Cavity Floquet engineering

Abstract

Floquet engineering is a promising tool to manipulate quantum systems coherently. A well-known example is the optical Stark effect, which has been used for optical trapping of atoms and breaking time-reversal symmetry in solids. However, as a coherent nonlinear optical effect, Floquet engineering typically requires high field intensities obtained in ultrafast pulses, severely limiting its use. Here, we demonstrate using cavity engineering of the vacuum modes to achieve orders-of-magnitude enhancement of the effective Floquet field, enabling Floquet effects at an extremely low fluence of 450 photons/μm². At higher fluences, the cavity-enhanced Floquet effects lead to 50 meV spin and valley splitting of WSe₂ excitons, corresponding to an enormous time-reversal breaking, non-Maxwellian magnetic field of over 200 T. Utilizing such an optically controlled effective magnetic field, we demonstrate an ultrafast, picojoule chirality XOR gate. These results suggest that cavity-enhanced Floquet engineering may enable the creation of steady-state or quasi-equilibrium Floquet bands, strongly non-perturbative modifications of materials beyond the reach of other means, and application of Floquet engineering to a wide range of materials and applications.

Fig. 1. The principle and experimental system of cavity-enhanced optical Stark effect (OSE).

a Illustration of the red-detuned chiral OSE. The circularly polarized pump leads to Floquet states ($∣\mathrm{X}-\mathrm{h}\nu ⟩$ and $∣\mathrm{g}+\mathrm{h}\nu ⟩$), which hybridize with the ground and excited state of excitons $∣\mathrm{g}⟩$ and $∣\mathrm{X}⟩$, resulting in blueshifted dressed states $∣{\mathrm{g}}^{\prime }⟩$ and $∣{\mathrm{X}}^{\prime }⟩$. A weak probe pulse measures the shifted transition energy. b Schematic of the half-wavelength cavity with a monolayer WSe₂ at the antinode. The simulated field distribution at the cavity resonance is plotted on the left side, showing a 200-fold enhancement of the resonant field at the antinode. Optical measurements are performed through transparent Sapphire. c Reflectance spectra of the $\mathrm{SiN}/{\mathrm{SiO}}_2$ bottom distributed Bragg reflector (DBR) (blue) with a side-band minimum at the exciton resonance, the ZnS/MgF₂ top DBR (red) with high reflectance at the laser and exciton energies, and the complete cavity (black) showing cavity (1.67 eV) and exciton (1.74 eV) resonances. The solid/dashed curves are the measured/simulated results, respectively.

Fig. 2. Extreme Floquet engineering.

a, b Co-circularly polarized probe reflectance spectra R₊₊(t) at different delay time t relative to the pump. a The monolayer is in a cavity, and pump fluence P = 18 fJ/μm². b The monolayer is on a DBR, and P = 460 fJ/μm². The dashed white curves are fitted exciton resonances. The same amount of Stark shift of  ~2.5 meV is observed when the pump and probe overlap in time, while the pump fluences differ by 26 times between the cavity and DBR devices. c Weak pump, co-circular differential reflectance spectra ΔR₊₊(t)/R₊₊(t) for the cavity device with a pump intensity of 33 kW/cm², or a fluence of 0.12 fJ/μm². d Strong pump, co-circular reflectance spectra R₊₊(t) of the cavity device with a pump intensity of 330 MW/cm² or fluence of 1.2 pJ/μm², showing a very large blueshift at zero time delay. e Cross-circular probe R₊₋(t) under the same pump as in (d), showing a redshift at zero time delay. f Energy difference between K and $K^{\prime }$ valley excitons when the same pump as in (d, e) is turned on (red) and off (blue) at time = 0, showing effective Zeeman splitting of 50 meV. g Co-circular Stark shift vs. the pump fluence and intensity for a monolayer in a cavity (red), on a DBR (black), and in free space (purple). Symbols are measurement results. The red and black dashed lines are fits using Eq. (2). The red solid line is a fit including both two-photon absorption and cavity-shifting. The blue dashed line is a fit only with cavity-shifting. The purple dashed line is an extrapolation from DBR to vacuum. h Dressing induced valley splitting vs. the pump fluence and Intensity for the cavity device. Symbols are measurement results. The black dashed line is model prediction.

Fig. 3. A chirality all-optical switch.

a The truth table of an XOR gate. When control is co-circular to the signal, a decrease in the reflected signal is defined as “0” output. When control is cross-circular to the signal, an increase in the reflected signal is defined as “1” output. b Left axis: The modulation amplitude of the probe reflectance by a 120 fJ/μm² (2 pJ) pump at zero time delay: dR_±  = R_±(on) −  R_±(off), where  ± denotes the co- (blue) and cross-circularly (red) polarized probe. Right axis: The corresponding extinction ratio Θ  = 10log₁₀(∣dR₊  −  dR₋∣/δ) (green). c Time-dependence of the modulation amplitudes for the co- (blue) and cross-polarized (red) probe and the corresponding extinction ratio Θ (green) at E = 1.75 eV, showing an ultrafast switch time of about 0.4 ps, reflecting the coherent nature of the Floquet effect.

Fig. 4. Optimal Floquet shift.

Calculated saturated Floquet shift vs. the pump detuning for our current device (black), a device with 1 meV exciton linewidth (red), and a device with 1 meV exciton linewidth and an order of magnitude smaller TPA (blue). Other parameters are the same as the current device.