. For experimentally accessible velocity distributions, the inclusion of the van der Waals interaction gives therefore a more stringent limit to interferometer scalability than previously discussed by Schmiedmayer et al. [2]. For an experimental implementation, it seems therefore more promising to implement the second grating as an optical phase grating like the one we recently demonstrated for fullerenes [13]. Exploiting the position-dependent ionization and fragmentation probability of molecules in a strong standing light wave, one could also replace the first and third gratings by laser light, thereby eliminating the fragile free-standing microstructures [22]. We acknowledge help in the design of the experiment by Gerbrand van der Zouw and William Case. This work has been supported by the European TMR network, Contract No. ERBFMRXCT960002, by the Austrian Science Foundation (FWF), within the projects F1505 and START Y177, and by the Marie Curie Fellowship HPMF-CT- 2000-00797 of the European Community (B. B.). *Email address: zeilinger-office@exp.univie.ac.at www.quantum.univie.ac.at [1] Special issue edited by A. Aspect and J. D. Dalibard, C.R. Acad. Sci. Ser. IV 2, No. 4 (2001). [2] Atom Interferometry, edited by P. R. Berman (Academic Press, San Diego, 1997). [3] H. Rauch and S. A. Werner, Neutron Interferometry (Oxford University Press, New York, 2000). [4] T. L. Gustavson, P. Bouyer, and M. A. Kasevich, Phys. Rev. Lett. 78, 2046 (1997). [5] D. S. Weiss, B. C. Young, and S. Chu, Phys. Rev. Lett. 70, 2706 (1993). [6] M. Arndt, O. Nairz, and A. Zeilinger, in Quantum [Un]Speakables, edited by R. Bertlmann and A. Zeilinger (Springer, New York, 2002). [7] C. Bordé, N. Courtier, F. D. Burck, A. Goncharov, and M. Gorlicki, Phys. Lett. A 188, 187 (1994). [8] M. S. Chapman, C. R. Ekstrom, T. D. Hammond, R. A. Rubenstein, J. Schmiedmayer, S. Wehinger, and D. E. Pritchard, Phys. Rev. Lett. 74, 4783 (1995). [9] J. F. Clauser and S. Li, Phys. Rev. A 49, R2213 (1994). [10] L. Deng, E. W. Hagley, J. Denschlag, J. E. Simsarian, M. Edwards, C. W. Clark, K. Helmerson, S. L. Rolston, and W. D. Phillips, Phys. Rev. Lett. 83, 5407 (1999). [11] J. Clauser, in Experimental Metaphysics, edited by R. Cohen et al. (Kluwer, Dordrecht, 1997). [12] M. Arndt, O. Nairz, J. Voss-Andreae, C. Keller, G. V. der Zouw, and A. Zeilinger, Nature (London) 401, 680 (1999). [13] O. Nairz, B. Brezger, M. Arndt, and A. Zeilinger, Phys. Rev. Lett. 87, 160 401 (2001). [14] H. F. Talbot, Philos. Mag. 9, 401 (1836). [15] M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, Phys. Rev. A 51, R14 (1995). [16] K. Patorski, Prog. Opt. 27, 1 (1989). [17] These specific values refer to Fig. 3. [18] M. K. Oberthaler, S. Bernet, E. M. Rasel, J. Schmiedmayer, and A. Zeilinger, Phys. Rev. A 54, 3165 (1996). [19] R. E. Grisenti, W. Schöllkopf, J. P. Toennies, G. C. Hegerfeldt, and T. Köhler, Phys. Rev. Lett. 83, 1755 (1999). [20] G. Vidali, G. Ihm, H. Y. Kim, and M. W. Cole, Surf. Sci. Rep. 12, 133 (1991). [21] M. S. Dresselhaus, G. Dresselhaus, and P. C. Eklund, Science of Fullerenes and Carbon Nanotubes (Academic Press, San Diego, 1998), 2nd ed. [22] Combinations of absorptive and phase gratings have been discussed for the quite different case of atoms by B. Dubetsky and P. R. Berman [Phys. Rev. A 59, 2269 (1999)]. 100404-4 100404-4