图 1  朗道超流判据　(a) 无相互作用粒子的能量-动量色散关系, 当粒子以速度 $ \boldsymbol{v} $(红色实线斜率)关于环境(如容器壁、晶格)匀速运动时, 粒子的能谱(黑色实线)发生多普勒位移(蓝色实线), 与$ \boldsymbol{v} $ 同方向的激发态发生红移, 与$ \boldsymbol{v} $ 反方向的激发态发生蓝移; 当某个态的能量红移至0时(严格来说0 + k_BT 范围内), 环境与粒子之间的热涨落会使粒子自发产生该状态激发, 粒子因此不再处于基态, 玻色凝聚以及超流因此遭到破坏; (b) 氦原子间的范德瓦耳斯作用力使液氦展现出集体振荡模式, 在长波极限(Q → 0)下, 其色散呈线性关系, 保证超流液氦以该斜率以下的速度匀速运动时不会自发产生声子激发, 超流液氦的临界速度 $ {\boldsymbol{v}}_{{\mathrm{c}}} $ 实际对应的是系统自发产生更高波矢量的roton激发的相速度^[5]

Figure 1.  Landau criteria for superfluidity: (a) Energy-momentum dispersion relation of a non-interacting particle. When a particle travels at a velocity $ \boldsymbol{v} $ (corresponding to the gradient of the red solid line) with respect to its environment (e.g. container, lattice), its excitation spectrum (black solid line) is Doppler-shifted (i.e. to the blue solid line): excitations with wavevectors parallel to $ \boldsymbol{v} $ are red-shifted while excitations with wavevectors anti-parallel to $ \boldsymbol{v} $ are blue-shifted. When a state is red-shifted to 0 energy (strictly speaking k_BT from 0), thermal fluctuations between the environment and the particle in consideration may cause the particle to be spontaneously excited to that state. The particle no longer remains in its ground state, therefore the Bose-Einstein condensation and the superfluidity will be destroyed along the way. (b) The van der Waals interaction between helium atoms causes liquid helium to exhibit collective excitations. In the long-wavelength limit (Q → 0), these collective modes exhibit a linear dispersion. As long as superfluid helium flows at a speed smaller than the gradient of such acoustic mode, phonons will not be spontaneously excited. The critical velocity of superfluid helium corresponds to the phase velocity of the roton excitations^[5].

图 2  超导体的集体模式　(a) 超导转变温度(T_c)以下, 超导序的自由能 $ F $ 关于序参量 $ \psi =\left|\psi \right|{{\mathrm{e}}}^{{\mathrm{i}}\phi } $ 展现墨西哥帽状的势能面; 在平衡态下, 序参量自发破缺周向的连续U(1) 对称; 序参量沿周向的运动对应超导相位 $ \phi $ 的涨落, 沿径向的运动对应超导振幅 $ \left|\psi \right| $ 的涨落, 这两个模式的能量分别为零和大于零, 因此可以看作无质量和有质量的准粒子激发; (b) 超导相位模式和振幅模式的色散关系^[6], 其中零质量的相位模式(蓝色虚线)通过Anderson机制获得质量, 其能量提升至超导等离子体频率($ {\omega }_{{\mathrm{p}}} $); 有质量的振幅模式 (红色实线)在长波极限下展现出2Δ的能隙

Figure 2.  Collective modes of a superconductor: (a) Below the superconducting transition temperature (T_c), the free energy of a superconductor exhibits a Mexican hat-like potential surface with respect to the superconducting order parameter $ \psi=\left|\psi\right|\mathrm{e}^{\mathrm{i}\phi} $. In thermal equilibrium, the continuous U(1) symmetry in the order parameter phase $ \phi $ is spontaneously broken. The collective motion of the order parameter along the circumferential direction corresponds to fluctuations in the order parameter phase; the collective motion along the radial direction corresponds to fluctuations in the order parameter amplitude $ \left|\psi \right| $. The energy of these two modes are zero and non-zero respectively. They can be described as massless and massive quasiparticles. (b) Dispersions of the phase mode and the amplitude mode of the superconducting order parameter^[6]. Here, the massless phase mode (blue dotted line) becomes massive due to the Anderson mechanism. Its frequency is lifted to the superconducting plasma frequency ($ {\omega }_{{\mathrm{p}}} $). The massive amplitude mode (red solid line) is gapped with an energy of 2Δ

