1. Introduction

It is well known that optical spectroscopy of materials has provided invaluable information about electrons, phonons and their interactions. Further insights into nonequilibrium and transport properties can be obtained by using ultrafast laser pulses. In general, these optical properties are characterized by the spatially uniform phase and polarization of the electric field in the temporal and/or frequency domains. On the other hand, recent progress in singular optics has enabled generation of spatially varying electric field including wavefront singularities. Typical examples are the phase [1] and polarization vortices [2]. An optical phase/polarization vortex has a wavefront distribution of phase/field vectors that rotates around a vortex axis over an angle proportional to the (polarization) topological charge (the winding number of phase/electric vector field around the singularity point). These characteristics have recently attracted considerable interest because of their increasing applications in a wide variety of fields, such as optical manipulation [3, 4], imaging [5, 6], geometrical diagnostics [7, 8], and communication and information processing [9, 10]. However, their potential use for spectroscopic measurements, especially on electronic materials, is still in an early developmental stage [11].

What we can obtain by using the optical vortices for electronic excitations is the information on closed-loop phase and/or polarization coherence of electrons (and phonons) that plays a crucial role in ring-shaped materials (structures). A typical example is the quasi-one-dimensional (1D) transition metal (M) chalcogenides (X) of the type MX3 that are known to have various types of closed-loop crystals [12]. Their 1D nature (Peierls instability) gives rise to charge-density-wave (CDW) phase transitions driven by electron-phonon interactions. Since the correlation length of CDWs in MX₃ is on the order of several µm [13], the coherent CDWs within a closed-loop 1D chain provides an opportunity to investigate the characteristic quantum effects such as the Aharonov-Bohm effect [14].

In this paper, we demonstrate a global evaluation of closed-loop CDWs in a ring-shaped quasi-1D crystal using polarization vortex pulses. We first introduce how to generate the polarization vortex pulse (Sect. 2), and then describe the quasi-1D CDW samples with their underlying physics (Sect. 3). In Sect. 4, we present an overview of our experimental approach using a pump-probe measurement with polarization vortex pulses, and then compare the single particle (SP) relaxation observed between azimuthal and radial polarization excitations (Sect. 5). We also compare the results with those in the whisker sample, and show that the ring-shaped CDWs exhibit a well-defined closed-loop polarization.

2. Generation of the polarization vortex

For the global measurement of the closed-loop 1D electrons, we utilize the polarization vortex pulse. The schematic polarization distributions of the vortex beam are shown in upper part of Fig. 1 (a). The left and right sides of the figures show the radial and azimuthal polarization vortices, respectively. The conventional distribution of the linearly polarized beam is also shown for comparison. The field distribution in conventional beams is spatially uniform while the field of the vortex beam varies around a singular point. The polarization vortices can be directly obtained from a lasing mode using a combination of thermal lensing and birefringence effects [15], and also be delivered from the conventional beam by using polarizing computer generated holograms [16] and liquid crystal polarizers [17]. In this study, we employed a conversion method using a radial polarizer (RP) as a polarization converter [18, 19]. This method is advantageous for generating the ultrashort vortex pulse owing to the spatial-dispersion-free configuration described in detail below.

The optical setup which converts the linearly polarized Gaussian beam to the polarization vortex is shown in Fig. 1 (b). This setup consists of a polarizer (P), a quarter wave plate (QWP), a RP, and a pair of half-wave plates (HWPs). The RP is made of a photonic crystal and purchased from Photonic Lattice Inc. The linearly polarized beam is converted into the circularly polarized one by the P and QWP. Then, we obtain the beam with radial polarization after passing through the RP, where the RP is considered to be a set of polarizers whose polarization axes continuously rotate about the optical axis. Let us consider the conversion process using Jones vectors and matrices. The matrix of RP is yielded by $\left(\begin{array}{cc}{\cos }^2\varphi & \frac{1}{2}\sin 2\varphi \\ \frac{1}{2}\sin 2\varphi & {\sin }^2\varphi \end{array}\right),$, where ϕ is the azimuthal angle in the beam cross section. The circularly polarized beam expressed by the vector $\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ i\end{array}\right)$ is then converted to $\frac{1}{\sqrt{2}}{e}^{\mathrm{i\varphi }}\left(\begin{array}{c}\cos \varphi \\ \sin \varphi \end{array}\right)$ indicating a radially polarized beam. The local polarization is linear but its orientation varies with respect to the azimuthal coordinate ϕ. Such a geometry thus allows to excite the topologically polarized electrons in materials. On the other hand, the term of exp(iϕ) constitutes a phase variation depending on ϕ, which characterizes the beam as an optical phase vortex as well as a polarization vortex. The rotation of the local polarization directions about themselves can be realized by a pair of achromatic HWPs (HWP1 and HWP2) in which the axes of HWP1 can be set to be in an arbitrary direction. Using a relative rotation angle α between two HWPs, the combination matrix is given by $\left(\begin{array}{cc}\cos 2\alpha & -\sin 2\alpha \\ \sin 2\alpha & \cos 2\alpha \end{array}\right)$. The Jones vector of the passed beam is thus given by ${\frac{1}{\sqrt{2}}e}^{\mathrm{i\varphi }}\left(\begin{array}{c}\cos \left(2\alpha +\varphi \right)\\ \sin \left(2\alpha +\varphi \right)\end{array}\right)$. As a result, we can select the azimuthal variation of the local vector fields by changing α. In the case of α=π/4 (0), we obtain the polarization distribution of the beam as $\frac{1}{\sqrt{2}}{e}^{\mathrm{i\varphi }}\left(\begin{array}{cc}-\sin \varphi & \left(\cos \varphi \right)\\ \cos \varphi & \left(\sin \varphi \right)\end{array}\right),$, indicating an azimuthally (radially) polarized beam. It should be emphasized that Jones matrices of components used in our system are independent of wavelength, resulting in capability of spatial-dispersion-free optical vortex generation even with an ultrabroadband pulse.

