Introduction

Floquet theory suggests that the application of a periodic perturbation with frequency \(\varOmega\) to an electronic system creates new states at the energy positions separated from an original electronic level by integral multiples of \({\hslash}{\varOmega}\)1,2,3. These states are called “dressed states”. A prototypical example is a photon-dressed state4,5,6,7,8,9,10, which is created by the interaction of an electronic state with an oscillatory electric field of light. In the case that a coherent light–electron interaction occurs, ultrafast optical switching and wavelength conversion associated with photon-dressed states can be achieved based on optical Stark effects4,7,11,12,13,14 and sideband emissions5,8,10,15,16,17,18,19,20,21, respectively. Using a terahertz or mid-infrared pump pulse, sub-cycle detection of photon-dressed states is possible8,19, although it is generally difficult because the photon-dressed states disappear within the duration of the pump pulse.

Several theoretical studies indicate that a dressed state can also be created by applying a periodic potential of phonons to an electronic system via electron–phonon coupling3,22,23. Such a dressed state, referred to as a “phonon-dressed state”, can exist during the lifetime of the phonon, providing an opportunity to investigate the formation and annihilation dynamics of optically allowed dressed states by detecting optical radiation from each dressed state in the time domain. An effective way to produce a phonon-dressed state is to resonantly excite an infrared-active phonon with an intense laser pulse. In Fig. 1a, we exemplify optical radiations from phonon-dressed states. Here, |φa〉 (|φf〉) is an original one-photon-allowed (forbidden) state. |φa, n〉 and |φf, n〉 show the n-phonon-dressed states of |φa〉 and \({{{{{\rm{|}}}}}}\)φf〉, respectively. When the excited phonon is infrared-active, \({|}{\varphi }_{{{{{\rm{a}}}}}},{{{{{\rm{even}}}}}}\; {\rm {number}}{{\rangle }}\) and \({|}{\varphi }_{{{{{\rm{f}}}}}},{{{{{\rm{odd}}}}}}\; {{{{{\rm{number}}}}}}{{\rangle}}\) are one-photon-allowed states, and thus radiative transitions from those states can occur as shown by the colored arrows in Fig. 1a.

Fig. 1: Observation of phonon-dressed states in a dimerized organic Mott insulator of K-TCNQ.
figure 1

a Schematic energy levels of phonon-dressed states. |φa〉 (|φf〉) is an originally one-photon-allowed (forbidden) excited state and |GS〉 is a ground state. |φa,f, n〉 is an \(n\)-phonon-dressed state. Colored arrows show light emissions from phonon-dressed states. b Schematic of the mid-infrared (MIR)-pump visible (VIS)-reflectivity-probe measurement. c A 1D dimerized TCNQ− stack in K-TCNQ. d Schematic of an out-of-phase intramolecular vibration (\({a}_{{{{{{\rm{g}}}}}}}\) mode) in a dimer, which is infrared-activated via the electron-intramolecular-vibration (EMV) coupling. This out-of-phase vibration modulates singly occupied molecular orbital (SOMO) levels and induces a partial charge transfer \(\left(-\triangle \rho \, < \, 0\right)\) between two molecules in a dimer. HOMO represents the highest occupied molecular orbital.

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Here, we report the sub-cycle observation of phonon-dressed states using a pump-probe technique. There are two favorable conditions for creating phonon-dressed states: a phonon mode with a large oscillator strength and a strong electron–phonon coupling, which cause effective mixing of electron wavefunctions in the original excited states. Considering these conditions, we choose the organic molecular compound potassium-tetracyanoquinodimethane (K-TCNQ)24. In this compound, totally symmetric (ag) modes of intramolecular vibrations of each molecule are strongly coupled to intermolecular charge transfer, which results in infrared-activated with large oscillator strength. By resonantly exciting these phonons with a femtosecond laser pulse, we observe sideband emission ascribed to the formation of phonon-dressed states associated with the intramolecular transition using the sub-cycle spectroscopy (Fig. 1b). This achievement of the ultrafast creation and detection of phonon-dressed states opens a possibility for Floquet engineering with phonon degrees of freedom.

