Absolute absorption cross sections from photon recoil in a matter-wave interferometer

We measure the absolute absorption cross section of molecules using a matter-wave interferometer. A nanostructured density distribution is imprinted onto a dilute molecular beam through quantum interference. As the beam crosses the light field of a probe laser some molecules will absorb a single photon. These absorption events impart a momentum recoil which shifts the position of the molecule relative to the unperturbed beam. Averaging over the shifted and unshifted components within the beam leads to a reduction of the fringe visibility, enabling the absolute absorption cross section to be extracted with high accuracy. This technique is independent of the molecular density, it is minimally invasive and successfully eliminates all problems related to photon-cycling, state-mixing, photo-bleaching, photo-induced heating, fragmentation and ionization. It can therefore be extended to a wide variety of neutral molecules, clusters and nanoparticles.

Optical spectroscopy, essential in the early development of quantum theory, is now a ubiquitous tool in the natural sciences [1,2]. In many cases it is desirable to access the electronic, vibrational or rotational properties of free molecules. Gas-phase spectroscopy has therefore been established in various systems, ranging from gas cells to free molecular beams or ion traps. Absorption measurements in vapor cells [3] yield relative spectral information -i.e. the positions of absorption lines. However, this requires the analyte particles to be sufficiently volatile and to form a gas of sizable opacity. For many complex molecules gas phase data are lacking because the number densities are too weak. This holds in particular for non-fluorescent molecules. Gas phase spectra are therefore often derived from solvent analysis [4]. In addition, measuring absolute values for absorption cross sections conventionally requires accurate knowledge of the vapor pressure. This is notoriously difficult to determine for complex molecules.
Combining absorption spectroscopy with light-induced fluorescence offers a higher signal-to-noise, and therefore sensitivity. This method has been refined to matrixisolated samples down to the level of single molecules [5,6]. However, in free molecular beams the number of photons scattered by complex molecules is intrinsically limited. The fluorescent light is typically red-shifted with respect to the absorption wavelength, with the energy difference remaining in the molecules. Frequent repetition of the absorption cycle rapidly leads to heating, photobleaching or destruction of the particles. This problem can be alleviated by keeping the analyte particles in a dilute buffer gas, which serves as a heat bath. Although this has been successfully exploited with isolated trapped biomolecular ions [7,8] it remains challenging for neutral molecules.
Here we present a method to measure absolute absorption cross sections which circumvents repeated photoncycling and which can be applied to extremely dilute beams. We exploit photon recoil in a Kapitza-Dirac-Talbot-Lau (KDTL) matter-wave interferometer [9] to measure the reduction in quantum interference contrast as a function of the position and intensity of a recoil laser. From this we extract the absorption cross section σ abs (λ k ) at the laser wavelength λ k [10].
The delocalization of complex molecules [11,12] has been studied in a variety of experimental arrangements, from far-field diffraction [13,14] to near-field interferometry [9,15,16]. The KDTL interferometer has enabled observation of quantum interference with the most massive nanoparticles to date, exceeding 10 4 AMU [17]. The interference fringes in the KDTL interferometer are free flying periodic nanostructures formed by the molecular number density distribution. If this nanostructure is exposed to a uniform external force the internal particle properties determine the resulting shift of the interference pattern. This method has enabled measurements of electric polarizabilities [18], susceptibilities and dipole moments [19], thermally induced conformational dynamics [20] as well as the distinction of structural isomers [21]. Here we consider the case of photon recoils associated with the Poissonian statistics of a molecule either absorbing, or not absorbing, a photon. In particular where internal conversion dissipates the photon energy and suppresses any secondary emission. Instead of shifting the interference pattern this has the effect of reducing the observed interference contrast. Photon-induced decoherence, caused by the random emission of spontaneous or thermal photons has previously been studied in atom [22,23] and molecule [24] interferometry, respectively. Photon absorption in Ramsey-Bordé interferometry [25] has also previously been used to determine the fine structure constant [26] and molecular data. Here we make use of fringe averaging in KDTL interferometry to derive precise absorption spectra, as suggested in reference [10].