图 3  2H-NbSe₂的拉曼散射实验　(a) 在Sooryakumar和Klein的实验中^[17], 通过施加外界磁场抑制超导态, 超导集体模式(~20 cm^–1)与电荷密度波集体模式(~40 cm^–1)的拉曼散射谱权重发生交换; (b) Méasson等^[55]通过改变温度, 同样观测到超导集体模式与电荷密度波集体模式的拉曼散射谱权重发生交换, 在6 K附近, 两者的谱权重大致相等

Figure 3.  Raman scattering experiments on 2H-NbSe₂: (a) In the experiment by Sooryakumar and Klein^[17], the superconductor collective mode (~20 cm^–1) and the charge-density-wave collective mode (~40 cm^–1) exchange spectral weight under the application of an external magnetic field that suppresses superconductivity. (b) By varying temperature, Méasson et al. ^[55] also observe an spectral weight transfer between the superconductor collective mode and the charge-density-wave collective mode. Around 6 K, the two modes have approximately equal spectral weight.

图 4  Higgs模式的时域自由振荡^[21]　(a) Matsunaga等^[21]采用单周期太赫兹泵浦 - 单周期太赫兹探测手段测量了Nb_1–xTi_xN超导薄膜的超快透射率; (b) 在单周期太赫兹脉冲对超导基态进行扰动后, 样品的太赫兹透射率呈现频率f～2Δ的自由振荡, 通过增大泵浦光的能量密度, 振荡频率 f 逐步降低、并伴随自由振荡的阻尼系数b逐步增强

Figure 4.  Time-domain free oscillations of the Higgs mode^[21]: (a) Using single-cycle terahertz pump – single-cycle terahertz probe technique, Matsunaga et al. ^[21] studied the ultrafast terahertz transmission of Nb_1–xTi_xN superconducting thin films. (b) After the superconducting ground state is perturbed by the single-cycle pump pulse, the terahertz transmissivity of the sample is probed and is found to exhibit a free oscillation with a frequency f ～2Δ, with increasing pump pulse energy density, the oscillation frequency f decreases, accompanied also by an increase in the damping constant b.

图 5  多周期太赫兹光电场驱动的Higgs振荡^[22]　(a) Matsunaga等^[22]利用太赫兹带通滤片过滤单周期太赫兹脉冲, 产生窄带多周期太赫兹光电场, 作为泵浦驱动NbN超导薄膜; 另一束单周期太赫兹脉冲作为探针测量样品的透射率; (b) 多周期太赫兹泵浦的波形及其模平方波形; (c) 在多周期太赫兹泵浦驱动下, 样品在T_c以上及以下的太赫兹透射率演化: 在T_c以上, 太赫兹泵浦仅诱导系统热化、带来透射率的单调变化; 而在T_c以下, 太赫兹泵浦诱导超流体展现出二倍驱动频的相干振荡, 起源于Higgs模式的受驱振荡

Figure 5.  Higgs oscillations driven by multi-cycle terahertz pulses^[22]: (a) Matsunaga et al. ^[22] pump the NbN superconducting films using multi-cycle terahertz pulses filtered from the single-cycle terahertz pulses (i.e. by using bandpass filters), and probe the transmissivity of the sample using another single-cycle terahertz pulse. (b) The waveform of the multi-cycle terahertz pulse and its modulus square. (c) Under the drive of the multi-cycle terahertz pulses, the terahertz transmissivity of the sample exhibits characteristic time-evolutions above and below T_c. Above T_c, the terahertz pump induces only thermal heating of the sample, manifested as a monotonic time-evolution of the terahertz transmissivity; below T_c, the terahertz pump induces a coherent oscillation of the superfluid at twice the driving frequency, originating from the driven Higgs oscillation.

