## Abstract

Periodic temporal modulation of Hamiltonians can induce geometrical and topological phenomena in the dynamics of quantum states. Using the interference between two lasers, we demonstrate an off-resonant optical lattice for a polariton condensate with controllable potential depths and nearest-neighbour coupling strength. Temporal modulation is introduced via a gigahertz frequency detuning between pump lasers, creating a polariton ‘conveyor belt’. The breaking of time-reversal symmetry causes band structures to become non-reciprocal and acquire a universal tilt given by Planck’s constant and the frequency of modulation (*h*Δ*f*). The non-reciprocal tilting is connected to the non-trivial topology of the Floquet–Bloch bands, which have a finite Chern number. Detailed characterization of the lattice potential depth and its dynamics highlights the role of high-energy carriers in the formation of optical potential landscapes for polaritons, demonstrating the possibility of modulation faster than the polariton lifetime and opening a pathway towards Floquet engineering in polariton condensates.

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## Main

The adiabatic modulation of a quantum Hamiltonian around a closed loop in parameter space causes eigenstates to acquire a geometric phase—the Berry phase^{1}. For periodically driven systems, the additional accumulation of geometric phase can lead to the formation of novel topological invariants^{2} and induce topologically protected transport and edge modes^{3,4,5,6}. Combined with a spatially periodic potential, eigenstates can be defined on a Floquet–Brillouin zone^{7,8}, and topological winding around this zone can cause non-reciprocal transport and has been observed in a wide range of spatio-temporally modulated physical systems, both classical and quantum^{9,10,11,12,13}. Non-reciprocity also emerges in the shearing and tilting of the momentum-space band structure, which is proportional to the Chern number of the Floquet–Bloch bands^{13,14,15,16}. In this Article, the formation of non-reciprocal topological band structures is demonstrated in an exciton–polariton condensate.

Exciton–polaritons are mixed light–matter quasiparticles that emerge from the strong coupling of semiconductor microcavity photons and quantum well excitons^{17}. Under strong non-resonant pumping, there is a phase transition from a thermal state to a quantum state with long-range spatial and temporal coherence—a polariton condensate^{18}. Lithographic methods for engineering the polariton potential landscape have been commonly exploited to emulate the physics of many-body two-dimensional systems using polariton condensates^{19,20,21,22}. While there have been proposals on how to achieve non-reciprocity with such static potentials^{23,24,25}, experimental demonstrations have so far been absent. An alternative to lithographic approaches for creating static potentials uses spatially patterned non-resonant optical fields. Drawing from the unique light–matter properties of polaritons, a carrier distribution proportional to the optical field interacts nonlinearly with the condensate. The created potential has both real^{26} and imaginary (non-Hermitian)^{27} components that have been used to create optical traps^{28,29}, to control the hopping between individual sites^{30,31,32} and to create non-Hermitian lattices^{33,34}.

While optical fields are typically patterned via spatial phase or intensity modulation of laser wavefronts, this work uses the angled interference between two lasers to create a spatio-temporally periodic laser intensity pattern—an optical conveyor belt (Fig. 1a). We experimentally demonstrate the formation of Floquet–Bloch bands and connect the emergence of non-reciprocity in a moving conveyor belt to the Berry phase acquired through the temporal modulation and the corresponding Chern number. We additionally quantify the limit on the speed at which optical potentials can be used to modulate polariton condensates, for example, to create Floquet topological lattices^{35}, to rotate polariton condensates^{36,37} or to control recently proposed polariton qubits^{38}.