 

Fig. 1. (a) Polarization distribution at the beam cross section of the polarization vortex (upper). α is relative angle between the axes of the half-wave plates in (b). Polarization distribution of the linear polarization are also shown for comparison (lower). (b) Optical setup for generating polarization vortex. (P: polarizer, QWP: quarter-wave plate, RP: radial polarizer, HWPs: half-wave plates)

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3. Sample properties

3.1. Characteristics of CDW

In this subsection, we briefly introduce characteristic properties of CDW. For more comprehensive and detailed reviews on CDW, see Refs. [20] and [21]. A CDW is a modulation of the electron density associated with a lattice modulation of same periodicity. In 1930, R. Peierls pointed out that these modulations could appear in quasi-1D conductors at low temperatures. Generally, electronic band gaps in solids arise from a Fourier component of the lattice potential. In quasi-1D metal consisting of chains of equally spaced atoms with the lattice constant a (Fig. 2(a) left), a modulation of the lattice with wavelength λ=π/k_F can produce SP gaps (2Δ_g) at ±k_F by reducing the conduction electron energy below E_F, where E_F and k_F are the Fermi energy and wavevector, respectively (Fig. 2(a) right). At low temperatures, the gain in the conduction electron energy overcomes the energy cost of the lattice deformation. As a result, quasi-1D metals undergo a phase transition (metal to insulator transition) when cooled below a certain temperature (T_c). This is a so-called CDW phase transition. Unlike normal insulators (and semiconductors), the ground states of CDW are characterized by two types of collective excitations: phase mode (PM) and amplitude mode (AM). Both modes are schematically illustrated in the box of Fig. 2(a). The PM determines the position of the CDW relative to the underlying lattice, and the AM determines the amplitude of the CDW. It should be noted that in realistic quasi-1D metals the three-dimensional (3D) correlation of the 1D CDWs is essential to reduce their fluctuations. This means that the 3D ordered CDWs exist below T_c.

 

Fig. 2. (a) The band structures and the electron-density distributions on the lattice above (left) and below (right) T_c. A schematic illustration of collective excitations is shown in the box. PM and AM are the displacive and amplitude modes of CDW, respectively. The electron micrographs of the (b) whisker and (c) ring shaped NbSe3 crystals. The red arrows indicate the directions of the conduction b axis of the crystals.

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3.2. Ring-shaped NbSe3 crystal

The samples studied here is NbSe3, which is one of the MX₃ family and is the most extensively studied quasi-1D compound. This compound undergoes CDW phase transitions at T _c1=145 K and T _c2=59 K [22]. These two transitions are known to take place on two of three types of chains with different nesting conditions in k-space [23]. The other chains remain metallic down to the lowest temperatures. The basic properties of NbSe₃ including its 1D nature have been well studied by various conductivity [22], X-ray [24], and optical measurements [25, 26]. The local CDW properties in real-space have been investigated by scanning tunneling microscopy and spectroscopy [27]. Since the CDW with long-range-order requires 3D correlation of electron density between 1D chains, their coherent dynamics attracts considerable interest and has been studied using time-resolved measurements [28, 29], which will be described in the next section.

Figure 2(b) and (c) show the scanning electron micrographs of the whisker and ring samples used in this study, respectively. The crystals were grown by the chemical vapor transport method and usually show a whisker structure. Under controlled conditions, thin whiskers naturally form closed-loop crystals by bending and joining [12]. Several ring crystals are stacked layer by layer, and form a disk-like structures (Fig. 2(c)). It should be noted that T_c’s (T _c1=140.8K and T _c2=57.4 K) in the ring are similar to those in the whisker [30]. The whisker crystal has a length of a few mm along the conducting b axis (chain axis) and width of 50 µm. The disk-like ring crystal with a diameter of ~50 µm has a bending b axis along the azimuthal direction in the disk. The circumference of the center hole of the disk is around 10 µm, which is comparable to the correlation length of CDWs (ξ‖_b >2.5µm [13]) in this compound. The details of the growth mechanism for the ring and its structural analysis based on X-ray diffraction were described elsewhere [31].