A schematic of the experimental setup is shown in Fig Absolute absorption spectroscopy in the KDTL matter-wave interferometer. a) Grating G1 produces a spatially coherent source of molecules from the thermal beam emitted by an oven. After a distance D the molecules encounter the recoil laser where the probability of absorbing n photons, of wavelength λ k , is described by a Poissonian distribution, Pn. At a distance L − D further the molecules diffract at the standing light wave G2. The resulting interference pattern is read out by translating grating G3 laterally before a quadrupole mass spectrometer. b) The effect of some molecules absorbing a photon from the recoil laser results in a reduction of the observed visibility V = A/µ → V = A /µ when averaged over the molecular velocity distribution. Here A (and A ) are the unperturbed (perturbed) amplitude and µ is the mean of the near-sinusoidal quantum interference curve.
temperature of 654 • C and fly through three gratings, G1, G2 and G3 before detection in a quadrupole mass spectrometer. Gratings G1 and G3 are etched into SiN with a period d = 266 nm and an opening fraction of 0.42. Grating G2 is a standing light wave formed by a laser with wavelength λ g = 532 nm and waists w x 20 µm and w y 500 µm. Adjacent gratings are spaced by a distance L 10.5 cm [9]. The collimation and velocity selection of the molecular beam is provided by a series of slits. This results in a beam height of about 200 µm and a divergence less than 1 mrad. The velocity distribution, measured using a time-of-flight technique, is approximated by with a mean velocity v 0 = 210.3(7) ms −1 and standard deviation σ vx = 38.4(5) ms −1 . Here quoted errors are the 1σ uncertainty estimates. The molecular beam is initially in-coherent. However, the openings of G1 represent a comb of slits which act as narrow and therefore spatially coherent sources of transmitted wavelets [11]. The molecules interact with G2 through the optical dipole force. The spatially periodic variation of the laser intensity imprints a spatially varying phase onto the transmitted matter wave, which evolves into a modulation of the molecular density distribution. A Quadrupole Mass Spectrometer (QMS) then counts the number of molecules transmitted by G3. The molecular density pattern arriving at G3 has a near-sinusoidal shape. This is revealed in the QMS count rate when G3 is laterally translated as illustrated in Fig. 1b. The visibility of this curve V = A/µ is determined by the fringe mean µ and amplitude A. Decoherence, dephasing or phase averaging in the interferometer reduce the fringe visibility V. Under controlled conditions this reduction can be used to extract detailed information about the interaction between the molecules and their environment.
Consider a particle which absorbs n optical photons from a recoil laser whose k-vector is parallel to the grating vector. The laser is positioned at a distance D from G1, where D < L. If a molecule absorbs a photon it receives a momentum recoil ∆p = h/λ k . In the plane of G3 this recoil shifts the position of the absorbing molecule relative to the unperturbed interference pattern by a distance s = ∆pD/mv x = λ dB D/λ k . The probability of a molecule absorbing n photons while crossing the recoil laser is described by the Poissonian distribution P n (n 0 ) = n n 0 exp (−n 0 )/n! where is the mean number of absorbed photons. A relative dephasing within the total ensemble occurs because not all molecules acquire the same momentum recoil. For a given molecular velocity v x this results in a reduced interference contrast V = RV where [10] R = exp (−n 0 [1 − cos (2πs/d)]). However, for a realistic molecular beam with a finite spread in velocities the observed reduction in contrast must be averaged over the velocity distribution. This results in the relation, V = R vx V, where the reduction in visibility is now described by [10] The recoil laser is superposed with the molecular beam by translating it along theŷ-axis. The reduction in fringe visibility is then well described by a Gaussian with a waist w k = 1.23(2) mm. In order to measure σ abs we record V /V for a variety of different laser positions between 3.5 < D < 5.5 cm. The recoil laser is fixed at a power of P k = 17.4(2) W and a wavelength of λ k = 532.2 nm, with a drift on the order of 10 MHz over the course of our measurements. Figure 2 shows how the observed reduction in fringe visibility varies with D. From a fit of equation (2) to this data we determine the absorption cross section to be σ abs (532 nm) = 1.97(6) × 10 −21 m 2 , without any other free parameter. The quoted uncertainty in σ abs is the 1σ estimate derived from the variation of the χ 2 for different values of σ abs . The systematic uncertainty is dominated by knowledge of the recoil laser power and waist, which for these experiments enters at the level of 4 × 10 −23 m 2 . In these experiments, the average number of absorbed photons is n 0 0.14.