Results

Steady-state polarized reflectivity spectra

In K-TCNQ, an electron is transferred from K to TCNQ, and TCNQ− anions are one-dimensionally stacked along the a-axis as shown in Fig. 1c. The \(\pi\)-electron band consisting of a singly occupied molecular orbital (SOMO) is half-filled (Fig. 1d). Because of the large on-site Coulomb repulsion energy \(U\) on each TCNQ− overcoming the electron transfer integral \(t\) between two neighboring molecules, this compound behaves as a Mott insulator25. Spins (S = 1/2) of electrons localized on each molecule form a one-dimensional (1D) antiferromagnetic spin system. Due to the spin-Peierls mechanism, 1D molecular stacks are dimerized even at room temperature24,25 (Fig. 1c).

To excite the phonons, we focus on the ag modes of intramolecular vibrations of TCNQ, which are infrared-inactive in an isolated molecule, but activated in a TCNQ− dimer. This is because each ag mode modulates the energy of the SOMO of TCNQ− via electron-intramolecular-vibration (EMV) coupling, and an out-of-phase combination of ag modes of two molecules gives rise to an intermolecular charge transfer within the dimer, generating a large dipole moment along the a-axis26 as illustrated in Fig. 1d.

In Fig. 2a, we show the steady-state reflectivity (\(R\)) spectra with electric fields \(E\) of light parallel (//) and perpendicular (⊥) to the a-axis. The broad structure around 1 eV for \(E\)//a is a charge-transfer transition, (TCNQ−, TCNQ−) → (TCNQ0, TCNQ2−)27. The structure around 2 eV for \(E\)⊥a is the intramolecular transition from highest occupied molecular orbital (HOMO) to SOMO in TCNQ− (see the inset)27,28, which creates a Frenkel exciton. Two sharp structures observed at the frequencies of \({\varOmega }_{1}\) = 1178 cm–1 and \({\varOmega }_{2}\) = 1330 cm–1 in the mid-infrared region are ascribed to the above-mentioned ag modes of TCNQ− (modes 1 and 2, respectively) activated via EMV coupling29,30. The atomic motions of each mode obtained from ab initio calculations are illustrated in Fig. 2c (see Supplementary Note 1).

Fig. 2: Optical responses associated with phonon-dressed states generated by a mid-infrared pulse in K-TCNQ.
figure 2

a Polarized reflectivity spectra of K-TCNQ. SOMO and HOMO represent a singly occupied molecular orbital and a highest occupied molecular orbital, respectively. b The spectra of mid-infrared pulses for pump I and pump II, and of a visible-probe pulse. c Schematics of atomic motions in \({a}_{{{{{{\rm{g}}}}}}}{{\nu }}_{5}\) mode with \({\varOmega }_{1}\) = 1178 cm−1 and \({a}_{{{{{{\rm{g}}}}}}}{{\nu }}_{4}\) mode with \({\varOmega }_{2}\) = 1330 cm−1. d The intensity envelope of the visible-probe pulse. e–j The electric-field waveform of the mid-infrared pump pulse (e, h), the time characteristic of the optical response, \(\Delta {I}_{{{{{{\rm{R}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\), at 2.07 eV (f, i), and the time characteristic of the high-frequency oscillatory component (g, j) for (e, f) off-resonant excitation (pump I) and (h–j) resonant excitation of \({a}_{{{{{{\rm{g}}}}}}}\) mode vibrations (pump II). The black broken line shown in j indicates the fitting curve.

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Mid-infrared pump pulse and visible-probe pulse

Figure 2b shows the spectra of the phase-stable mid-infrared pump pulses in two conditions, pump I and pump II; pump II, with a central frequency of 1211 cm–1, resonantly excites two ag modes, whereas pump I, with a central frequency of 970 cm–1, does not. The electric-field waveforms of mid-infrared pulses for pump I and pump II are shown in Fig. 2e, h, respectively.

To detect phonon-dressed states, we use an ultrashort visible-probe pulse polarized perpendicular to the a-axis. Its spectrum, drawn in iridescent in Fig. 2b, covers the intramolecular transition of TCNQ−. We aim to detect optical radiation from its phonon-dressed states in the reflection configuration as illustrated in Fig. 1b. Figure 2d shows a temporal profile of the probe pulse intensity. Its width is approximately 8 fs, much shorter than the periods of ag modes 1 (28 fs) and 2 (25 fs). This condition facilitates sub-cycle detection of optical radiation.

