Dynamical interplay between superconductivity and charge density waves: A nonlinear terahertz study of coherently driven $2H\text{−}{\mathrm{NbSe}}_2$

$2H\text{-NbS}{\mathrm{e}}_2$ is an archetypal system in which superconductivity and charge density wave (CDW) coexist and compete macroscopically with each other. In particular, this interplay also manifests in their dynamical fluctuations. As a result, the superconducting amplitude fluctuation (i.e., Higgs mode) is pushed below the quasiparticle continuum, allowing it to become a coherent excitation observable by Raman scattering. In the present study, we coherently drive the collective oscillations of the two orders and visualize their interplay in the time domain. We find that both collective modes contribute to terahertz third-harmonic generation (THG) and their THG signals interfere below $T_{\mathrm{c}}$, leading to an antiresonance of the integrated THG signal. The dynamical Ginzburg-Landau model suggests that around the antiresonance a periodic energy transfer between the driven Higgs oscillations and the driven CDW oscillations is possible. Our results illustrate the roles of collective modes in the terahertz THG process, revealing a close connection of this technique to Raman scattering. In systems where the different collective modes are coupled, our experimental scheme also illustrates a paradigm for realizing coherent control via such couplings.

Figure 1

Terahertz third-harmonic generation (THG) in $2H\text{-NbS}{\mathrm{e}}_2$. (a) Terahertz transmission from the $2H\text{-NbS}{\mathrm{e}}_2$ thin film on ${\mathrm{SiO}}_2$ substrate pumped by 0.3-THz light at several representative temperatures. The $2H\text{-NbS}{\mathrm{e}}_2$ sample is approximately 70 nm thick, with $T_{\mathrm{c}}\sim 7.2$ K as determined from resistivity measurement (see Appendix pp2). (b) THG waveforms extracted from raw data in (a) by applying 0.9-THz Fourier bandpass filter. For the three waveforms below $T_{\mathrm{c}}$, we fit the data to two individual Gaussian-enveloped sinusoidal functions with fixed time centers (black lines). Near 6 K, a clear interference pattern is seen, as marked by black arrows. (c) Fourier intensity spectrum of raw data in (a) below $T_{\mathrm{c}}$. Maroon- and yellow-shaded areas highlight the spectra of the 0.3-THz driving pulse and the THG response. THG intensity is integrated from yellow-shaded frequency window. THG noise floor is estimated from gray-shaded frequency windows. Around 6.2 K a noticeable splitting of 0.9-THz peak is observed. (d) Temperature dependence of THG intensity across $T_{\mathrm{c}}$ and up to $T_{\mathrm{CDW}}\sim 33$ K. A noticeable dip in the integrated THG intensity is manifested at 6.2 K (white arrow), similar to previous reports of THG antiresonance in cuprate superconductors [18, 19]. Dotted line marks the THG noise level estimated from the FFT spectra. Inset: Detailed temperature sweep performed around 6.2 K confirming the dip in THG intensity, signifying the antiresonance of THG.

Figure 2

Different origins of THG. Additional measurements are performed on a $2H\text{-NbS}{\mathrm{e}}_2$ thin film exfoliated on a ${\mathrm{SiO}}_2/\mathrm{Si}$ substrate (black dots), where the Si layer is $n$-doped. In this sample THG increases significantly above $T_{\mathrm{CDW}}$. Measurements on the bare ${\mathrm{SiO}}_2/\mathrm{Si}$ substrate reveal that there is indeed a significant onset of THG above $\sim 35$ K (dark-green dots). We ascribe this THG contribution to the terahertz-driven nonlinear current of mobile carriers in the $n$-doped Si layer, which are thermally activated above 35 K. Below 35 K, this contribution falls below the noise level. It is also negligible compared to the additional THG source originating in the CDW phase.

Figure 3

Polarization dependence of THG. THG parallel to the incoming linearly polarized terahertz light is measured from $2H\text{-NbS}{\mathrm{e}}_2$ on ${\mathrm{SiO}}_2/\mathrm{Si}$ as the terahertz polarization is rotated. At low temperatures, (a) THG originating from the Higgs fluctuations and (b) THG originating from the CDW fluctuations are mostly isotropic, consistent with the $A_{1\mathrm{g}}$ symmetry of the two collective modes. (c) THG originating from the mobile carriers’ nonlinear current measured at 75 K exhibits a strong anisotropy suggestive of a fourfold rotational symmetry that is consistent with the crystalline structure of the Si layer. Note that waveplates in the terahertz frequency range are not readily available. Therefore, these measurements are performed by rotating the wire-grid polarizers in front of sample to set incoming polarization while maintaining the same field strength on the sample. THG from the sample is measured along two orthogonal directions by setting the polarizer after sample to $0^{\circ }$ and ${90}^{\circ }$. We then calculate the polarization axis of the resulting THG from two orthogonal THG components, and project the THG field onto the incoming polarization axis (i.e., they are not necessarily parallel). Random and systematic errors in these operations may contribute a slight modulation of the THG intensity as a function of the polarization angle, for example in (a) and (b). The Degree of variation of the THG intensity in (a) and (b) is consistent with our previous characterization of the isotropic THG response of ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ in Ref. [18], which is in stark contrast to (c) where the maximum and minimum THG intensities differ by more than one order of magnitude.

Figure 4

Interference between two sources of THG. (a)–(c) Raw THG waveforms reproduced from Fig. 1. (d)–(f) Coherent sums of the oscillations of two driven coupled modes as calculated from the dynamical Ginzburg-Landau (GL) model described in text. (g)–(i) Oscillatory response of the individual coupled modes under a Gaussian-enveloped periodic drive. In (i), the heavily damped CDW mode (highlighted in thick) is undergoing an antiresonance. A visible dip is manifested near the middle of its response envelope (i.e., around 20 ps). It originates from the periodic energy transfer between the two coupled modes around the antiresonance of the heavily damped CDW mode. (j)–(l) Values used for the parameters of the dynamical GL model. In (j), the frequencies of the Higgs mode (blue) and the CDW amplitude mode (red) from Raman-scattering studies are shown along with their uncertainties (error bars). The frequencies of the Higgs mode and the CDW amplitude mode after interaction in the dynamical GL model are shown as blue and red dots. The 0.6-THz driving frequency is shown as the dotted line in (j), (k). In (k), the bare Higgs mode and CDW amplitude mode frequencies before interaction are shown along with the square root of the coupling constant $g\equiv g_{\mathrm{\Psi },\mathrm{\Phi }}=g_{\mathrm{\Phi },\mathrm{\Psi }}$. In (l), the driving forces for the two modes are shown for each temperature.

Figure 5

Schematic of the terahertz third-harmonic generation setup.

Figure 6

Determination of $T_{\mathrm{c}}$ via resistance measurement.