3.3. Nonequilibrium SP dynamics in optical pump-probe spectroscopy

When we investigate the globally correlated electron systems such as closed-loop CDWs, noncontact and non-destructive measurements are preferable. Therefore, it is advantageous to probe the CDW dynamics with photoexcitation. A pump-probe method is usually employed in this kind of time-resolved experiments, where optical responses of photoexcited electrons are traced by a probe pulse immediately after an intense pump pulse with a delay time between the two pulses. The pump pulse is utilized to excite the electronic system to a nonequilibrium state while the probe detects changes of the system through the optical properties, such as reflection and transmission. It is significant to note that the probe polarization determines the transition probability of the nonequilibrium electrons, especially in narrow gap systems [32].

Recently, comprehensive experimental studies of various correlated electron systems involving superconductors have established a successful theoretical model for the ultrafast optical response of phase transition materials[33, 34], which have provided information that complements the transport and steady-state optical measurements. Here, we describe the typical transient optical responses in CDW conductors. As mentioned in Sect. 3.1, the 3D-correlation of the CDWs forms a gap around E_F below T_c. Therefore one can treat the SP dynamics in CDW conductors as those in semiconductors, namely, SP relaxation time (τ_sp) reflects the relaxation across the band gap (while τ_sp becomes instantaneous above T_c, reflecting the intraband relaxation in the metallic phase). Another characteristic transient in the CDW conductors is a coherent oscillation that evidences the collective excitations of the CDW (see the box of Fig. 2(a)). The instantaneously photoexcited SPs change the CDW’s equilibrium distributions and result in starting the collective motion. This occurs in the same manner as displacive excitation of coherent phonons [28, 29]. In our previous measurements, a coherent oscillation with ν_AM~1.1 THz was observed in NbSe₃ and was ascribed to the AM of the CDW [28, 29].

4. Experimental setup

The measurements were carried out using a two-color pump-probe setup (Fig. 3), in which the coaxial configuration between the pump and probe beams allows to excite the precise position of the crystals even with the vortex beam. This optical configuration is also advantageous for the polarization spectroscopy, in which the distinction between the probe reflectance and the scattered pump was done by the dichroic beam splitter. The light source that we used was a mode-locked 100 fs Ti-sapphire laser oscillator operating at a repetition rate of about 76 MHz, which synchronously pumped an optical parametric oscillator (OPO). The probe pulse was extracted from the fundamental at ~1.5 eV (wavelength ~800 nm), and the pump was delivered by the OPO at ~1.1 eV (wavelength ~1160 nm). Both energies are much higher than the gap energies of the CDWin NbSe3. It is important to note that the probe energy that exceeds the pump energy allows to avoid contributions from the higher excited states. In another word, this two-color combination enhances the optical response associated with the change of the conduction bandedge (i.e., CDW gap formation below T_c). Also note that transient signal was independent of the pump energy and polarization [35].

The linearly polarized probe beam with a Gaussian spatial profile was converted into polarization vortex by passing through the vortex generation system described in Sect. 2. The pump and probe beams were combined by a dichroic mirror and focused by using an achromatic objective lens for the near-infrared region with a nominal magnification factor of 10, a working distance of 34 mm, and a numerical aperture of 0.28. The focal spots of the two laser beams were adjusted to obtain a complete spatial overlap on the sample surface. Especially for the photoexcitation on the ring-shaped crystal, we carefully adjusted the vortex axis to place the center hole of the sample. The overlap between two beams and the position on the sample surface were monitored using a charge coupled device camera, and they were kept at a fixed position during the measurements. The samples were mounted on a cold finger of the helium-flow cryostat and all measurements are performed at ~20 K.

To measure the transient response, the pump pulse was mechanically delayed with respect to the probe pulses using a translation stage. The pump pulse was chopped at a frequency of 2 kHz and the reflectivity change ΔR of the probe was detected with a lock-in amplifier. The cross-correlation between the pump and probe pulses in a nonlinear optical crystal showed a temporal resolution of ~200 fs. The total fluence was kept lower than 50 µJ/cm², and the pump/probe ratio was 2:1. The steady state heating caused by the laser was accounted for by measuring the excitation-power dependence of the transient signal which shows anomalies near the CDW transition temperatures [28, 29].

 

Fig. 3. Experimental setup for the two-color pump-probe measurement. The polarization vortex generation system is inserted on the path of the probe pulse.

