Enabling multiple intercavity polariton coherences by adding quantum confinement to cavity molecular polaritons

Significance A new microcavity infrastructure, with lateral confinements, was designed and built for molecular vibrational polaritons, which are hybrid half-light, half-matter quasiparticles. The newly implemented photonic confinement to the Fabry-Perot cavity created additional “quantized” cavity modes and enabled the formation of a polaritonic multi-qubit systems, also called qudits. This new photonic structure enabled multiple coherences that were robust against environmental fluctuations, addressing a bottleneck in applying molecular polaritons for quantum technology. This work not only served as an important step for developing molecular vibrational polaritons for quantum information technology, such as simulating light harvesting complex, but also had significant implications in realizing topological polariton systems and quantum light spectroscopy for molecular systems.


Figure S1 (A) The optical microscope image of the Ge/ZnO DBR window that composes the confined cavity. (B) experimental and (C) simulated linear spectra at different IR beam positions. The relation between linear spectra and corresponding beam positions is indicated by the color in (A).
The photon mode could be described by the wave equation of electric field (− ∥ 2 ∥ 2 + ⊥ 2 ( d( ) ) 2 ) ( ) = 2 ( ) .d(r) is the cavity thickness at position r considering the lowest cavity confinement in z-dimension (surface normal). ∥ and ⊥ are the effective velocity in in-plane and perpendicular directions. Then it could be discretized on a 1x400 grid as ( ∥ 2 2 (2 , − , +̂− , −̂) + ⊥ 2 ( ) 2 , ) = 2 and solved by diagonalization with an open boundary condition. The eigenvalues are cavity modes and eigenvectors are corresponding electric field distributions. The IR beam is represented by a Gaussian of which the center corresponds to the IR focused position and its imaginary part corresponding to the angle of the IR beam. The peak intensity is calculated by the convolution of cavity mode electric field spatial distribution and beam profile. By changing the center of the beam (Fig.S1A), we can simulate the linear spectra of a confined cavity (Fig.S1C) which reproduced most features from the experimental ones in Fig.S1B. Thus, we confirmed that the doublet peaks that decrease as the IR beam shift outward the trench correspond to the modes in the confined area, while the single peak at higher frequency originates from the cavity outside the trenched area. From the simulation, we observed three more peaks at higher frequencies at 2015cm -1 , 2041cm -1 and 2071cm -1 , which were missing in the experimental data. These peaks were not resolved due to that they were too small and hindered by the background.
Notably, the P mode is missing in both experimental and simulated spectra. The P mode should be between S and D modes. P mode is not a bright mode in our experiment because that P mode is an odd function in space (Fig.S2), and when convoluted with the laser beam mode, the net signal is neglectable. However, when the beam is off-centered, with a smaller beam diameter (~20 µm, 60% smaller compared to experiment condition), and with a high cavity Q value, P modes is resolved (yellow trace in Fig.S2b). The requirements are too strenuous for the current experimental condition, which makes P and other odd modes invisible in our experiment.
To simulate the dispersion curve, we varied the imaginary components of the Gaussian profile, which reflected the plane wave component that is parallel to the cavity as the beam incidence angle changed.  The separations between confined cavity modes are functions of media refractive index and the lateral dimension. Specifically, from equation (1), the separation between S and D modes is ∆ =

2.Experimental data for more confined cavities
. In different media with same lateral dimension (Fig. S3A), the modes separation ratio is is refractive index of the media inside the cavity. To examine this relationship, we compare the cavity peak separation when the air and hexane were hosted in the cavity. The refractive index of Hexane is 1.375 and air is 1, respectively. This leads to the peak separation ratio to be 1.9 between modes in air and hexane. Experimentally, the peaks separation is 13cm -1 in Hexane and 25cm -1 in the air (Fig. S3A), agreeing with the model prediction well.
The peak separation between modes in same media is where ∥ is the lateral dimension. We tested this relationship by measuring cavity spectra of 100, 50 and 25 µm. At 100 µm (Fig.S3B), only one peak is observed and therefore, the cavity modes did not experience lateral confinement. Two modes started to be resolvable at lateral size of 50 µm. Based on a Gaussian peak fitting, the peak separation was 7cm -1 . In contrast, the peaks were well resolved with a lateral size of 25 µm, with a separation of 27 cm -1 . Using the relationship at the beginning of this paragraph, the ratio of peak separations between 25 and 50 µm cavities should be ~4, agreeing with the experimental determined value 27 7 well. We note that this measurement was done using the cavity confined in both lateral dimensions. However, because the in-plane momentum of laser beam was parallel to one of the dimensions, it only probes confinements in one dimension. Thus, the 2D confined cavity followed the same trend and the 1D confined cavities, evidenced by that their cavity mode peak separation remained similar experimentally.

