Floquet engineering enabled by charge density wave transition

TL;DR

This work addresses how time-periodic driving can coherently modify electronic structures in a system with spontaneous spatial order. By combining time-resolved photoemission with mid-infrared pumping on the CDW material 1T-TiSe2 and a four-band CDW model, it demonstrates phase-dependent Floquet engineering: near-resonant pumping induces an instantaneous, resonance-enhanced downshift of the valence band maximum only in the CDW phase, while a slower blueshift accompanies CDW melting, vanishing above $T_{ ext{CDW}}$ or off-resonance. The findings reveal a direct interplay between spatial symmetry breaking and Floquet band renormalization, establishing a framework for electronic-phase-selective engineering in CDW and related materials. This approach paves the way for exploiting spontaneous symmetry breaking to tailor non-equilibrium electronic states via tailored light fields.

Abstract

Floquet engineering has emerged as a powerful approach for dynamically tailoring the electronic structures of quantum materials through time-periodic light fields generated by ultrafast laser pulses. The light fields can transiently dress Bloch electrons, creating novel electronic states inaccessible in equilibrium. While such temporal modulation provides dynamic control, spatially periodic modulations, such as those arising from charge density wave (CDW) order, can also dramatically reconstruct the band structure through real-space symmetry breaking. The interplay between these two distinct forms of modulation-temporal and spatial-opens a new frontier in electronic-phase-dependent Floquet engineering. Here we demonstrate this concept experimentally in the prototypical CDW material 1T-TiSe$_2$. Using time- and angle-resolved photoemission spectroscopy (TrARPES) with mid-infrared pumping, we observe a striking pump-induced instantaneous downshift of the valence band maximum (VBM), which is in sharp contrast to the subsequent upward shift on picosecond timescale associated with CDW melting. Most remarkably, the light-induced VBM downshift is observed exclusively in the CDW phase and only when the pump pulse is present, reaching maximum when pumping near resonance with the CDW gap. These observations unequivocally reveal the critical role of CDW in the Floquet engineering of TiSe$_2$. Our work demonstrates how time-periodic drives can synergistically couple to spatially periodic modulations to create non-equilibrium electronic states, establishing a new paradigm for Floquet engineering enabled by spontaneous symmetry breaking.

Figures (7)

• Figure : Fig. 1 ${\mid}$ Schematics of Floquet engineering in the CDW phase of 1T-TiSe$_2$.a,b, Schematics of electronic structures in the normal phase ($T>T_{\text{CDW}}$, panel a) and in the CDW phase ($T<T_{\text{CDW}}$, panel b), respectively. Red and blue curves represent the VB at the $\Gamma$ point and CB at the M point, which are folded by the CDW wavevector q$_\text{CDW}$ in the CDW phase. c, Schematic illustration of the CB (light blue curve) and VB (light red curve) in the CDW phase, light-induced CB sideband with n = -1 (dashed gray curve), and the renormalized CB sideband (thick blue curve) and VB (thick red curve) upon resonance pumping. d, Schematic illustration of TrARPES setup with MIR pumping.
• Figure : Fig. 2 ${\mid}$ Observation of Floquet-induced band renormalization in CDW phase of 1T-TiSe$_2$.a-f, TrARPES dispersion images measured at different delay times along the M-$\Gamma$-M direction by using p-pol. pump ( a-c), and corresponding second derivative images for direct visualization of the bands ( d-f). The pump photon energy is 248 meV and the fluence is 2.0 mJ/cm$^{2}$. g-i, Extracted dispersions of VB from ( a-c) by fitting EDCs. The dotted black curves are extracted dispersion at -1000 fs. j-k, EDCs extracted at momenta of $k_1$ and $k_2$ (marked in c) at different delay times. Black, red and blue tick marks are the corresponding peak positions. l, Intensity map obtained by plotting EDC at momenta $k_1$ as a function of delay time. m, TrARPES intensity of n = 1 VB sideband as a function of delay time. n, Extracted band shift at momenta $k_1$ as a function of delay time. Solid curves are fitting results, while red and blue shaded regions represent the downshift induced by Floquet engineering and upward shift indicating CDW melting.
• Figure : Fig. 3 ${\mid}$ Evolution of Floquet-induced band renormalization when tuning the pump photon energy from resonance to off-resonance.a-e, Second derivative TrARPES dispersion images measured at $\Delta t$ = 0 when tuning the pump photon energy from 207 to 620 meV. The pump fluence is fixed at 2.0 mJ/cm$^2$. f-j, Time-resolved energy shift of n = 0 VB extracted at the $\Gamma$ point at different pump photon energies. k-l, EDCs at the $\Gamma$ point at different pump photon energies for delay time of 0 ( k) and 200 fs ( l). The red and blue tick marks are the corresponding peak positions. m, Extracted energy shift at the $\Gamma$ point for different pump photon energies, where the error bar is determined from the curve fitting results in f-j. n-p, Schematic summary of transient electronic structures upon below-gap pumping, resonant pumping and above-gap pumping.
• Figure : Fig. 4 ${\mid}$ Interplay between the CDW order and Floquet band renormalization.a,b, Schematic dispersion of TiSe$_{2}$ in the CDW phase ( a) and the normal phase ( b). c,d, Second derivative dispersion images measured at $\Delta t$ = -1000 fs at 80 K ( c) and at 250 K ( d). e,f, Second derivative dispersion images measured at $\Delta t$ = 0 fs at 80 K ( e) and at 250 K ( f). The pump photon energy is 248 meV with p-pol. and a pump fluence of 2 mJ/cm$^{2}$. g,h, Extracted dispersions of VB at 80 K ( g) and 250 K ( h) by fitting EDCs from c,e and d,f respectively. i, Time-resolved energy shift of VBM at the $\Gamma$ point measured at different sample temperatures. j, Extracted Floquet-induced energy shifts (red circles) and CDW-melting-induced energy shifts (blue squares) at the $\Gamma$ points as a function of temperature.
• Figure : Fig. 5 ${\mid}$ Numerical results from the four-band model for 1T-TiSe$_2$.a,b, The dispersion of the equilibrium electronic structure in the normal phase ($T>T_{\text{CDW}}$) along the $\Gamma$-M$_2$ direction. The CBs ( a) are from three M points and the VB ( b) is at $\Gamma$ point. The dashed line in ( a) represents a doubly degenerate CB. The inset is the Brillouin zone with three CB pockets (black ellipses) at three M points. c, Dispersion of the equilibrium VB (black curve) and folded CBs (dashed curves) in the CDW phase ($T<T_{\text{CDW}}$). The inset is the Brillouin zone with three CB pockets folded to the $\Gamma$ point. d,e, Dispersion of the VB upon pumping in the normal phase ( d) and CDW phase ( e) compared to the equilibrium VB (black dashed curve). The pump photon energy is 220 meV. f-h, TrARPES intensity with the photon energy smaller than ( f, 180 meV), almost equal to ( g, 240 meV), and much larger than ( g, 400 meV) $\Delta_{\text{CDW}}$. The red arrow in ( g) indicates the energy shift of the VBM. The black (blue) dashed curve is the equilibrium VB (n = -1 replica of the CB) in the CDW phase. i, Energies of the Floquet electronic states (solid curves) as a function of the pump photon energy after hybridizing the VB and the n = -1 replica of the CB (thin dashed lines). The black dashed curve is a schematic interpolation between them.