In contrast to classical systems, quantum states exhibit a discrete spectrum similar to that of particles; these include lattice vibrational quanta (phonons), electron oscillational quanta (plasmons), and spin-wave quanta (magnons)1−3. Through traditional spectroscopic measurements, we can explore these quantum states by observing the resonant transitions between discrete levels, resulting in distinct resonance peaks whose lifetimes are determined by Fermi’s golden rule4,5. However, this spectroscopic understanding provides only limited information on the quantum states, necessitating the incorporation of spatial and temporal exploration. Additionally, environmental factors, particularly grain boundaries (GBs), potentially contribute to the destruction of quantum states, thereby constraining their sustainability in solids6−12. The prevailing conjecture is that the quantum nature of condensed matter can endure at extremely low temperatures, where phenomena such as superconductivity and superfluidity manifest13. As such, the direct observation of quantum coherence and interference has proven challenging at room temperature (RT), and to date, visualizing the spatiotemporal dynamics of quantum states remains elusive.
Here, we report the direct visualization of a hundred-micrometre quantum state in a 12-nm-thick GB-free Cu(111) thin film at RT. We prepared a film with a thickness of 12 nm (equivalent to 56 atomic layers) to ensure both single crystallinity and high THz transmission, enabling reliable data analysis (Extended Data Fig. 1 for the single crystallinity of our Cu(111) film)14,15. The Cu(111) film shows resistance against oxidation. This quasi-2D GB-free Cu(111) film hosts air-stable quantum well states (QWSs) where the illumination of a terahertz (THz) electromagnetic field induces optical transitions between these states. We employ a tip-assisted THz probe to visualize the emergence of coherent charges associated with the optical transition. Two key criteria of our experiments are provided below. First, compared to the time window of the picosecond (ps) THz probe, a longer coherence time of the QWS needs to be maintained. By employing a GB-free Cu(111) film, we ensure a long coherence time, even in ps. Second, the implementation of the cross-polarization is essential for capturing the coherent charges with vorticity, as in the Hall measurements. By employing crossed polarizations for both the THz source and detector, we filter out the contribution from background Drude charges sensitive to co-polarization, enabling focused observation of the coherent charges.
At RT, we observe intriguing quantum oscillations in the GB-free Cu(111) film. We employ tip-assisted THz spectroscopy under cross-polarization conditions (Fig. 1a), where the input THz field (ETHz,in//x) and the output THz field (ETHz,out//y) are orthogonal to each other. Figure 1b shows THz mapping images of the Exy signal over a 200 µm × 200 µm film at specific times (t1 − t3). The presence of the Exy signal reflects a rotation of the E field from the x- to y-directions in the GB-free Cu(111) film. Due to the in-plane isotropy (Extended Data Fig. 1) and lack of net magnetization of Cu(111), the observed rotation of the field polarization originates from the charges induced by THz excitation. Specifically, the THz-induced charges at a given position transversely flow into neighbouring positions, similar to the Hall effect, leading to the Faraday-like rotation of the THz E field over the sample. Upon an examination of the snapshots, we observe a pronounced quadrupole-like charge distribution near the edges (edge mode) and a comparatively weaker monopole-like charge distribution near the centre (centre mode). As time progresses, the charges near the edges undergo a dramatic evolution, transitioning from weak to strong intensities and then back to weak intensities. On the other hand, the charges near the centre exhibit a more complex and less straightforward progression (see Extended Data Fig. 2 for more detailed snapshots).
Remarkable quantum oscillations, oscillations with a fixed frequency, appear in the time-domain signal of GB-free Cu(111) at RT. Figure 1c depicts the time-domain signal measured near the edges (marked as P1 and P2 in left panel of Fig. 1b); here, the highest intensity occurs within the time range of −1 ps ≤ t ≤ 12 ps, with the time zero set to the peak time of ETHz,in.
