Raman gas self-organizing into deep nano-trap lattice

Trapping or cooling molecules has rallied a long-standing effort for its impact in exploring new frontiers in physics and in finding new phase of matter for quantum technologies. Here we demonstrate a system for light-trapping molecules and stimulated Raman scattering based on optically self-nanostructured molecular hydrogen in hollow-core photonic crystal fibre. A lattice is formed by a periodic and ultra-deep potential caused by a spatially modulated Raman saturation, where Raman-active molecules are strongly localized in a one-dimensional array of nanometre-wide sections. Only these trapped molecules participate in stimulated Raman scattering, generating high-power forward and backward Stokes continuous-wave laser radiation in the Lamb–Dicke regime with sub-Doppler emission spectrum. The spectrum exhibits a central line with a sub-recoil linewidth as low as ∼14 kHz, more than five orders of magnitude narrower than conventional-Raman pressure-broadened linewidth, and sidebands comprising Mollow triplet, motional sidebands and four-wave mixing.


Supplementary Figure 7.
Evolution with the product of pump input power and pressure of the linewidth of forward Stokes (blue points) and backward Stokes (red points). The solid lines are a fit for eye-guidance for the linewidth evolution of the forward Stokes (blue line) and backward Stokes (red line). Here the pump power and pressure range was limited to the case where the 2 nd Stokes is not generated. Experimental set-up to measure the longitudinal-motion sidebands. The forward Stokes beam is diffracted using a diffractive grating with a groove spacing of 833 nm, and is recorded using a photodetector placed at a distance L=11 m from the diffractive grating. The input pump power was ~20 W, and the gas pressure: 15 bar. Supplementary note 1: Experimental set-up for optical and RF spectral measurements.

Supplementary
The 20 m long fibre used here is a home-made photonic bandgap (PBG) guiding HC-PCF and fabricated using the stack and draw technique. The fibre is filled with molecular hydrogen at a controllable pressure, by placing the two fibre-ends in gas cells. The gas pressure is kept uniform along the whole length of fibre by monitoring it with pressure gages placed at both cells. The gas cells are equipped with AR coated windows at both sides to avoid laser back reflection. The fibre has a core radius of r = 3.2 m (see top of The hydrogen-filled fibre is pumped with a randomly polarized 1061 nm wavelength Yb-fibre CW laser that could emit up to a maximum of 100 W of optical power, with a linewidth of only ~400 kHz. The optical spectrum from both fibre-ends is monitored using an optical spectrum analyser to record the FS and BS spectra. Furthermore, the experimental set-up also comprises a portion with a delayed self-heterodyne interferometer to measure the linewidth of both forward and backward propagating beam spectral components. The self-heterodyne system consists of a delay arm made of a 6 km long optical singlemode fibre (SMF) at wavelength longer that 1 µm, and a short modulation arm comprising an acousto-optic modulator (AOM) operating at ~211 MHz. The beat signal between the delayed optical beam and the AOM-frequency down-shifted signal is detected using a fast photo-detector (~1 GHz bandwidth) and recorded using an RF spectrum analyser (Rohde&Schwarz FSEA 30). The RF spectrum analyser resolution bandwidth is set at 10 KHz when the span bandwidth is ~150 MHz, and 1 kHz for a narrower span of less than 6 MHz. The RF spectral traces of the pump, FS and BS are then recorded for different pump laser powers and gas pressures over the two above mentioned bandwidth span.

Supplementary note 2: Pump linewidth spectrum
The linewidth of the transmitted residual pump has been monitored for different input pump powers and gas pressures. Supplementary figure 1 shows that the pump linewidth remains unchanged with input power increase.