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5. Results and discussions

The left row of Fig. 4 shows transient ΔR measured by the probe pulse with various polarization distributions and angles. All data were plotted as a function of delay time between pump and probe pulses. First, we overview the result for the whisker sample using linearly polarized pulses (Fig. 4(a)). The polarization angle θ is defined from the b axis (chain axis) of the whisker. As we mentioned in the Sect. 3.3, the signal below T_c (below T _c2) is characterized by two features; exponential decay with a decay time of ~1 ps and a damped sinusoidal oscillation with a frequency of 1.12 THz. We thus attribute the exponential part (ΔR_sp) to the SP relaxation across the CDW gap and the oscillation part (ΔR_AM) to the AM of the CDW. The exponential decay exhibits a significant polarization anisotropy, which can be associated with the 1D nature of the sample. This SP component was completely absent when the polarization is perpendicular to the b axis (θ=90°). In contrast, the oscillation component is nearly independent of the polarization θ, and is consistent with the fact that the AM is the fully symmetry A1 mode [32]. For the estimation of ΔR_sp and ΔR_AM, we use the following fitting function:

where we neglect the damping of the AM that is much longer than the time scales observed here.

Next, we consider the results for the ring sample using the polarization vortex pulses (Fig. 4(b)), where the θ′ indicates the rotation angle of the local field polarization; θ′=0° and 90° correspond to azimuthal and radial polarizations, respectively. By using such a topological polarization coordinate, the time evolution and its polarization dependence are related with those reported above in the whisker. Roughly speaking, Fig. 4(b) is in good agreement with Fig. 4(a); the SP component decreases with increasing θ′ and vanishes at θ′=90° while the oscillation component is constant with θ′. It is significant to remember that the relative position between polarization vortex and center hole of the ring is critical for the ΔR polarization. We carefully adjusted the optical vortex onto the center hole of the ring. In order to quantitatively compare the data sets, we plot the polarization of the SP component ΔR_SP as a function of θ′(θ) in the right row of Fig. 4, where we estimate the SP polarization from the ratio between ΔR_sp and ΔR_AM in Eq.(1). Because of the fully symmetry of the AM oscillation, this method allows to cancel the remnant polarization of the experimental setup, thus being more accurate than the comparison of the absolute values of ΔR. The results in both data (Fig. 4(d) and (e)) clearly indicate the polarization of the SP component whose magnitudes $\frac{I_{0°}-I_{90°}}{I_{0°}+I_{90°}}$ are near unity. Therefore we conclude that the ring sample exhibits a well-defined azimuthal SP polarization. Although this conclusion is easily deduced from, for example, the geometrical structure, our measurement verifies that the CDW electrons in the ring maintain their 1D character, and so that the technique using polarization vortex has the ability to evaluate the electron dynamics in closed-loop systems that plays an important role in the quantum correlation phenomena.

Finally, to enhance the SP polarization presented above, we also show the results for the whisker using the polarization vortex pulses (bottom part of Fig. 4). In contrast to Fig. 4(a) and (b), the ΔR in (c) exhibits no polarization. The constant SP polarization is also shown in Fig. 4(f), where we can verify that the SP magnitudes are intermediate between the oscillatory magnitudes observed for the above two cases (Fig. 4 (d) and (e)). This means that the polarization vortex with radial symmetry selects the local projection of the SP polarization along the azimuthally distributed θ′. In another words, the photoexcited 1D electrons act as a linear polarizer to evaluate the radial polarization distribution of the vortex pulses. The standard deviation of the magnitudes in Fig. 4(f) is 1.03, showing the high quality of the polarization vortex achieved in our time-resolved measurement.

 

Fig. 4. Transient ΔR for (a) a whisker crystal measured by the linear polarization with various θ, (b) a ring crystal measured by the polarization vortex with various θ′, and (c) a whisker crystal measured by the polarization vortex with various θ′. (d)–(e) show the corresponding polarization angle (θ or θ′) dependence of the SP polarization in (a)–(c), respectively, where the polarization is deduced from ΔR_SP normalized by the symmetric ΔR_AM.

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6. Summary

In summary, we demonstrated the global evaluation for the topologically polarized SP dynamics in the CDW condensate using polarization vortex pulses. For the measurement, we realized a spatial-dispersion-free vortex pulse generation with RP in a two-color pump-probe setup. In the ring crystal, the transient ΔR probed by the polarization vortex shows a large anisotropy depending on the azimuthal polarization distributions, indicating that the photoexcited electrons are polarized azimuthally and globally. These topologically-polarized electrons cannot be defined globally using the conventional photoexcitation (uniformly polarized optical pulses). Our demonstration thus provides a new spectroscopic technique for diagnosing the closed-loop coherence of carriers in solids, which should play an important role in the quantum correlation characteristics such as the Aharonov-Bohm effect.