Main

The adiabatic modulation of a quantum Hamiltonian around a closed loop in parameter space causes eigenstates to acquire a geometric phase—the Berry phase1. For periodically driven systems, the additional accumulation of geometric phase can lead to the formation of novel topological invariants2 and induce topologically protected transport and edge modes3,4,5,6. Combined with a spatially periodic potential, eigenstates can be defined on a Floquet–Brillouin zone7,8, 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 quantum9,10,11,12,13. 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 bands13,14,15,16. 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 excitons17. 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 condensate18. Lithographic methods for engineering the polariton potential landscape have been commonly exploited to emulate the physics of many-body two-dimensional systems using polariton condensates19,20,21,22. While there have been proposals on how to achieve non-reciprocity with such static potentials23,24,25, 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 real26 and imaginary (non-Hermitian)27 components that have been used to create optical traps28,29, to control the hopping between individual sites30,31,32 and to create non-Hermitian lattices33,34.

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 lattices35, to rotate polariton condensates36,37 or to control recently proposed polariton qubits38.

Fig. 1: Band structures of a polariton conveyor belt.
figure 1

a, A diagram of the sample excitation. The angle between the lasers (Δklaser = 0.3–0.7 μm−1) controls the fringe periodicity, while the frequency offset (|Δf| < 7 GHz) controls their speed and movement. The normal-incidence laser has P1 = 1.2Pth with a condensation threshold Pth of 100 mW. The angle-offset laser has Pm = 1−110 mW. b, Example real-space tomography (intensity normalized) just below and just above the condensation threshold. The red-coloured diagram on top corresponds to the intensity of the laser interference pattern. c, Example band structures (intensity normalized) at zero frequency offset for different lattice periods and depths. Thin white lines show the results of spectral peak finding and clustering as described in Methods. Black dashed lines correspond to the fitted Bloch modes. The energy width of the ground state (in terms of nearest neighbour hopping J) is marked in green. d, The average size of the first (∆k0) and second (∆k1) BZs as functions of ∆klaser (between 7 and 52 samples). Shaded regions correspond to the expected edge of the first BZ (purple-magenta, ∆k0 ≈ π/a), where a is the fringe period) and of the boundaries of the second BZ in the extended scheme (blue-grey, 2π/a < ∆k1 < 4π/a). Error bars are estimated by averaging datasets with different Pm.

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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 Δf ≈ 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 lattice29,39 and the stability of the interference pattern, the band formation itself is robust and can be controlled through the angle between the lasers (Δklaser) and the relative power of the angled beam (Pm). Shallow (small Pm) and wide (small Δklaser) 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/Δklaser (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 microcavities40.

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 carriers28 are reduced to an effective potential Veff for the exciton field χ, expressed as

$$\begin{array}{c}i{{\hslash}}{\partial}_{{\rm{t}}}\psi =-\frac{{{{\hslash}}}^{2}{\nabla}^{2}}{2{m}_{{\rm{eff}}}}\psi +{\hslash}{\Omega}_{\mathrm{R}}{\chi} -\frac{\delta}{2}\psi +{V}_{{\rm{shift}}}\psi \\ i{{\hslash}}{\partial}_{{\rm{t}}}\chi ={V}_{{\rm{eff}}}\cos \left({\Delta k}_{{\rm{laser}}}x-2{\uppi} {\Delta} {ft}\right)\chi +{\hslash}{\Omega}_{\mathrm{R}}\psi +\frac{\delta }{2}\chi +{V}_{{\rm{shift}}}\chi, \end{array}$$
(1)

where ψ is the photon field, meff is the photon effective mass, δ is the exciton–photon detuning, Vshift is the average energy of excitons and photons and 2ℏΩR 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 Veff, after which the s band energy was used to estimate Vshift. 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 |Δf| > 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 ∆klaser. For positive (negative) ∆f, the conveyor belt moves in the same (opposite) direction as ∆klaser. 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, ΔE = E(π⁄a) – E(−π⁄a) (Fig. 2a, inset), which is linearly proportional to ∆f. The constant of proportionality is universal and independent of ∆klaser and Pm and corresponds to Planck’s constant (Fig. 2b).

Fig. 2: Universal band tilting in a moving conveyor.
figure 2

a, An example band structure of a moving conveyor. The thin light-grey line is the result of spectral peak finding and clustering. The dashed grey lines are linear fits to this clustered data. The energy shift at the BZ boundary ∆E (inset) is extracted from the linear fit. b, The average (from 50–200 samples) energy shift at the BZ boundary as a function of the frequency for different modulation laser angles. Errors (shaded regions) are estimated by averaging over datasets with different Pm and over different bands in each band structure. c, Experimental band structures for a fixed ∆klaser and Pm as a function of frequency between the pump lasers. d, The simulated far-field emission of Floquet–Bloch solutions to the fitted polariton Hamiltonian for the parameters in c.

