Strong Coupling of Light with Collective Terahertz Vibrations in Organic Materials

We demonstrate for the first time strong coupling between a terahertz cavity and collective, intermolecular vibrations in organic crystals. Beyond observing the Rabi splitting, we directly measure the vacuum Rabi oscillations using time-domain THz spectroscopy. © 2019 The Author(s)

When light is compressed into a region comparable to its wavelength, its interaction with matter can overcome all the incoherent and dissipative processes, which profoundly changes its nature. In this regime, known as strong coupling, the wavefunctions of the photons and the material excitations are coherently mixed to form cavity polaritons, which are hybrid light-matter quantum states 1 . This fascinating phenomenon has been observed in many different types of material systems, such as cold atoms [2][3][4] , excitons in semiconductors 5, 6 , electronic spins in nitrogen-vacancy centers 7 , phonons in inorganic crystals [8][9][10][11][12] and many others. Among these, strong coupling with organic molecules 13,14 has seen an ever-increasing interest in recent years, both in conventional Fabry-Perot microcavity systems as well as in plasmonic structures 15 . Interestingly, the creation of the polaritonic wavefunctions under strong coupling and the modification of the energetic landscape of the molecules can have a significant influence on the physical and chemical properties of the molecules 1,16,17 , affecting the rates and yields of chemical reactions [18][19][20][21][22][23] , their emission properties [24][25][26] , electronic and excitonic transport [27][28][29][30][31] and more. This new field, known as polaritonic chemistry, is currently under intense study, both experimentally and theoretically. While traditionally organic strongly-coupled systems involved the coupling of an optical resonance to electronic transitions in molecules (Frenkel excitons), recently, vibrational strong coupling has been introduced as a new paradigm [32][33][34][35][36][37] . In such systems, a particular intramolecular, optically-active vibrational transition is coupled to a mid-infrared resonator, creating hybrid excitations termed "vibro-polaritons". As has been demonstrated over the past few years, the creation of such vibro-polaritons allows the manipulation of molecular processes occurring at the electronic ground-state, by targeting a specific bond in the molecules 19,38,39 .
Here we demonstrate, for the first time, strong coupling of collective vibrations in ensembles of organic α-lactose molecules, occurring at THz frequencies (10 11 -10 13 Hz). Unlike the previously studied vibrational strong coupling, here the cavity mode is coupled to inter-molecular vibrations in molecular crystallites, i.e. oscillatory motion of the hydrogen-bonded molecules with respect to one another. Interestingly, we observe the Rabi-splitting typical of strong coupling and coherent Rabioscillations at room temperature, despite the fact that the energy of the collective vibrational transition (ℏߥ ௩~2 meV), as well as the light-matter interaction strength are much lower than k B T (~25 meV).