图 6  NbN超导体的太赫兹三次谐波响应^[22]　(a) Matsunaga等^[22]使用多周期太赫兹驱动光入射在NbN超导薄膜上, 从样品透射的太赫兹光谱中可以观测到明显的三次谐波响应; (b) 超导序参量2Δ的温度依赖以及两倍驱动频率 (ω = 0.3, 0.5, 0.7 THz) 的对比; (c) 不同频率的太赫兹场驱动下三次谐波强度的温度依赖, 可以看出当满足2Δ(T) = 2ω条件时, Higgs模式的受驱振荡与驱动场发生共振, 导致三次谐波信号的发散

Figure 6.  Terahertz third harmonic generation (THG) response of NbN superconductors^[22]: (a) Using multi-cycle terahertz pulses to drive the NbN superconducting thin film, Matsunaga et al. ^[22] observe a clear terahertz THG response from the sample below T_c. (b) The temperature-dependent superconducting order parameter 2Δ compared to twice the terahertz driving frequency (ω = 0.3, 0.5, 0.7 THz). (c) The temperature dependence of the THG intensity under different terahertz driving frequencies. At temperatures that satisfy 2Δ(T) = 2ω, the driven Higgs oscillation becomes resonant with the driving field, leading to a divergence of the THG signal.

图 7  多周期太赫兹电场驱动Anderson pseudospin的进动过程^[22]　(a), (c) BCS超导态下, 费米面附近的$ ({\boldsymbol{k}}_{\uparrow }, -\boldsymbol{k}_{\downarrow})$ 电子对与$({\boldsymbol{k}}_{\uparrow }, -\boldsymbol{k}_{\downarrow})$ 空穴对相干叠加形成库珀对, 用pseudospin = 1/2 (–1/2)来定义该电子对的占据态(空态), 则BCS超导态可以表示为这两个pseudospin态的相干叠加. 投影到这两个pseudospin态定义的Bloch 球上, BCS基态对应Anderson pseudospin指向xy平面内; (b)对于半填充的金属而言, 费米面附近的$ ({\boldsymbol{k}}_{\uparrow}, -\boldsymbol{k}_{\downarrow}) $ 电子对与$({\boldsymbol{k}}_{\uparrow }, -\boldsymbol{k}_{\downarrow }) $ 空穴对无相干配对, 费米面由pseudospin = 1/2 态直接过渡到pseudospin = –1/2 态; (d) 在周期太赫兹场驱动下, Anderson pseudospin发生进动行为, 意味超导序参量发生涨落

Figure 7.  Precession of the Anderson pseudospin under a multi-cycle terahertz drive^[22]: (a), (c) In the BCS state, pairs of electrons at $ ({\boldsymbol{k}}_{\uparrow }, -\boldsymbol{k}_{\downarrow }) $ and pairs of holes at $ ({\boldsymbol{k}}_{\uparrow }, -\boldsymbol{k}_{\downarrow})$ are coherently superposed to form Cooper pairs. The occupancy of the pair of states at $ ({\boldsymbol{k}}_{\uparrow }, -\boldsymbol{k}_{\downarrow }) $ can be used to define a pseudospin = 1/2 state (occupied state) and a pseudospin = –1/2 state (empty state). Thus, the BCS state can be expressed as a superposition of the two pseudospin states. When projected to the Bloch sphere defined by the two states, the BCS state corresponds to Anderson pseudopsin pointing in the xy-plane. (b) In the normal metallic state, there is no coherence between the electrons at $({\boldsymbol{k}}_{\uparrow }, -\boldsymbol{k}_{\downarrow}) $ and the holes at $ ({\boldsymbol{k}}_{\uparrow }, -\boldsymbol{k}_{\downarrow }) $. Therefore, the Fermi surface represents a disrupt transition from the pseudospin = 1/2 state to the pseudospin = –1/2 state. (d) Under the periodic drive of a terahertz pulse, the Anderson pseudospin undergoes a precession, signifying collective fluctuations in the superconducting order parameter.

图 8  拉曼散射的微观机制　(a) 非共振光子-电子散射(也称作双光子散射、带内散射、diamagnetic scattering)诱导电子-空穴对密度涨落, 散射前后光子、电子的能量、动量均发生变化; (b) 共振光子-电子散射(也称作带间散射、paramagnetic scattering)过程中光子被电子完全吸收、产生电流、再辐射出; (c) 非共振弹性(Rayleigh)散射过程; (d) 共振弹性(Rayleigh)散射过程; (e) 太赫兹三次谐波生成过程; (f) 非共振拉曼/太赫兹三次谐波散射过程; (g) 两种玻色子共同参与的非共振拉曼/太赫兹三次谐波散射过程; 这里波浪线为光子传播子, 实线为电子传播子, 红色双虚线和黑色双波浪线为玻色子传播子