Supplementary note 3: The theoretical model
We consider the propagation of continuous-wave pump and first order Stokes radiation in the fundamental mode of a photonic bandgap guiding photonic crystal fibre, neglecting the excitation of higher-order Stoke and anti-Stokes lines, as well as the energy transfer to higher-order transverse modes of the fibre.
The propagation characteristics of Stokes and pump, such as the wavenumbers , as well as losses  s ,  p , were calculated using the JCMwave finite-element Maxwell solver with high precision, using the tabulated data of the fused silica refractive index and the transverse cross-section of the fibre determined by the microscopy.
To derive the propagation equations, we first consider the steady-state values of the density matrix , (1) where the Rabi frequencies  11 ,  22 and  12 are defined by Here a s , b s , a p , b p , d s being constants related to dipole moments. The quantities  12 and  12 are the population decay rate and the Raman gain linewidth, correspondingly.
The Stokes component in the considered case consists of the forward-propagating part E S (f) and , which satisfy two distinct propagation equations. The values  12 and D allow calculating the polarizations P SF , P SB and P P as where N is the concentration of the molecules.
The propagation equations then are written as where is the parameter which accounts for the microscopic spatial distribution of the gain. In the propagation equations, we have ignored the change of the refractive index which arises from microscopic density modulation, as detailed below. The analytical expression for is cumbersome and is not given here.
The origin of the backward Stokes component is the reflection from the input and output fibre interfaces back into the fibre due to mismatch of the effective refractive index of the fibre mode and of the free space, imperfections at the fibre ends, grating formed by the modulation of particle density, and fibre roughness, with a reflection coefficient estimated by a total value of 1% in energy.
Therefore, the boundary conditions for the Stokes field are where r is the energy reflection coefficient and L is the fibre length. We note that the model doesn't take into account the reflection from the molecular lattice index modulation. Although neglecting the reflection off the Stokes optical lattice will have an effect on the exact magnitude of FS and BS, it doesn't significantly impact the results reported here.
In our model, we consider the frequency difference between the pump and scattered Stokes to be equal to the Raman transition frequency, and the initial population is in the ground state (i.e. ). and is inversely proportional with the fibre length in the delay arm of the unbalanced interferometer. Here c is the speed of light, n is the fibre core material index and L fib is the fibre length. A large fraction of the measured linewidths with was found to be lower than , which raises questions on the accuracy or validity of the measured linewidth for the values below 34.5 kHz. To address this question, we first recall that this rule is only valid for a white frequency noise where the phase noise is assumed to be a zero-mean stationary random Gaussian process, which is not necessary the case of our experiment, where the Stokes radiations are emitted in a Lamb-Dicke regime. For a non-white phase noise structure, the above limit condition doesn't necessarily hold. Furthermore, reported works 1,2 show that linewidth as low as 6 kHz could be measured with a fibre length of only 2 km long by extracting the spectral lineshape from the phase spectral noise (PSD). We compared this PSD technique of line-shape extraction to our nonlinear fit by reconstructing the line-shape of the beat signal from its temporal trace (recorded by an oscilloscope with 2 ms integration time) and comparing it to the nonlinear fitted RF spectrum. We found that the two techniques give similar results within less than 10% of relative discrepancy. In order to further ensure the validity of our linewidth measurements, for values less than 34.5 kHz, we have estimated our fittings for spectral data points with frequency offset from the centre in the range of 50 to 200 kHz to avoid the interferometer effect on the linewidth. We found that all the experimental data at the RF frequency range exhibits a strong signal to noise ratio and fit extremely well with the multiple peak Lorentzian function. A final test on the validity of the linewidth measured was done by proceeding with linewidth measurements with different delays, and we found that the linewidth values are independent with delay variation within 20%.

Supplementary Note 5: FS and BS linewidth spectrum over a broader spectral span
Here, we re-examine the linewidth traces and their evolution with input power and gas pressure, but Furthermore, we observe that FS and BS exhibit a difference in the frequency of their sidebands in the range of sideband frequency of 1.2-2 MHz due to the overall motion of the molecular lattice. The second family of sidebands are located in the range of ~7-8 MHz, which is roughly half that of the Rabi sideband frequency. Moreover, higher order Rabi sidebands are also observed. We attribute this effect to four wave mixing (FWM) between Stokes central peak and the two sidebands and with a frequency given by ( ) ( )=( ). The 2 nd order Rabi sideband signal will be determined by the nonlinear susceptibility at its frequency ( ) via ( ) ( ) ( ) ( ).