## Results

Using one laser to create the interference pattern, the spatially periodic carrier distribution can be directly resolved in real space (Fig. 1b). Below threshold (top panel), the polariton ground state has a periodically modulated energy, with the highest energy at the interference maxima. Above threshold (bottom panel), there is strong emission from the positions of maximum gain and a decrease in the background occupation, allowing us to resolve the quantized, trapped states occurring at the intensity minima where the potential energy is minimum. To achieve frequency modulation, two different lasers were used, which washes out the spatial structure over experimental imaging time scales, even for Δ*f* ≈ 0 (Supplementary Section II). However, the polariton emission in momentum space still shows the formation of quantized states (Fig. 1c). While the exact band in which a condensate(s) forms will depend on the periodicity of the lattice^{29,39} and the stability of the interference pattern, the band formation itself is robust and can be controlled through the angle between the lasers (Δ*k*_{laser}) and the relative power of the angled beam (*P*_{m}). Shallow (small *P*_{m}) and wide (small Δ*k*_{laser}) fringes lead to more closely separated bands, and vice versa (Fig. 1c), intuitively corresponding to a potential proportional to the fringe pattern. A quantitative demonstration that the periodicity of the laser interference is transferred to the polariton wavefunction is given by the formation of Brillouin zones (BZs) in the dispersion images. The distance between emission peaks of the first excited state (*p* band) falls within the expected second BZ in the extended scheme (Fig. 1d, blue-grey), and the distance between energy maxima of the ground state (*s* band) corresponds to the expected edge of the first BZ of a lattice with period *a* of 2π/Δ*k*_{laser} (Fig. 1d, purple-magenta). The presence of ground-state curvature additionally proves that hopping between adjacent sites is non-negligible and that interfering laser beams can provide a simpler alternative to typical wavefront shaping for the simulation of spatially periodic Hamiltonians in microcavities^{40}.

The measured energy mode structure was compared with an exciton–photon mean-field model, where gain and polariton nonlinearity are neglected for simplicity and where the nonlinear interactions with the pump-generated carriers^{28} are reduced to an effective potential *V*_{eff} for the exciton field *χ*, expressed as

where *ψ* is the photon field, *m*_{eff} is the photon effective mass, *δ* is the exciton–photon detuning, *V*_{shift} is the average energy of excitons and photons and 2ℏ*Ω*_{R} is the photon–exciton Rabi splitting. The exciton mass can be safely neglected as it is much larger (by five orders of magnitude) than that of the photon. The Bloch bands of equation (1) were calculated numerically, and a least-squares fitting to the experimental splitting between *s* and *p* bands (Fig. 1c, thin light-grey lines) was used to estimate *V*_{eff}, after which the *s* band energy was used to estimate *V*_{shift}. The results of this fit are superimposed on Fig. 1c (dashed black), displaying good agreement in the band energies and momentum structures of the model and the dispersion spectra.

When |Δ*f*| > 0, the laser interference pattern that creates the lattice potential moves along the surface of the microcavity. The direction of movement of this conveyor belt is controlled by the signs of ∆*f* and ∆*k*_{laser}. For positive (negative) ∆*f*, the conveyor belt moves in the same (opposite) direction as ∆*k*_{laser}. Bands are still present in the far-field emission of the microcavity, but in response to the lattice motion they become tilted with respect to the momentum axis (Fig. 2a). These bands are non-reciprocal in that the energy of a Bloch wave will depend on its direction of motion, with higher-energy emission coming from waves moving in the same direction of motion as the conveyor belt, which can be intuitively understood as the Doppler effect. This non-reciprocity is quantified by measuring the tilt of the bands, extracted from the energy difference between the forward and backward edges of the BZ, that is, Δ*E* = *E*(π⁄*a*) – *E*(−π⁄*a*) (Fig. 2a, inset), which is linearly proportional to ∆*f*. The constant of proportionality is universal and independent of ∆*k*_{laser} and *P*_{m} and corresponds to Planck’s constant (Fig. 2b).

This universality can be understood by considering the space–time Floquet–Brillouin zone where band tilting arises from topological winding around the 1 + 1D unit cell^{8}. It is known that the tilting energy in classical waves is connected to the Chern number *C*_{n} of the band Δ*E* = *C*_{n}*h*Δ*f* (ref. ^{41}), but the argument is generally applicable to any wave in a spatio-temporally periodic medium.