3
Our results extend the applicability of polaritonic chemistry to other organic large-scale systems, such as biological macromolecules 40 , polymer chains 41 and energetic materials with low lying collective vibrations 42 .
Lactose, which is found in milk, is a disaccharide composed of galactose and glucose. In this study we use α-lactose monohydrate, which is one of the anomers formed upon the crystallization of lactose, with the chemical structure shown in Fig. 1a. The α-lactose powder used in this study (Sigma-Aldrich) is comprised of small, polycrystalline particles, a few tens of microns in size, as shown in Fig. 1b. To measure its THz absorption spectrum, we prepared a ~1.3 mm-thick pellet of pristine αlactose using a pressing die (see Methods Section), and measured its absorption spectrum using terahertz time-domain spectroscopy (THz-TDS). The result is presented in Fig. 1c, showing a sharp absorption peak at 0.53 THz (17.7 cm -1 ) and a width of 21 GHz FWHM. This absorption line corresponds to a collective, intermolecular vibration in the molecular crystal, in which the molecules move with respect to each other as a rigid body [43][44][45] . An additional weaker absorption peak is observed within our usable THz bandwidth, at 1.2 THz. In order to demonstrate strong coupling of the collective vibrational mode at 0.53 THz, we utilized the open-cavity geometry 46 depicted in Fig. 1d (see Method Section for further details). The cavity is composed of two Au mirrors prepared by sputtering ~6 nm Au layers on 1 mm-thick quartz substrates. The reflection amplitude of the mirrors was measured to be ~90% for the THz field (81% reflectivity). In the open cavity geometry, one of  the mirrors is fixed, while the other is mounted on a computer-controlled translation stage, parallel to the fixed mirror, such that the cavity length d can be varied continuously.
The measurements were performed using a home-built, time-domain terahertz spectrometer, shown in the schematic diagram in Fig. 2a. In a typical measurement, an ultrashort laser pulse (100fs pulse duration, 800nm central wavelength) from a Ti:Sapphire chirped pulse amplifier (Legend Duo, Coherent Inc.) is split to form a strong optical beam for THz generation and a weak readout pulse for time-resolved electro-optic sampling of the THz field 47,48 . A single-cycle THz pulse is generated via tilted pulse-front optical rectification in LiNbO 3 (LN) 49 and focused through the sample (S), which is placed at the focal plane of a 4-f setup composed of two off-axis parabolic reflectors. The THz field and the readout pulse are combined by a pellicle beam-splitter (PBS) and focused at the electro-optic detection crystal (Gallium Phosphate, GaP), following which the probe beam is analyzed for its    shown in Fig. 3b. As can be seen, the resonant Fabry-Pérot cavity modes are clearly visible, with their frequencies obeying the relation ݂ = ଶௗ ݉, where c is the speed of light, d is the cavity length (the distance between the mirrors) and m is the mode number (assigned in Fig. 3b). Specifically, for a cavity length of 640 µm, for which the second-order mode is close to the α-lactose absorption line, the resonant modes have a transmission peak of 0.5-1%, and a linewidth of 14 GHz, matching the calculated Finesse for mirrors with reflectivity of 81%. In addition, we performed T-matrix calculations for the 640 µm cavity to simulate the spectral response of the cavity (solid green line in Fig. 3b), which agree with the experimental measurement.
Next, we examined the response of the cavity with the α-lactose pellet placed between the mirrors. We prepared an α-lactose pellet of 250 µm in thickness, attached it to the fixed mirror and adjusted the total cavity length to ~350 µm. Under such conditions, the effective optical length of the cavity (given by ݀ ௧ = ݀ ఈ ݊ ఈ + ݀ with ݀ ఈ being the pellet thickness, ݊ ఈ =1.8 the background refractive index of α-lactose 52 and ݀ ~1 00 µm is the thickness of the air-gap) is ~550 µm, such that the second-order cavity mode is resonant with the collective vibrational mode of the α-lactose at 0.53 THz. The time-resolved THz field exiting the cavity is presented in Fig. 4a. We observe a similar exponentially-decaying oscillation, as for the empty cavities, but here the signal is modulated by a periodic envelope. This periodic modulation corresponds to Rabi-oscillations in the cavity, signifying the strong coupling between the collective vibrations of the α-lactose crystallites and the cavity. The transmission spectrum of the α-lactose cavity, obtained using the Fourier transform of the signal in Fig. 4a, is shown in Fig. 4b (blue solid line). Furthermore, by fitting the results to T-matrix calculations (using the experimentally-measured refractive index of α-lactose 52 ), shown by the green solid line, we obtain a thickness of ݀ ఈ =250 µm for the α-lactose pellet and ݀ =97 µm for the air gap thickness. As can be seen in both the experimental measurement and the simulations, the hybrid cavity/α-lactose system exhibits a clear splitting in the spectral response around the collective vibration frequency and the formation of two THz vibro-polariton states, at 0.50 and 0.56 THz, indicating once again that the hybrid system is indeed within the strong coupling regime. In addition 7 to the polaritonic modes, the first (m=1) and third (m=3) order cavity modes at 0.25 and 0.83 THz are also located within the bandwidth of the input pulse. Therefore, the single-cycle pulse excites a coherent superposition of the polaritonic modes as well as the non-coupled cavity modes, which gives rise to the seemingly-irregular dynamics seen in the time domain (Fig. 4a). To illustrate this, we numerically filter the time-domain signal by a band-pass filter, leaving only frequencies within the range of 0.38-0.68 THz. The filtered time-domain signal is presented in Fig. 4c. As seen, the Rabioscillations of the coupled system are clearly observed, demonstrating the reversible and coherent light-matter interaction taking place in the system. Interestingly, Rabi oscillations occurring under strong coupling of molecular excitons and plasmonic structures were previously observed by probing 8 the excited state population, using ultrafast pump-probe spectroscopy 53 . However, here we are able to observe the Rabi-oscillations in the emitted field directly, including its oscillating phase.
Next, we varied the position of the moveable mirror, repeated the measurement and calculated the spectral response as shown in Fig. 4b, for several different cavity lengths. In Fig. 4d we plot the resulting transmission spectra (blue lines), as well as the simulated T-matrix results (green lines). In our simulations, we fix all the parameters except for the thickness of the air-gap between the pellet surface and the moveable mirror, which is extracted by fitting the simulated transmission to the experimental data. Using these simulations, we can extract the second-order cavity resonance for each value of the cavity thickness, by removing the contribution of the vibrational resonance to the refractive index of the α-lactose, only taking into account its background index of refraction. Finally, we use these results to plot the dispersion of the hybrid molecular/cavity system, i.e. the measured vibro-polariton frequencies as a function of the cavity resonance frequency. As seen in Fig. 5, the dispersion shows the formation of the characteristic polariton branches around the absorption frequency of the α-lactose collective vibration. We fit these measurements to the dispersion resulting from the coupled-oscillator model, given by where Ω ୖ is the (angular) Rabi frequency, ߥ ௩ = 0.53 THz is the collective vibration frequency, ߥ is the cavity frequency (of the second-order mode), and Δߥ ௩ =21 GHz and Δߥ =14 GHz are their linewidths. By fitting Equation (1) to the measured data, we obtain a Rabi frequency value of Ω ோ 2ߨ ⁄ = 68 GHz. This value is higher than the decay rates of both the cavity and the uncoupled vibration, confirming that our system is indeed within the coherent, strong coupling regime.
Moreover, this value is about 13% of the bare vibration frequency, placing this system close to the ultrastrong coupling regime.