Figure 8.  Microscopic mechanisms for Raman scattering in solids. (a) Non-resonant photon-electron scattering (also known as two-photon scattering, intraband scattering, diamagnetic scattering) induces electron-hole density fluctuations. During the scattering, the energy and momentum of the photon and the electron both change. (b) In the resonant photon-electron scattering process (also known as interband scattering, paramagnetic scattering), a photon is absorbed by an electron and generates a current, which later re-radiates out a photon. (c) Microscopic process for non-resonant elastic (Rayleigh) scattering. (d) Microscopic process for resonant elastic (Rayleigh) scattering. (e) Microscopic process for terahertz third harmonic generation (THG). (f) Microscopic process for non-resonant Raman scattering/terahertz THG. (g) Microscopic process for non-resonant Raman scattering/terahertz THG involving two different bosons. Here wavy lines represent photon propagators, solid lines represent electron propagators, red double dotted lines and black double wavy lines represent boson propagators.

图 9  铜基高温超导体的太赫兹三次谐波响应^[50]　(a)—(c) 最佳掺杂La_2–xSr_xCuO₄, DyBa₂Cu₃O_7–x, YBa₂Cu₃O_7–x中太赫兹三次谐波强度的温度依赖; (d)—(f) 这3个样品中太赫兹三次谐波相位的温度依赖, 黑色虚线表示T_c, 红色实线表示三次谐波相位发生 –π跳跃的温度(T_π), 与(a)—(c)中振幅响应的局域最低点重合, 对应耦合双谐振子模型中的反共振(anti-resonance)或法诺共振(Fano resonance)现象, 插图为T_π 温度以上及以下两组三次谐波时域信号的对比

Figure 9.  Terahertz THG response of cuprate high-temperature superconductors^[50]: (a)–(c) Temperature dependence of the terahertz THG intensity from optimally-doped La_2–xSr_xCuO₄, DyBa₂Cu₃O_7–x, YBa₂Cu₃O_7–x thin films. (d)–(f) Temperature dependence of the terahertz THG phase from the three samples. Black dotted line denotes T_c. Red solid line marks the temperature (T_π) at which the THG phase shows an abrupt jump of –π, which coincides with the local minimum in the THG amplitude response in (a)–(c). These features are well-described by the anti-resonance of a coupled oscillators model or the Fano resonance. Insets show the comparison of two representative time-domain THG responses below and above T_π.

图 10  铜基高温超导体太赫兹三次谐波响应的磁场依赖^[28]　(a) 超导体Higgs模式与另一集体模式耦合示意图, 在T_π温度以下(以上)两者的受驱振荡展现出相同(相反)相位, 该相位翻转符合离散模式与连续激发态之间的法诺共振特征; (b) 一般谱学实验观测到的法诺共振体现在激发态关于不同驱动频率/探测波长(ω_drive)的非对称振幅/相位响应; 在太赫兹三次谐波实验中, 集体模式的共振频(ω_osc1)随温度发生变化, 而太赫兹驱动频率保持不变, 因此法诺共振体现在集体模的振幅/相位式关于ω_osc1 (或超导体温度)的非对称响应; (c), (e) 最佳掺杂及过掺杂La_2–xSr_xCuO₄样品在不同磁场环境下的太赫兹三次谐波强度的温度依赖; (d), (f) 相应的太赫兹三次谐波相位的磁场及温度依赖; 该实验中磁场平行于样品的c轴, 可以看出, 沿c轴方向的磁场抑制了三次谐波信号随温度展现出的法诺共振现象