This can be proven by considering the quasi-energy of the Floquet state in terms of the instantaneous normalized eigenfunctions *ψ*_{k}(*t*) and energies *ϵ*_{k}(*t*) (ref. ^{42}). In the case of adiabatic evolution, it can be expressed as

where the first term is the dynamical phase acquired from eigenstate evolution and the second is the Berry phase acquired from the specific path taken in parameter space during the adiabatic evolution. The difference between the *n*th band quasi-energies at the two BZ edges can be written as the winding of the Berry phase around the Floquet–Brillouin zone thus

which is equal to the Chern number *C*_{n} of the band, so

Numerical calculation of the Chern numbers using the direct diagonalization eigenfunctions of equation (1) confirms that the Chern number of all bands is the same, that is, *C*_{n} = ±1 = sign(Δ*f*), and that topological edge modes are present under closed boundary conditions (Supplementary Section III). Additional agreement comes when comparing the experimental far-field patterns as a function of frequency (Fig. 2c) with the simulated far-field emission pattern of randomly occupied Floquet–Bloch bands (Fig. 2d). There is good agreement in the band splitting, anharmonicity and momentum structure in addition to the tilting magnitude and direction of the bands. For larger lattice periods (Fig. 1c, middle and Supplementary Section IV), the theory starts to deviate from experiment with regard to the anharmonicity of higher-order bands, with experimental bands being more closely separated than the theory predicts. An extension of equation (1) to include complex potentials^{27,43} can improve the agreement between theory and experiment (Supplementary Section IV), but nonetheless, the present theory accurately captures the observed band tilting for all parameters studied.

An additional feature in the frequency dependence of the bands is the reduction of the separation between them, indicating a reduction in the confining potential and setting a limit on the speed at which non-resonant optical potentials can be used to modulate polariton condensates. To understand the microscopic behaviour behind the potential formation and its dynamical time scales, we present the results of the fitting procedure of equation (1) as a function of modulation depth, lattice period and frequency of the conveyor belt. For a static conveyor belt, the power and period dependence of *V*_{eff} and *V*_{shift} are shown in Fig. 3a,b. For lattices with large period (|Δ*k*_{laser}| < 0.5 μm^{−1}), *V*_{shift} is linearly proportional to *P*_{m} (Fig. 3b, dash-dotted) while *V*_{eff} has a square root dependence on *P*_{m} (Fig. 3a, dark-grey dash-dotted). As the lattice period gets smaller, both *V*_{shift} and *V*_{eff} decrease and have a less pronounced dependence on pump power. At the smallest periods (|Δ*k*_{laser}| > 0.5 μm^{−1}), *V*_{eff} shifts from a square root to a fourth root dependence on *P*_{m} (Fig. 3a, light-grey dash-dotted).

The detailed power dependence described above sets limits on the underlying microscopic processes that create the optical potential. These have typically been considered to be a single phenomenological excitonic reservoir, proportional to the pump laser and with its density pinned by stimulated scattering into the condensate, which implies that the shape and depth of the potential are pinned at threshold^{28,30,31,32,34}. This explanation can capture the two different functional dependencies of the effective potential depth only by considering the relaxation processes from the free-carrier plasma (*n*_{eh}). At typical estimated experimental plasma densities (~10^{9} cm^{−2}; Supplementary Section V), excitons form through both geminate (∝ *n*_{eh}) and bimolecular processes (\(\propto {n}_{\mathrm{eh}}^{2}\)) (ref. ^{44}). Given that the plasma density is proportional to the laser intensity \({I}_{{\rm{laser}}}={P}_{1}+{P}_{\mathrm{m}}+2\sqrt{{P}_{1}{P}_{\mathrm{m}}}\cos \left({\Delta k}_{{\rm{laser}}}x\right)\), it follows that the exciton reservoir density has both square and fourth root terms (\({n}_{\mathrm{R}}\propto {n}_{\mathrm{eh}}^{2}\propto \sqrt{{P}_{\mathrm{m}}},\sqrt[4]{{P}_{\mathrm{m}}}\); Supplementary Section V), which qualitatively and intuitively explain the observed potential’s dependencies on increasing modulation power in Fig. 3. Additional processes such as exciton diffusion, the non-Hermitian potential^{27,43} and direct interaction with the electron–hole plasma^{45} need to be included to explain the simultaneous reduction in potential depth with the change in power dependence. However, despite the potential for additional refinement, there is order-of-magnitude agreement between the directly measured nearest-neighbour hopping *J* (Fig. 1c, green label and Fig. 3c, solid lines) and the hopping from of the fitted model (Fig. 3c, dashed lines). As expected for tightly bound sites, the hopping strength decreases approximately exponentially as a function of *P*_{m} and increases with smaller lattice period, while the measured hopping is of magnitude comparable to that inferred in previous experiments using optical confinement^{30,31,46}.