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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 cell8. It is known that the tilting energy in classical waves is connected to the Chern number Cn of the band ΔE = CnhΔf (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 ψk(t) and energies ϵk(t) (ref. 42). In the case of adiabatic evolution, it can be expressed as

$${\Omega}_{\rm{k}}=\frac{1}{T}{\int}_{0}^{\rm{T}}{dt}{\epsilon}_{\rm{k}}(t)+\frac{1}{T}{\rm{Im}}\left[{\int}_{0}^{{T}}{dt}\left\langle {\psi}_{\rm{k}}\; |\; {\partial}_{\rm{t}}{\psi}_{\rm{k}}\right\rangle \right],$$

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 nth band quasi-energies at the two BZ edges can be written as the winding of the Berry phase around the Floquet–Brillouin zone thus

$$\Delta {E}_{\rm{n}}={\hslash}\left({\Omega}_{\uppi {\rm{/L}}}-{\Omega }_{-\uppi {\rm{/L}}}\right)=\frac{{{\hslash }}}{T}\mathop{\oint }\limits_{{BZ}}\mathrm{d}k{\partial }_{\rm{k}}{\rm{Im}}\left[{\int }_{0}^{\rm{T}}\mathrm{d}t\left\langle {\psi }_{\rm{k}}\;|\; {\partial }_{\rm{t}}{\psi }_{k}\right\rangle \right]$$

which is equal to the Chern number Cn of the band, so

$$\Delta E=h\left|\Delta f\,\right|{C}_{\rm{n}}.$$

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, Cn = ±1 = sign(Δf), 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 potentials27,43 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 Veff and Vshift are shown in Fig. 3a,b. For lattices with large period (|Δklaser| < 0.5 μm−1), Vshift is linearly proportional to Pm (Fig. 3b, dash-dotted) while Veff has a square root dependence on Pm (Fig. 3a, dark-grey dash-dotted). As the lattice period gets smaller, both Vshift and Veff decrease and have a less pronounced dependence on pump power. At the smallest periods (|Δklaser| > 0.5 μm−1), Veff shifts from a square root to a fourth root dependence on Pm (Fig. 3a, light-grey dash-dotted).

Fig. 3: Formation dynamics of a non-resonant potential.
figure 3

a, The average fitted effective potential depth Veff as a function of Pm for different ∆klaser values (colour key). Dash-dotted dark-grey and light-grey lines are the square root and fourth root fits for long and short period lattices, respectively. b, The average fitted potential shift Vshift as a function of Pm for different ∆klaser. The dash-dotted dark grey line is the linear fit to long-period lattices. c, The average ground-state bandwidth as a function of Pm for the experiment (solid) and fitted theory (dashed). d,e, the average effective potential depth as a function of ∆f, for varying Pm and ∆klaser values of 0.32 μm−1 (d) and 0.67 μm−1 (e). f, The average effective potential depth as a function of ∆f, for fixed Pm and varying ∆klaser. Errors (shaded) and averages (solid lines) are estimated by averaging up to five different realizations with the same parameters.

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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 threshold28,30,31,32,34. 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 (neh). At typical estimated experimental plasma densities (~109 cm−2; Supplementary Section V), excitons form through both geminate (∝ neh) 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 potential27,43 and direct interaction with the electron–hole plasma45 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 Pm and increases with smaller lattice period, while the measured hopping is of magnitude comparable to that inferred in previous experiments using optical confinement30,31,46.

The results of the least squares fitting for the potential depth as a function of Δf is shown in Fig. 3d,e (for fixed lattice periods) and Fig. 3f (for fixed modulation power). The datasets for other ∆klaser/Pm 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 (τdecay = 210 ± 80 ps) is independent of Pm and ∆klaser (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 τdecay and previous measurements of relaxation time scales47 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 Veff at high Δf is independent of ∆klaser (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 carriers45 before any relaxation processes and scatterings start to introduce spatial diffusion and scattering. Hence, both the decay and saturation of Veff 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 (Δf < 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 fields48,49. Given a typical hopping measured in this work (J = 5 μeV), the high-frequency regime is within experimental reach (Δf = 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 Cn = ±1 of the instantaneous eigenstates, which is analogous to Thouless pumping in the limit of a sliding lattice50,51,52. 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 frequencies53 and extensions into two-dimensional geometries54 that can result in higher Chern numbers41 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 condensates35 and other phase-coherent states in semiconductor microcavities55 and studying the interplay between polariton non-Hermicity, band topology and non-reciprocity.

Methods

Sample

The microcavity sample used consisted of 35 (30) AlAs/Al0.15Ga0.85As mirror pairs for the bottom (top) distributed Bragg reflectors (DBRs) forming a 5λ/2 cavity (λ being the resonance wavelength) with four sets of (3 × 13)-nm-wide GaAs/Al0.3Ga0.7As. As quantum wells, leading to a Rabi splitting of 2ℏΩ ≈ 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 = 1.2Pth) 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 ky and then clustered using the AgglomerativeClustering function in scikit-learn56. 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.