Two prominent oscillatory signals, each lasting for approximately 10 ps, are observed near the edges (the P1; red region, the P2; blue region). They show a similar lifetime (τ) of 4.8 ps according to the analysis (fitting curves); this time is definitely longer than the duration of our THz probe. This signal is similar to Rabi oscillation between two discrete levels in quantum systems through resonant excitation16−19. The red region shows polarization rotation towards the +y direction, while the blue region shows rotation towards the −y direction. Conversely, the oscillatory signal near the centre (P3) is weaker, and the lifetime is shorter, lasting only 3 ps (bottom panel). A distinction between the edge and centre modes likely indicates the emergence of two distinct quantum transitions near the edges and centre. However, in the polycrystalline Cu thin film (with the same dimensions), no distinct Exy signal is observed in either the real space image (Fig. 1d) or the time domain signals (Fig. 1e). These results confirm that quantum oscillation is exclusively observed in the GB-free Cu(111) films, which highlights the direct relevance of quantum oscillation to the coherence of electrons.
Individual QWSs in both the edge and centre regions are revealed through their characteristic resonances, providing a playground for the quantum oscillation. Figure 2a shows the Fourier transformed spectra of the time-domain signals in Fig. 1b; two distinct resonances are observed at 0.2 THz for the edge mode and 0.6 THz for the centre mode (dots: experimental data, solid lines: fitting curves). These resonances are analysed by fitting to a Gaussian model represented by \(\text{g}\left(\text{ω}\right)\text{=}\frac{\text{1}}{\text{γ}\sqrt{\text{2π}}}\text{exp}\left(\frac{\text{}\text{(ω}\text{}{\text{ω}}_{\text{0}}\text{)}}{\text{2}{\text{γ}}^{\text{2}}}\right)\), where ω0 is the resonance frequency, while a broadening parameter (γ ~ 0.21 THz for the edge mode and γ ~ 0.43 THz for the centre mode) is determined; these results are consistent with the decay time (4.76 ps for the edge mode and τ ~ 2.32 ps for the centre mode) of the quantum oscillation in Fig. 1b. In the following discussion, we address the differences between the experimental data and the fitting results, which potentially originate from the interference between the two resonances.
Real-space mapping images at resonance frequencies are obtained and provide a visualization of charge transport during the quantum transition. Through Fourier transformation20, the amplitude (\(\left|{E}_{\text{x}\text{y}}\right|\)) spectrum is calculated, and by accumulating these values, we generate a real-space image of \(\left|{E}_{\text{x}\text{y}}\right|\) at the specific frequencies of 0.2 and 0.6 THz (depicted in the top panels of Fig. 2b). The \(\left|{E}_{\text{x}\text{y}}\right|\) image at 0.2 THz prominently exhibits a quadrupole-like strong intensity concentrated at the corners of the edge, and the image at 0.6 THz displays a monopole-like weak intensity. These observations align with the findings from the time-domain mapping results and demonstrate the consistency between the two experimental approaches. Furthermore, the phase (Φ) image reveals distinct patterns at different frequencies (depicted in the bottom panels of Fig. 2b). At 0.2 THz, the phase image clearly shows separation into four sections with alternating signs, indicating the presence of phase differences in the edge mode. However, for the centre mode at 0.6 THz, the signal at the centre becomes dominant, resulting in a blurred phase map with less discernible separation of the four sections. Notably, our tip-assisted THz measurement technique enables real-space visualization of quantum objects with characteristic frequencies.
The experimental data at 0.35 THz provide compelling evidence of quantum interference between the edge and centre modes. Figure 2c displays mapping images at 0.35 THz, where the spectral overlap between the two resonances is maximized (refer to Fig. 2a). The presence of quantum interference is indicated by the fan-like feature observed near both the edge and the centre regions. More concrete evidence of quantum interference emerges from the comparison of the difference spectra (ΔExy, Fig. 2C top right panel) between the experimental data (dots) and the corresponding fitting curves (lines) in Fig. 2a; these data reveal identical periodic oscillations for both the edge and centre regions. Furthermore, their first derivatives exhibit corresponding crossing points with the x-axis, validating the presence of the quantum interference in the GB-free Cu(111) film (Fig. 2d bottom right panel). In contrast, no significant spectra were obtained for the polycrystalline Cu film, and no carrier separation were observed for the differentiation of the edge or centre modes (Fig. 2d); these results confirm the absence of GBs and demonstrate quantum behaviour. We also found the spectral weight transfer from longitudinal charges (Exx) to transverse charges (Exy), showing the reliability of our data and analysis based on fundamental charge conservation (Extended Data Fig. 3). Therefore, the presence of quantum states is intrinsic to GB-free Cu(111). Note that the quantum states are sustainable for more than one year due to the oxidation resistance of our GB-free Cu(111); this was confirmed experimentally.
