Manipulating nonlinear exciton polaritons in an atomically-thin semiconductor with artificial potential landscapes

Exciton polaritons in atomically thin transition-metal dichalcogenide microcavities provide a versatile platform for advancing optoelectronic devices and studying the interacting Bosonic physics at ambient conditions. Rationally engineering the favorable properties of polaritons is critically required for the rapidly growing research. Here, we demonstrate the manipulation of nonlinear polaritons with the lithographically defined potential landscapes in monolayer WS2 microcavities. The discretization of photoluminescence dispersions and spatially confined patterns indicate the deterministic on-site localization of polaritons by the artificial mesa cavities. Varying the trapping sizes, the polariton-reservoir interaction strength is enhanced by about six times through managing the polariton–exciton spatial overlap. Meanwhile, the coherence of trapped polaritons is significantly improved due to the spectral narrowing and tailored in a picosecond range. Therefore, our work not only offers a convenient approach to manipulating the nonlinearity and coherence of polaritons but also opens up possibilities for exploring many-body phenomena and developing novel polaritonic devices based on 2D materials.


S2 Coupled oscillator model and simulation of trapped polariton state in a potential
Exciton polariton can be formed when the interaction between semiconductor excitons and cavity photons is faster than their individual decay rates.For a monolayer WS2 in a planar cavity, the two hybridized polariton branches, i.e., the LPB and UPB can be described by coupled oscillator model 1 : Where / / k represents the in-plane wavevector,   is the energy of the exciton resonance (cavity mode), is the detuning defined as    , and  is the Rabi splitting.The cavity mode can be expressed: , where C m is the effective mass of the bare cavity, which is typically on the order of 10 -4 exc m 1 .With the known Hopfield coefficients C and X in the polariton branch, the effective mass and dispersion of LPB and UPB polaritons can be expressed as: To simulate the discrete polariton state in pillar microcavity shown in Fig. 1f, we calculate the dispersion relation via solving Schrodinger's equation 2 for polaritons in cylindrical mesa structures: where the trapping of polaritons is achieved by confining their photonic part C  rather than their excitonic part X  .The confinement potential   V r has the same profile as the cylindrical mesa structure with a finite depth of 183 meV (determined by atomic force microscope image shown in Fig. 1c).The time-independent eigenstate and eigenvalue can be obtained by separating the time-dependent evolution terms.And then we use Finite Difference Method (FDM) to calculate the matrix in real space assuming the Boltzmann distribution of polaritons 3 .

A. Obtaining the diffusion length and exciton density in the mesa cavity
To experimentally determine the exciton diffusion length in our system, we measure the imaging of the excitation spot (Fig. S3a) and the steady-state photoluminance (Fig. S3b) for the heterostructure hBN/WS2 on the bottom DBR, which can be considered as the convolution between the laser's gaussian profile and the diffusion-length-dependent Bessel function K0, that is, where n(r) is exciton distribution, Lx is the exciton diffusion length, and w is related to the excitation profile (w = 486 nm).As shown in Fig. S3c, the PL linewidth is broader than that of the laser due to the exciton diffusion, and the diffusion length is fitted to be 0.472 m, coincident with the values reported in previous literature 5,6 .
Those results indicate that the excitons almost localize in the mesa region, and the spatial distribution or effective density of excitons keeps unchanged for different trap dimensions.So the exciton distribution area is all the same in different mesa cavities.For calculating the exciton-polaritons interaction strength in the trapping mesa cavities, we need to estimate the exciton density under the excitation of a 532 nm CW laser.The effective 2D exciton density eff x n is represented by 7 : where pump P is the time-averaged pump power in the microcavity after considering the transmission coefficient of the cavity.S is the exciton distribution area and  represents the absorption of WS2 at 532 nm (   3.3% pump E   ) 8 .To determine the exciton lifetime in our system, we performed the time-resolved photoluminescence measurement for the heterostructure hBN/WS2 on the bottom DBR with PMMA coated.With a 532 nm laser (pulse width of 6 ps with a repetition rate of 5 MHz), the exciton density is about 1.25×10 3 m -2 and the exciton lifetime is deduced to be 233 ps ( Fig. S4).To calculate the exciton distribution area, we used the width corresponding to the e -2 of the PL image profile as the exciton diameter.Then the S is calculated to be 1.41 m 2 after considering the exciton diffusion.Our calculation method results in effective 2D exciton densities from 10 2 to 10 3 m -2 , far below the Mott density of the monolayer TMD 9 .

B. The dependence of nonlinear interaction constant of polariton on trapping size by resonant reflectivity measurement
In this section, we measured the nonlinear response of the artificial mesa cavities by resonant excitation to further reveal the controllable nonlinearity by confining polarions.To improve the depth of the reflection dips for the resonant measurement, we redesigned the mesa cavity with reduced pairs of SiO2/TiO2 for both the top and bottom DBR and utilized a femtosecond laser with a duration of about 145 fs and a repetition rate of 5 kHz to resonantly pump the system 10 .Fig. S5a shows the blueshift from low pump fluence to high pump fluence where we resonantly excite the 5.5 m pillar microcavity.To quantitatively analyse the polariton-polariton interaction, the polariton density has been calculated by the following equation 10,11 : as the orange solid squares in Fig. S5b, showing a linear increase with the polariton density.To further reveal the nonlinear response to the trapping size, the blueshifts of the peak position in different mesa sizes with similar detuning were also plotted for comparison, indicating that the smaller the mesa cavity is, the more the peak shifts.
The polariton-polariton interaction, that is, the slope of the polariton-density-dependent energy shift, is extracted and plotted in Fig. S5c and fitted by an inversely quadratic function (red line), which is in agreement with the theoretical expectation 12 .Additionally, compared with a planar microcavity with similar detuning (Fig. S6), the value of polariton-polariton interaction of the mesa cavity shows an increase of nearly an order of magnitude.These results undoubtedly demonstate that confining polaritons can enhance the polariton-polariton interaction strength 13 .