The interaction potential at G2 depends on the optical polarizability of the molecules. Although they enter the interferometer in the electronic ground state S 1 a small fraction can be excited to the triplet state T 1 by the recoil laser. If some remain in T 1 at G2 this will be seen as a change in V /V as the light grating power varies. For a distance of (L−D) 5 cm and a velocitiy of 210 ms −1 the molecules take approximately 240 µs to reach G2 after the recoil laser. This is significantly longer than the expected triplet lifetime of C 70 in the gas phase which has been measured to be τ 41 µs [27]. Figure  3 shows the results of experiments where we measured V /V for light grating powers ranging from 1 − 6.5 W. The recoil laser here is positioned 4.85 cm from G1 to en-sure a reasonable reduction in contrast while maintaining a good signal-to-noise ratio at low light grating powers. We see no significant change in V /V and conclude that the excited triplet state plays no significant role in our absorption cross section measurements. A faster method to measure the absorption cross section can be employed with similar precision and only slightly reduced accuracy. For a fixed velocity P (v x ) = δ(v x − v 0 ) the first minimum in the reduction of contrast occurs at D (1) min = dmv 0 λ k /2h and the absorption cross section is described by Close to D = D (1) min the effect of finite velocity spread is small for all values of v 0 . The data point highlighted as a red triangle in Fig. 2 is the nearest to this minimum we could achieve in our current setup, with a position D = 3.5 cm. If we ignore the effects of velocity spread and use equation (4) to determine the absorption cross section we find a value σ abs = 1.7 × 10 −21 m 2 , which underestimates the value determined from our detailed analysis by approximately 14%. Although this type of measurement is subject to systematic shifts towards lower absorption cross sections the effect is small compared to the change in absorption cross section occurring close to molecular resonances which can span decades. A further simplification can be made for molecules with a very low mean velocity or for short recoil laser wavelengths. The spatial period of V /V is 2D (1) min = dmv 0 λ k /h for a monochromatic beam, varying only slightly when realistic velocity distributions are considered. When D is very large compared to 2D (1) min the reduction factor is described by R vx exp (−n 0 ) and the sensitivity to D is removed.
In summary, we have demonstrated a precise and accurate method to measure the absolute absorption cross section of molecules in dilute beams from quantitative observations of the fringe visibility as a function of the intensity and position of a probe laser. Our experimental result for C 70 at 532 nm is in excellent agreement with previous [3] and more implicit [28,29] measurements. We realise an absolute uncertainty of 3% which can be used to anchor relative spectra to our fixed frequency value at 532 nm with the same absolute accuracy. Further improvements are conceivable, especially for beams with a narrower velocity spread. This method can be applied to many molecular species and is of particular interest to extremely dilute beams whose vapor pressures cannot be known a priori. This applies especially to neutral biomolecular beams which could be studied as isolated particles or embedded in nanosolvent environments. One may also consider multiphoton experiments with higher time resolution for various interferometer configurations [11,16]. Absorption spectroscopy through photon recoil in matter-wave interferometry will be important in future research at the interface between quantum optics, physical chemistry and biomolecular physics.
We acknowledge valuable discussions with S. Nimmrichter and K. Hornberger. We are grateful for financial support from the FWF through projects Z149-N16 and DK CoQuS W1210-2, the European commission through project ERC AdvG PROBIOTIQUS(320694) and the Vienna ZIT communication project (957475). JPC is supported by a VCQ fellowship.