The results of the least squares fitting for the potential depth as a function of Δ*f* is shown in Fig. 3d,e (for fixed lattice periods) and Fig. 3f (for fixed modulation power). The datasets for other ∆*k*_{laser}/*P*_{m} are qualitatively similar, and all share two main features: the potential depth decreases with increasing frequency offset, and it saturates at a non-zero value at high frequency. Firstly, the functional form of the decay changes with lattice period, with large period lattices (Fig. 3d) having a sharper decay curve as compared with short period lattices (Fig. 3e). Nonetheless, the decay constant (*τ*_{decay} = 210 ± 80 ps) is independent of *P*_{m} and ∆*k*_{laser} (Supplementary Section VI) and corresponds to the inability of the exciton reservoir to adiabatically follow the movement of the laser fringes. The similarity between the measured *τ*_{decay} and previous measurements of relaxation time scales^{47} supports the conclusion that the reduction in potential depth is due to the slow relaxation from the free-carrier plasma to the exciton reservoir. Secondly, the saturation value of *V*_{eff} at high Δ*f* is independent of ∆*k*_{laser} (Fig. 3f) and is 20–75% of the value at ∆*f* = 0 (see also Supplementary Section VI). The fact that this remaining potential lattice is independent of the lattice period suggests that it arises directly from interaction of the polaritons with the free carriers^{45} before any relaxation processes and scatterings start to introduce spatial diffusion and scattering. Hence, both the decay and saturation of *V*_{eff} with increasing modulation speed can be explained within the same phenomenological framework: a free-carrier plasma that instantaneously follows the laser intensity and an exciton reservoir that is thermally populated from this plasma and feeds the condensate via stimulated scattering. This means that, while slow modulation (Δ*f* < 1/210 ps) will always be strongest, it is possible to use the electron–hole plasma to modulate the energy of GaAs polaritons faster than their thermal relaxation time scale. This is crucial to achieve effective Floquet Hamiltonians in the high-frequency approximation, which would allow the generation of artificial gauge fields^{48,49}. Given a typical hopping measured in this work (*J* = 5 μeV), the high-frequency regime is within experimental reach (Δ*f* = 10 GHz corresponds to 40 μeV modulation).

## Discussion

We have demonstrated the formation of spatio-temporal potential lattices in exciton–polariton condensates. Using the simple interference between two lasers, we have created potential lattices with a controllable depth (up to ~4 meV) and tunable nearest-neighbour hopping. Detailed power and frequency dependencies of the potential depth have highlighted the crucial role of the electron–hole plasma in the formation of the non-resonant potential, demonstrating that modulation is possible at speeds higher than the thermal relaxation. Finally, we observed universal tilting of bands in the adiabatic modulation limit and directly linked it to non-trivial Chern numbers *C*_{n} = ±1 of the instantaneous eigenstates, which is analogous to Thouless pumping in the limit of a sliding lattice^{50,51,52}. The non-zero Chern number demonstrates the topological non-triviality of the non-reciprocal tilting. The simplicity of the technique opens the door to future experiments using additional modulation frequencies^{53} and extensions into two-dimensional geometries^{54} that can result in higher Chern numbers^{41} and lead to the experimental confirmation of topologically protected edge states in modulated polariton lattices. Additionally, the possibility of modulation at non-adiabatic speeds using the non-resonant plasma is a steppingstone towards creating artificial gauge fields and opens the door to studying Floquet topological phase transitions in polariton condensates^{35} and other phase-coherent states in semiconductor microcavities^{55} and studying the interplay between polariton non-Hermicity, band topology and non-reciprocity.

## Methods

### Sample

The microcavity sample used consisted of 35 (30) AlAs/Al_{0.15}Ga_{0.85}As mirror pairs for the bottom (top) distributed Bragg reflectors (DBRs) forming a 5*λ*/2 cavity (λ being the resonance wavelength) with four sets of (3 × 13)-nm-wide GaAs/Al_{0.3}Ga_{0.7}As. As quantum wells, leading to a Rabi splitting of 2ℏ*Ω* ≈ 8.5 meV. At *k* ≈ 0, the cavity energy was 7 meV below the exciton energy and the lower polariton lifetime was 22 ps.