Sub-cycle detection of oscillatory signals in optical radiation from phonon-dressed states

In Fig. 2f, i, we show the time profiles of optical radiation, \(\Delta {I}_{{{{{{\rm{R}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\), at 2.07 eV by the mid-infrared pump pulses for the conditions of pumps I and II, which were measured with the pump-probe experiments illustrated in Fig. 1b. \({t}_{{{{{{\rm{d}}}}}}}\) is the delay time of the probe pulse relative to the pump pulse. The electric-field amplitude of the mid-infrared pump pulses (Fig. 2e, h) is 6 MV cm−1 in common. Here, \({I}_{{{{{{\rm{R}}}}}}}\) is the intensity of the reflected probe pulse without a pump pulse, and \(\Delta {I}_{{{{{{\rm{R}}}}}}}({t}_{{{{{{\rm{d}}}}}}})\) is its intensity change with a pump pulse, which includes radiation from dressed states. In the off-resonant phonon excitation with pump I (Fig. 2e), \(\Delta {I}_{{{{{{\rm{R}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\) consists of a background component coincident with the envelope of the square of mid-infrared electric fields and a high-frequency oscillatory component. The latter is shown in Fig. 2g. Its frequency (2001 cm–1) is almost equal to double the central frequency of the pump pulse (970 cm–1). No delayed responses were observed. These ultrafast responses are ascribed to photon-dressed states.

In the resonant phonon excitation with pump II (Fig. 2h), \(\Delta {I}_{{{{{{\rm{R}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\) includes at least three types of long-lived components that survive after the pump pulse diminishes: the decrease in the reflected probe pulse with a long decay time, the low-frequency oscillatory component, \(\Delta {I}_{{{{{{\rm{LF}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\), with a period of ∼200 fs, and the high-frequency oscillatory component, \(\Delta {I}_{{{{{{\rm{HF}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\), with a period of ∼14 fs. The extracted time profile of \(\Delta {I}_{{{{{{\rm{HF}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\) is shown in Fig. 2j, in which a prominent waviness is seen. This indicates that \(\Delta {I}_{{{{{{\rm{HF}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\) consists at least of two oscillations with similar frequencies.

To clarify the origin of the high- and low-frequency oscillatory components, we perform Fourier transformations of \(\Delta {I}_{{{{{{\rm{R}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\). The obtained power spectra are shown by red circles in Fig. 3a, b. Figure 3d exhibits steady-state absorption (\({\varepsilon }_{2}\)) for ag modes 1 and 2, while its horizontal scale is half that in Fig. 3b, c. The power spectrum in the mid-infrared region shows three oscillations labeled as A, B, and C, the frequencies of which are \({\varOmega }_{{{{{{\rm{A}}}}}}}\) = 2370 cm–1, \({\varOmega }_{{{{{{\rm{B}}}}}}}\) = 2527 cm–1, and \({\varOmega }_{{{{{{\rm{C}}}}}}}\) = 2685 cm–1, respectively. \({\varOmega }_{{{{{{\rm{A}}}}}}}\), \({\varOmega }_{{{{{{\rm{B}}}}}}}\), and \({\varOmega }_{{{{{{\rm{C}}}}}}}\) correspond to the sum of the frequencies of modes 1 and 2, which are \(2{\varOmega }_{1}\) = 2356 cm–1, \({\varOmega }_{1}+{\varOmega }_{2}\) = 2508 cm–1, and \(2{\varOmega }_{2}\) = 2660 cm–1, respectively. The power spectrum in the far-infrared region shows an oscillation labeled as D. Its frequency \({\varOmega }_{{{{{{\rm{D}}}}}}}\) = 158 cm–1 is almost equal to the subtraction between the frequencies of modes 1 and 2, \({\varOmega }_{2}-{\varOmega }_{1}\) = 152 cm–1. These results indicate that the high- and low-frequency oscillations are caused by the second-order terms of the ag-mode vibrations. The first-order terms become zero because K-TCNQ is invariant for the a-axis inversion.