Modeling of the cavity thickness dispersion curve with 4x4 Hamiltonian matrix
From the experiment result Fig.3C, we know there should be 4 polariton modes corresponding to 4x4 matrix in the following form: Since the polariton modes were formed in solvent hexane, from Fig.S3A we knew in the matrix d-s=13 cm -1 (Fig. S3A). In this way, we could solve the coupling strength b1 and b2 at different cavity thickness with the polariton modes position from Fig.3C. We found that b1=b2=21cm -1 could optimize the modeled dispersion curve to match with the experimental ones. The modeled dispersion curve was plotted in Fig.3C.

Figure S4 Scheme of two-dimensional infrared experimental setup.
Two-dimensional infrared (2D IR) spectroscopy 1 was applied to characterize the confined cavity polariton system. The setup scheme is shown in Fig. S3. 800-nm laser pulses (~35 fs, ~5 W, 1 kHz) generated by an ultrafast Ti:Sapphire regenerative amplifier (Astrella, Coherent) were sent into an optical parametric amplifier (OPA) (TOPAS, LightConversion) which output tunable frequency near-IR pulses. The near-IR pulses were converted to mid-IR pulses through a difference frequency generation (DFG) process by a type II AgGaS2 crystal (Eksma). After DFG, a CaF2 wedge split the mid-IR pulse into two parts: the 95% power transmitted part was sent into a Ge-Acoustic Optical Modulator based mid-IR pulse shaper (QuickShape, PhaseTech) and was shaped to double pulses with tunable temporal separation t1, which formed the pump beam arm; the 5% reflected was the probe beam arm. Both pump (~ 1.1 μJ) and probe (~ 0.2 μJ) were focused by a lens onto the sample.
The pulse sequence is shown in Fig. 3A. Two pump pulses and a probe pulse (pulse duration of 100~150 fs) interacted with samples at delayed times (t1, t2, and t3). After the first IR pulse, a polaritonic coherence was generated, which was converted into a subsequent population or coherence state by the second IR pulse and was characterized by scanning t1 (0 to 8000 fs with 32 fs steps) using the mid-IR pulse shaper. A rotating frame at f0 = 1716.3 cm-1 was applied to shift the oscillation period to make the scanning step meet the Nyquist frequency requirement. After waiting for t2, the third IR pulse (probe) impinged on the sample, and the resulting macroscopic polarization emitted an IR signal. The MCT detector (PhaseTech) experimentally Fourier transformed the signal, thus generating a spectrum along the ω3 axis. Numerical Fourier transform of the signal along the t1 axis was required to obtain the spectrum along ω1. The resulting 2D IR spectra were plotted against ω1 and ω3. The t2 time delay was scanned by a computerized delay stage, which was controlled by LabVIEW programs to characterize the dynamic features of the system. A rotational stage was mounted on the sample stage to choose the IR incidence angle and, therefore, the in-plane wavevector of the driven polaritons.

5.Tailoring pump pulse by pulse shaper
In our 2D IR experiments, the pulse shaper was used to turn the incoming transform-limited pulse into a double-pulse with controllable time separation, phase in the time domain, and spectral shape in the frequency domain. Specifically, we set the central frequency and bandwidth of the first and second pulses separately, so that we could control the polariton states being created after each light-matter interaction. For example, we can set the first pulse only centered at UPp (without exciting other states), and the second pulse to be centered at LPS, to create a specific coherence state. This method allowed us to create specific coherence without being interfered or overwhelmed by other large signals. Figure S5 Comparison between the 2DIR spectra of dual cavity polariton system(A) and confined cavity polariton system (B). Comparing to the 2DIR of dual cavity system, the intercavity cross peaks are much more well resolved in confined cavity system (labeled by dash square) which imply that more quantum pathways contributed to those peaks.

Figure S6 Energy diagram of Rabi oscillation coherence between |UPD〉〈LPS |(A), |UPD〉〈LPD| (C) and |UPA〉〈UPB| (E) corresponding experimental coherence signal along t2, (B) for |UPD〉〈LPS| (D) for |UPD〉〈LPD| and (F) |UPA〉〈UPB| in frequency domain and time domain respectively.
We tailored the first two IR pulses of 2D IR using pulse shaper to create targeted coherences and scanned t2 to examine whether any arbitrary coherence could be created. Besides the coherence we showed in Fig.4, we also tailored the first pump pulse to be centered at and the second pulse centered at and individually, to initiate the coherence | 〉〈 | and | 〉〈 | (Fig.S5). Similarly, both coherences were resolved well in frequency and time domain which could support that intercavity coherence was robust in confined cavity polariton system. We note that there were other coherence signals at lower frequencies, which did not match the frequency of the original coherences. Such a signal could be due to coherence transfer, which was out of the scope of this work. We also included coherence | 〉〈 | from previous work 2 .