Supplementary Note 6: Longitudinal-motion sidebands
The model predicts the existence of longitudinal-motion sidebands corresponding to the trapped molecules oscillation with the quasi-harmonic oscillator along the z-direction. The expression of the fundamental frequency of these oscillations is Furthermore, the trace shows asymmetry in the peaks heights, which shows the quantum nature of the motion. Further work and more detailed measurements with appropriate instruments are needed to analyse the spectral structure of these sidebands. The second set-up to corroborate the longitudinal sideband consists of sending the FS to an optical spectrum analyser (OSA) set at its highest sensitivity and resolution (10 GHz). We note, however, that when the OSA is set on high sensitivity, the real resolution of our OSA is probably below 10 GHz based on our previous experience with this tool. Supplementary figure 13 shows the typical spectrum with its multiple peaks fit from several recorded spectra. Similarly with the first set-up results, the spectrum shows three distinct peaks. Here, the frequency spacing is equal between the peaks and found to ~15 GHz, in qualitative agreement with the above results despite the insufficient resolution of the OSA.

Supplementary Note 7: Power scaling of ultra-narrow linewidth Stokes
This Supplementary note demonstrates the fibre power coupling handling and the extremely high quantum conversion to the first order Stokes.
Supplementary figure 14 shows the evolution of FS and BS with input power for a different fibre length from the one considered in the main manuscript. Here the fibre length was set to 7 m. With this length we have demonstrated a coupling with input power as high as 85.5 W. At this input power, the FS power was found to be 55 W, and BS power 3 W. With the estimated fibre coupling efficiency of 75%, we find ~97% of quantum efficiency.
Furthermore, in the main manuscript, the ultra-narrow linewidth obtained with the 20 m long fibre were limited to input powers less than 30 W. Above this input power level the generation of the second order Stokes strongly alters the SONS-GPM molecular lattice. This section is shown as a proof of concept that the lattice and hence the narrow linewidth can be obtained for higher input powers by simply shortening the fibre length and reducing the gas pressure. This will increase the input power onset at which the generation of the second-order Stokes occurs.
Supplementary figure 15 shows the linewidth traces of FS and BS generated from a 7 meter long PBG HC-PCF. The fibre is similar to the one used so far. The linewidth measurements were taken with input power up to 70 W, and the gas pressure was set to 10 bar and 20 bar. We obtained with a pressure of 10 bar, a generated and transmitted Stokes with a power level in the range of 30-50 W and a linewidth of ~100 kHz.

Supplementary Note 8: Effective Rabi frequency definition and dependence with input power
The effective two-photon Rabi frequency acts only in the nano-traps. Consequently, its expression deviates from , and its magnitude should be averaged over the wavelength, with the weight given by D. This gives the following expression:

Supplementary Note 10: The influence of the particles redistribution on the dynamics
As explained in the manuscript, the change of the gas density due to the modulated expectation value of the Hamiltonian leads to the significant, above fourfold, increase of the density in the lentils. This higher density will lead to two major effects: firstly, the population decay rate and the coherence decay rate are going to be modified. Secondly, the higher density will result in a higher collision rate of the molecules. Both of these effects, in turn, lead to the modification of the expectation value of the Hamiltonian: the former one directly as described by the formalism shown in the paper, the second one indirectly, by modifying the time molecules dwelling in any given position under the influence of the field. Therefore the distribution of the molecules over the position, velocities, and the quantum state described by the  12 and D should be calculated self-consistently, including the above effects. This calculation is not included in the current simplified version of theory; however, even such a simplified version gives quantitative agreement to the experimental values.