Fig. 3: High- and low-frequency oscillatory components in optical responses associated with phonon-dressed states in K-TCNQ.
figure 3

a The Fourier power spectrum of the low-frequency oscillation D in \(\Delta {I}_{{{{{{\rm{R}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\) (red circles) and the calculated oscillation obtained from the fitting analysis of the time characteristic of \(\Delta {I}_{{{{{{\rm{LF}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\) (the black line). b The Fourier power spectrum of the high-frequency oscillatory component \(\Delta {I}_{{{{{{\rm{HF}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\) (red circles) and the fitting curve (the black line). c Spectra of three oscillations A–C obtained by the fitting analysis of the time characteristic of the high-frequency oscillatory component \(\Delta {I}_{{{{{{\rm{HF}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\) in \(\Delta {I}_{{{{{{\rm{R}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\) shown in Fig. 2j. d The ε2 spectrum along the a-axis. The spectrum of the mid-infrared pump pulse (pump II) is shown by the red broken line. e Intensities of three oscillatory components A, B, and D, obtained from the fitting analyses of \(\Delta {I}_{{{{{{\rm{HF}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\) and \(\Delta {I}_{{{{{{\rm{LF}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\). The error bars, defined as one standard deviation of the least-squared fit, are smaller than the data points.

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Figure 3e shows how the integrated intensities of the power spectra, \({I}_{{{{{{\rm{A}}}}}}}\), \({I}_{{{{{{\rm{B}}}}}}}\), and \({I}_{{{{{{\rm{D}}}}}}}\), for oscillations A, B, and D, respectively, depend on the maximum electric-field amplitude of the mid-infrared pulse, \({{{{{\rm{max }}}}}}|{E}_{{{{{{\rm{MIR}}}}}}}({t}_{{{{{{\rm{d}}}}}}})|\). All the intensities are proportional to \(({{{{{\rm{max }}}}}}|{E}_{{{{{{\rm{MIR}}}}}}}({t}_{{{{{{\rm{d}}}}}}})|)^{4}\). This means that amplitudes of each oscillation are proportional to \(({{{{{\rm{max }}}}}}|{E}_{{{{{{\rm{MIR}}}}}}}({t}_{{{{{{\rm{d}}}}}}})|)^{2}\). These results are explained in the following way. The oscillatory signals of the reflectivity changes, \(\Delta {I}_{{{{{{\rm{LF}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\) and \(\Delta {I}_{{{{{{\rm{HF}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\), are due to the electronic state modulations by the out-of-phase combinations of each ag mode in each dimer. Therefore, they are expected to be proportional to \([{g}_{{{{{{\rm{S}}}}}}1}{Q}_{1}({t}_{{{{{{\rm{d}}}}}}})+{g}_{{{{{{\rm{S}}}}}}2}{Q}_{2}({t}_{{{{{{\rm{d}}}}}}})]^{2}\). Here, \({Q}_{1}\left({t}_{{{{{{\rm{d}}}}}}}\right)\) and \({Q}_{2}\left({t}_{{{{{{\rm{d}}}}}}}\right)\) are the normal coordinates of mode 1 with frequency \({\varOmega }_{1}\) and mode 2 with frequency \({\varOmega }_{2}\), respectively. \({g}_{{{{{{\rm{S}}}}}}1}\) and \({g}_{{{{{{\rm{S}}}}}}2}\) are the EMV coupling constant of mode 1 and mode 2 defined by \({g}_{{{{{{\rm{S}}}}}}1}=\sqrt{\frac{\hslash }{2{\varOmega }_{1}}}\frac{\partial {\epsilon }_{{{{{{\rm{S}}}}}}}}{\partial {Q}_{1}}\) and \({g}_{{{{{{\rm{S}}}}}}2}=\sqrt{\frac{\hslash }{2{\varOmega }_{2}}}\frac{\partial {\epsilon }_{{{{{{\rm{S}}}}}}}}{\partial {Q}_{2}}\), respectively, which determine how large the energy \({\epsilon }_{{{{{{\rm{S}}}}}}}\) of SOMO changes with the coordinate \({Q}_{i}\left({t}_{{{{{{\rm{d}}}}}}}\right)\) (i = 1, 2). Thus, the oscillatory signals include the components proportional to \({\left[{Q}_{1}\left({t}_{{{{{{\rm{d}}}}}}}\right)\right]}^{2}\), \({\left[{Q}_{2}\left({t}_{{{{{{\rm{d}}}}}}}\right)\right]}^{2}\), and \({Q}_{1}\left({t}_{{{{{{\rm{d}}}}}}}\right){Q}_{2}\left({t}_{{{{{{\rm{d}}}}}}}\right)\). As a result, the high-frequency signals, \(\Delta {I}_{{{{{{\rm{HF}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\), consists of oscillatory components with the frequencies of \(2{\varOmega }_{1}\), \(2{\varOmega }_{2}\), and \({\varOmega }_{1}+{\varOmega }_{2}\), and the low-frequency signal, \(\Delta {I}_{{{{{{\rm{LF}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\), is the oscillation with the frequency of \({\varOmega }_{2}-{\varOmega }_{1}\). Since \({Q}_{i}\left({t}_{{{{{{\rm{d}}}}}}}\right)\) (i = 1, 2) is proportional to the electric-field amplitude, \({{{{{\rm{max }}}}}}\left|{E}_{{{{{{\rm{MIR}}}}}}}({t}_{{{{{{\rm{d}}}}}}})\right|\), the amplitude of each oscillatory signal is proportional to the square of \({{{{{\rm{max }}}}}}\left|{E}_{{{{{{\rm{MIR}}}}}}}({t}_{{{{{{\rm{d}}}}}}})\right|\), and the integrated intensities \({I}_{{{{{{\rm{A}}}}}}}\), \({I}_{{{{{{\rm{B}}}}}}}\), and \({I}_{{{{{{\rm{D}}}}}}}\) are, therefore, proportional to \({{{{{\rm{max }}}}}}\left|{E}_{{{{{{\rm{MIR}}}}}}}({t}_{{{{{{\rm{d}}}}}}})\right|\) to the forth power as observed in Fig. 3e. Since all the oscillatory amplitudes in \(\Delta {I}_{{{{{{\rm{R}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\) are proportional to the square of \({{{{{\rm{max }}}}}}\left|{E}_{{{{{{\rm{MIR}}}}}}}({t}_{{{{{{\rm{d}}}}}}})\right|\), so that they can be also considered a kind of third-order nonlinear optical responses. Strictly speaking, the magnitudes of the oscillatory signals are affected by the effects of the EMV coupling associated with HOMO as well as SOMO, while the electric-field dependence does not change. This point is discussed in Supplementary Note 3.