Figure 10.  Magnetic field dependence of the terahertz THG response of cuprate high-temperature superconductors^[28]. (a) Illustration for the coupling of the Higgs mode to another collective mode. Below (above) T_π the driven motions of the two oscillators are in-phase (out-of-phase), consistent with the phase evolution of the Fano resonance between an underdamped mode and a continuous background. (b) In typical spectroscopy manifestations, Fano resonance manifests as an asymmetric profile in the amplitude and phase response of an oscillator as a function of the driving frequency (ω_drive) or probing wavelength. In the terahertz THG experiment, the resonance frequency (ω_osc1) of the collective mode is varied by varying temperature while the driving frequency is kept constant. In this case, the Fano resonance is manifested in the characteristic evolution of the amplitude and phase response of the collective mode as function of ω_osc1 or temperature. (c), (e) Temperature-dependent THG intensity from an optimally-doped and overdoped La_2–xSr_xCuO₄ thin film under different magnetic field strengths. (d), (f) Temperature-dependent THG phase of the two samples under different magnetic field strengths. The magnetic field is applied along the c-axis of the sample. It is clear that the c-axis magnetic field suppresses the Fano resonance manifested in the THG response as a function of temperature.

图 11  2H-NbSe₂的太赫兹三次谐波响应^[53]　(a) 超导及电荷密度波温区内的太赫兹三次谐波时域响应, 黑色箭头标注两个不同来源的三次谐波响应之间的干涉; (b) 太赫兹三次谐波强度的温度依赖, 在6 K附近, 可以看到两个不同来源的三次谐波响应之间的相消干涉导致三次谐波整体强度呈现局域最低点

Figure 11.  Terahertz THG response of 2H-NbSe₂^[53]: (a) Time-domain terahertz THG response of the sample throughout the superconducting and the charge-density-wave temperature range. Black arrow marks the interference between two distinct sources of THG. (b) Temperature dependence of the total terahertz THG intensity. Near 6 K, the destructive interference between the two sources of THG leads to a suppression of the total THG intensity. This in turn leads to a local minimum in the temperature dependence of the total THG intensity.

图 12  YBa₂Cu₃O_7–x的太赫兹三次谐波响应^[54]　(a) 最佳掺杂样品(T_c = 87.4 K)在85 K的太赫兹透射小波分析, 实验采用的驱动频率为0.5 THz, 从时域响应可以看出三次谐波(1.5 THz)响应展现出一前一后两个波包; (b) 样品在该温度下的三次谐波波形, 更加清晰地展示两部分波包; (c) 傅里叶分析显示频率相近的两个不同三次谐波信号成分; (d) 太赫兹三次谐波信号启始温度(T_THG)在YBa₂Cu₃O_7–x相图中的分布; T*为赝能隙启始温度, T_Kerr表示光学克尔实验报道的时间反演对称破缺温度, T_onset表示c轴红外光谱实验探测到的超导涨落启始温度

Figure 12.  Terahertz THG response of YBa₂Cu₃O_7–x^[54]. (a) The moving-wavelet Fourier analysis of the terahertz transmission of an optimally-doped sample (T_c = 87.4 K) under 0.5 THz periodic drive at 85 K. The time-domain THG response exhibits two individual wavelets separated in time. (b) The time-domain THG waveform. (c) Fourier analysis of the time-domain waveform shows two distinct components at the THG frequency. (d) The onset temperature for the terahertz THG response (T_THG) in the YBa₂Cu₃O_7–x phase diagram. T* denotes the onset temperature for the pseudogap. T_Kerr denotes the onset temperature for time-reversal symmetry breaking as reported by optical Kerr measurements. T_onset denotes the onset temperature for superconducting fluctuations as determined by c-axis infrared ellipsometry measurements.

图 13  交织序的研究策略: 平衡态与非平衡态　(a)对于交织序(如超导与电荷密度波), 过往研究往往依赖于平衡态物性表征: 譬如, 通过施加外界磁场、应力场抑制超导态, 再利用X射线衍射观察电荷密度波的关联长度、临界温度, 从而得出两个序参量之间相互独立、竞争或协作的结论; 除了提供相互作用以及其类型(吸引、排斥)的实验证据, 该手段无法提供更加具体、微观的耦合信息; 这类平衡态研究策略可以类比为拉动耦合双谐振子中的一个, 由于耦合弹簧的存在, 另一个谐振子也相应发生位移; 通过观察两个谐振子的静态位移, 可以获知两个谐振子之间存在耦合, 却无法得知耦合的具体形式、能量尺度等物理信息. (b) 当我们动态驱动这两个耦合谐振子中的一个时, 两个谐振子之间的能量、动量传递随弹簧系数的变化, 展现不同的动力学过程; 因此通过观察两个谐振子的运动, 可以获取耦合弹簧的具体参数信息; 同理对于交织序, 通过动态观测耦合序参量的动力学过程, 我们有望得出更加微观和具象的耦合机制、能量尺度等物理信息