Next, we investigate the spatial scale of quantum coherence or interference, which extends over 400 μm even at RT. Figure 3a displays phase mapping images, which are suitable for tracking coherence, for Cu(111) films of different sizes: 200 μm × 200 μm, 400 μm × 400 μm, and 800 μm × 800 μm (see Extended Data Fig. 4 for the complete data of each sample). Notably, the phase mapping image clearly distinguishes the four sections up to a size of 400 μm × 400 μm (see Extended Data Fig. 5 for reproducibility), whereas they are less well defined for the 800 μm × 800 μm sample. These results demonstrates a minimum quantum coherence length of 400 μm for quasiparticles in the quantum state of Cu(111). The polar plots in the bottom left two panels in Fig. 3a further support this deduction. These plots show the phase value of each position on the inner circles with a radius R = 0.4d (white dashed line) and the outer circles with a radius R = 0.9d (black dashed line), where d represents the length of the sample. Regarding quantum interference, we compare
mapping images at the interference frequency (0.35 THz) for the different sample sizes (Fig. 3b), revealing the macroscopic scale (~400 μm) of the quantum nature of Cu(111) at RT. This remarkable observation in our oxidation-resistant GB-free Cu(111) can facilitate new developments in quantum physics and technology, providing an unprecedented platform for macroscopic and RT quantum studies.
Finally, we attempt to understand the possible origin of the two distinct quantum resonances from the band structure for the Cu(111) film with a thickness of 12 nm (equivalent to 56 atomic layers). According to the STM image (Fig. 4a), our Cu(111) film exhibits a real-space triangular lattice of Cu atoms. Based on this experimental lattice structure, we calculated the Fermi surface of Cu(111) in the hexagonal Brillouin zone with highly symmetric Γ and K points (Fig. 4b). Additionally, the calculated band structure (Figs. 4c-d) shows several electronic states due to quantum confinement near the Γ and K points. Due to the confinement within the 12-nm-thick film, the degenerate electronic bands split, resulting in the QWS at the Γ and K points. The smallest energy spacings of the calculated QWS are found near the Fermi level (thus, THz accessible), and these spacing are approximately 16 meV (~ 3.9 THz) near the Γ point and approximately 2.3 meV (~ 0.56 THz) near the K point. These energy spacings near the K point have the same order of magnitude as the energies of THz resonances in our experimental data, indicating the relevance of hundred-µm quantum states to the K valleys of the GB-free Cu(111) film.
Furthermore, the THz resonances show decay times of approximately 5 ps for the edge mode and approximately 2 ps for the centre mode; both of these values are a few times greater than the probing duration (~ 1 ps) of our THz E field. Therefore, during the measurement, these hundred-µm quantum states have the potential to be coherent. Moreover, the Fermi surface plot (Fig. 4b) for the 12-nm single-crystal Cu(111) shows a Fermi velocity at the K valley, which is an order of magnitude greater than that of the Γ point. This calculation result indicates the possibility of assessing a longer coherence length at the K valley than at the Γ point. Based on both the coherence length and energy spacing, a reasonable interpretation is that the quantum state is linked to the K-valley electrons in our Cu(111) film. Our observations highlight the remarkable hundred-µm-scale coherence of GB-free Cu(111) with the K valley: thus, GB-free Cu(111) is an air-stable quantum platform with a valley degree even at RT.