### Laser interference

Two continuous wave lasers were used to simultaneously excite the microcavity: a single-mode, cavity-locked Ti:sapphire (~120 mW = 1.2*P*_{th}) and a single-mode, tapered amplifier (TA) semiconductor laser (<1 W, 20 GHz mode-hope free tuning, 1–90% of Ti:sapphire power). Both lasers were chopped into ~10 μs pulses synchronized with the camera capture using acousto-optic modulators to prevent sample heating and defocused with a cylindrical lens, providing a 50 × 16 μm elliptical condensate on the microcavity.

The TA was offset from the Ti:sapphire both in angle and in frequency, creating intensity fringes on the sample surface (*a* = 9–20 μm) that moved with speeds of up to 0.2 μm ps^{−1}. To achieve the angle offset, both lasers were initially aligned at normal incidence on the microcavity, and the TA was beam-walked to come in at a variable angle (0.3–0.7 μm^{−1}) while maintaining maximal spatial overlap. The frequency offset was controlled via the output grating of the laser diode of the TA and monitored using the beat signal between the two lasers measured on a fast photodiode. The modulation range was 20 GHz with an accuracy of 100 MHz (mostly limited by slow frequency drift).

### Measurement

The sample was held in a continuous-flow helium cryostat at 4 K. An objective with a numerical aperture of 0.4 was used to focus the lasers and collect the emitted photoluminescence, which was separated from the laser light using a dichroic mirror. The photoluminescence passed through a high-resolution spectrometer (with a minimum full-width at half-maximum of ~60 μeV) and was then imaged with a charge-coupled device camera.

### Band analysis

To extract the experimental band structures, images were smoothened, and energy peaks were detected for each value of momentum *k*_{y} and then clustered using the AgglomerativeClustering function in scikit-learn^{56}. The clustering metric prioritizes clustering along the mode momenta and takes into consideration the energy tilting for non-zero frequency offset between the lasers. Peak fitting and clustering were supervised to ensure consistent results for all experimental conditions.

## Data availability

Raw data is available on figshare at https://doi.org/10.6084/m9.figshare.25217435 (ref. ^{57}).

## Code availability

The code used to analyse data and perform simulations are both available via GitHub at https://github.com/YagoDel/microcavities (ref. ^{58}).

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## Acknowledgements

The authors acknowledge financial support by the Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research (KAKENHI; grant nos. JP17H04851 and JP19H0561), Japan Science and Technology Agency PRESTO grant no. JPMJPR1768, NTT Research and the State of Bavaria. T.C.H.L. was supported by the Singapore Ministry of Education via the AcRF Tier 3 Program ‘Geometrical Quantum Materials’ (MOE2018-T3-1-002). X.X. was supported by the National Natural Science Foundation of China (grant no. 12264061). E.A.O. was supported by the Australian Research Council Centre of Excellence Grant CE170100039. The Würzburg group acknowledges financial support by the German Research Foundation under Germany’s Excellence Strategy – EXC2147 ‘ct.qmat’ (project id 390858490).

## Author information

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### Contributions

Y.d.V.I.R. and M.D.F. conceived, designed and performed the experiments. Y.d.V.I.R. collected and analysed the experimental data. X.X. and T.C.H.L. conceived the numerical model. Y.d.V.I.R. and X.X. performed the computations. E.A.O., T.C.H.L., A.S. and R.T. provided the theoretical and analytical framework. C.S., S.D., S.K. and S.H. designed, fabricated and characterized the sample. S.T. and M.D.F. supervised the project. The paper was prepared by Y.d.V.I.R. and M.D.F., with review and editing carried out by all authors.

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Supplementary Sections 1–7, Fig. 1–6 and References.

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del Valle Inclan Redondo, Y., Xu, X., Liew, T.C.H. *et al.* Non-reciprocal band structures in an exciton–polariton Floquet optical lattice.
*Nat. Photon.* **18**, 548–553 (2024). https://doi.org/10.1038/s41566-024-01424-z

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DOI: https://doi.org/10.1038/s41566-024-01424-z