S5 The theory about manipulating nonlinear interaction strength
In a mesa cavity, the polariton-exciton interaction Hamiltonian is given by: is the field operator of the cavity photons and is the field operator of the excitons.In the quantum description, we can express the Hamiltonian | ( ) | ( ) We can now identify that the blueshift of a quantized mode (n-th mode) is given by 2 2 | ( ) | ( ) When polariton-reservoir interaction dominates the spectral shift under nonresonant excitation ( ), we find that the effective polariton-exciton interaction strength for s-state polariton is given by: Note that the exciton-exciton interaction strength x x g  is unaffected by the trap size, since the trap size in our study is several micrometers, much larger than the excitonic Bohr radius but comparable with the polaritonic spatial extension.So, in the presence of confinement, although the exciton density is independent of the trap sizes, the spatial overlap between the exciton and polariton is changed in different-size traps, which is mainly due to the photonic component of polaritons confined by the mesas and thus modifies the polariton-exciton interaction.In the calculation, the   ( Fig. S7b), then we can achieve the manipulation of polariton-reservoir interaction through tuning the trap sizes.It is important to note that the polariton-reservoir interaction in essence is from the exciton-exciton interaction, among which one exciton strongly couples with cavity photon and the other exciton is from the reservoir 4,14-16 .To compare the macroscopic coherence of polaritons with different linewidth, we use the model about atom lasers with interactions proposed in Ref. 17 Through this model, the first-order correlation function can be expressed as: where u represents polariton interaction and  is the decay rate of ground-state polariton.c n is related to the pump strength and s n is the gain saturation number.
For both cases, the last term represents the Schawlow-Towns formula which can describe the coherent characterization of the photon laser 18 .The middle term reveals the particle-pairs interaction leading to the dephasing of the polariton state.So when the coherence time is larger than the polariton lifetime, the first-order correlation function approaches an exponential form (sample 1).Otherwise, the first-order correlation function evolves to Gaussian form (sample 2).
S9 Linewidth of polariton spectrum at // 0 k  for different trap size

Fig
Fig. S1 (a) Angle-resolved reflection map of the bare cavity mode measured on the planar microcavity, showing a blueshift compared with that in hBN/WS2 area.(b) Extracted spectrum at // 0 k  to calculate the Q factor.Through Lorentzian fitting,

S3
Fig. S2 (a) PL intensity and (b) linewidth of polaritons at // 0 k  as a function of pump power for the mesa size of 3 m.They both linearly increase with pump powers, indicating that the excitation is far below saturation to avoid bleaching of exciton-photon interaction 4 .

Fig. S3 (
Fig. S3 (a) The real-space image of the excitation spot.(b) The PL image for the heterostructure hBN/WS2 on the bottom DBR substrate.(c) Excitation and PL intensity profiles (dots) extracted from (a) and (b), and their corresponding fitting curves (lines).

Fig
Fig. S4 Time-resolved photoluminescence (dark curve) fitted with an exponential decay model.

Fig
Fig. S5 (a) Polariton spectra at / / 0 k  for different excitation powers in 5.5 m mesa cavity, showing the blueshift from low pump fluence to high pump fluence.(b) Blueshift of s-state polariton as a function of polariton density for varied trap sizes from 3 m to 6.5 m.(c) The estimated polariton-polariton interaction strength pp g versus trap size.
of the s-state polariton by fitting the reflection spectra.The P and f is the average power and repetition rate of the laser, beam area.By fitting the reflection dips of the polariton with the Lorentzian function, we obtain the polariton energies plotted

Fig. S6
Fig. S6 Measurement of the polariton-polariton interaction in the planar cavity.(a) Reflection spectra of s-state polariton at k// = 0 for different pumping densities.(b)The spectra dip energies extracted from (a) as a function of polariton density.The

where
operators of the cavity photons and excitons respectively.r are the single-particle wave functions of the cavity photon and exciton, respectively.Under a mean-field approximation, we replace the exciton number operator Ntot is the total number of excitons which relates to the exciton density as/ x tot N N S , and S is the exciton distribution area.Then 2 2 be represented by the spatial distributions of the simulated photonic mode and photoluminance, as shown in Fig.S7a.As a result, the p x g  is dependent on the trap diameter d with a relationship of

Fig 1 0
Fig. S7 (a) The spatial distribution of the experimental photoluminance and the simulated s-state polariton in different traps.(b) The integral value of spatial overlap as a function of cylindrical radius.The black dot is the calculated result and the red curve is the fitting result with a function of d -0.73 .

Fig. S10
Fig. S10 Power-dependent coherence time of s-state polaritons in a 3 m trap in another mesa cavity (not the studied one in Figure 4) at the detuning of -69 meV.

Fig. S13
Fig. S13 Linewidth of s-state polaritons at // 0 k  as a function of trap size for the