Discussion
We have demonstrated the strong-coupling of the collective vibration of α-lactose crystallites and a provide an extended test-bed for studying the very basic underlying physics of the strong-coupling phenomenon. Furthermore, the relatively large (tens to hundreds of µm) length-scale of the THz cavity makes it accessible to additional stimulations, such as optical excitations, that may alter the molecular structure, as well as to structural patterning of the sample to manipulate the light-matter interaction within cavity.

Methods
Open microcavity configuration. The variable-length open cavity used in this work (see Fig. 2) is composed of a moveable mirror (CM1) and a fixed mirror (CM2). Both mirrors were produced by sputtering a thin layer (6 nm) of gold on a quartz substrate (1 mm thickness), resulting in a transmission amplitude of 90% across the whole usable bandwidth with no apparent spectral dependence. CM1 is mounted on a computer-controlled single-axis stage, with micrometer resolution (<2μm repeatability). By moving CM1 with respect to CM2 we control the length of the cavity and corresponding resonance frequency. CM1 and CM2 are set parallel to each other by coinciding the multiple reflections of a green diode laser from the mirrors at the far field. The α-lactose sample (white, round pellet) was prepared by placing 0.1 g of α-Lactose powder in a pressing die (20mm diameter) at a pressure of 220 kN for 15minutes, which yielded a ~250μm pellet. The pellet was then glued onto CM2 at a few points around its circumference.
Transfer matrix calculations. The simulated transmission spectra of the cavity were calculated using the T-matrix formalism 56 . In these simulations, we used the experimentally-measured refractive index of gold 57 to model the cavity mirrors and adjusted the thickness of the mirrors to match the measured reflectivity of 81%. We note that the fitted thickness of the mirrors was found to be 1.5 nm, which is lower than the actual thickness of 6 nm. This is most probably due to the fact that at such low thicknesses the sputtered metal film is not continuous, but rather composed of small Au islands. The dielectric function of the α-lactose pellet within our usable THz range can be accurately described by