Figure 13.  Experimental approaches for investigating intertwined orders: equilibrium versus non-equilibrium. (a) For investigating the relationship between intertwined orders (e.g. superconductivity and charge-density-wave), past studies have focused on their equilibrium manifestations. For example, by applying an external magnetic field or a strain field to suppress superconductivity, one may study the resulting changes to the charge-density-wave correlation length or ordering temperature using X-ray diffraction. While such an approach provides evidence for the coupling between the two orders and the type of such interactions (i.e. competing or co-operative), it does not provide further microscopic information about the interaction mechanism. Such an equilibrium strategy can be compared to displacing a coupled harmonic oscillator. Due to the presence of the coupling spring, the other oscillator is also displaced. By observing the static displacements of the two oscillators, one may infer that they are coupled. However, no more details about the coupling spring can be learnt using this approach. (b) When we dynamically drive one of the coupled oscillators, the energy and momentum transfer between the two oscillators depends sensitively on the coupling spring constant. Therefore, by studying the dynamics of both oscillators, we may obtain precise knowledge about the coupling spring constant. In a similar spirit, by studying the dynamics of the individual orders that are intertwined, we may obtain microscopic information about the energy-scale and the mechanism

图 14  二维光谱技术, 对于存在多自由度的物理体系而言, 不同自由度(包括它们的激发态)之间是否存在耦合, 是理解这类体系物性、相变的关键信息; 传统线性谱学技术(如吸收谱、反射谱、拉曼散射等)对于表征不同自由度的激发态已较为成熟, 然而这类谱学技术对于表征、验证不同自由度之间的相互作用却难以提供有效数据; 在这方面, 二维光谱技术具有巨大潜力, 该技术采用多束宽谱脉冲, 对样品中不同自由度的动力学过程进行探测; 第1束脉冲同时产生多个激发态: (中图)如果这些激发态源自同一个基态(同一个自由度), 或者(右图)这些激发态源自不同自由度但是它们之间存在耦合, 则这些激发态之间会产生相干性; 在一定时间延时τ后, 第2束以及第3束脉冲照射在样品上, 诱导该相干态的演化(rephasing或non-rephasing); 最后, 该相干态通过辐射, 使各自由度回落到基态. 通过采集不同延时τ下的辐射光谱, 并将其对τ做傅里叶变化, 可以获得以上示意的二维光谱信息, 二维光谱对角线上的峰对应各激发态的能量, 非对角线上的峰表示两个激发态之间的相干性(具体可以解读为源自同一基态或者存在能量弛豫通道等), 相较于线性谱学技术, 二维光谱可以提供更加直观、具体的耦合信息^[63]

Figure 14.  2D spectroscopy, for systems exhibiting many different degrees of freedom (DOFs), the coupling between these DOFs (including their excitations) is often the key to understanding the physical properties or phase transitions of these systems. While conventional linear spectroscopy techniques (e.g. absorption, reflection, Raman scattering) are good at capturing the excitations of different types of DOFs, evidencing or characterizing the interactions between different DOFs proves to be difficult with current techniques. In this regard, 2D spectroscopy has demonstrated significant potential. 2D spectroscopy uses several broadband pulses to study the dynamics of multiple DOFs. The first pulse induces multiple excitations: (middle panel) if the excitations derive from the same ground state (i.e. the same DOF), or (right panel) if they derive from different DOFs but there is a coupling in-between, then a coherence between these excitations will be established. After a variable time-delay τ, the second and the third pulse interact with the sample, inducing an evolution of the coherent state (e.g. rephasing or non-rephasing). Finally, the coherent state radiates and the system falls back to the ground state. By sampling the radiation spectrum at each τ, and Fourier transforming these results with respect to τ, one obtains a 2D spectrum as illustrated above. The peaks lying on the diagonal of the 2D spectrum represents individual excitations. The peaks lying on the off-diagonal represents coherence between two excitations (they encode information such as a common ground-state for the two excitations or a relaxation channel between the two excitations). Compared to traditional linear spectroscopy techniques, 2D spectroscopy provides more direct and detailed information about the interaction between different DOFs^[63].