The Fourier power spectrum of the oscillatory signals can also be deduced by the fitting analyses of the time evolutions of the oscillatory components using damped oscillators, which are reported in the Method section and Supplementary Note 2. The high-frequency oscillatory component is well reproduced by the fitting analyses as shown by the black solid line in Fig. 2j. The corresponding Fourier power spectrum is shown by the black solid line in Fig. 3a, b, which is in good agreement with that shown by the red circles in the same figures. The advantage of this fitting analysis is that we can deduce the spectrum of each oscillatory signal separately, which is also shown by the colored solid lines in Fig. 3c. This analysis is used to evaluate the probe-energy dependence of each oscillatory signal as discussed later.

Theoretical framework of optical radiation from phonon-dressed states based upon third-order optical nonlinearity

The optical radiation observed as the oscillatory signals in \(\Delta {I}_{{{{{{\rm{R}}}}}}}\left({t}_{{{{{{\rm{d}}}}}}}\right)/{I}_{{{{{{\rm{R}}}}}}}\) is expected to originate from phonon-dressed states of the intramolecular transitions illustrated in Fig. 1a. To clarify the relation between the oscillatory signals and the phonon-dressed states, we formulated the oscillatory signals using the three-level model and the framework of the third-order optical nonlinearity, the details of which are reported in Supplementary Notes 3 and 4, respectively. In this sub-section, we report the main part of the formulation.

In our formulation, we consider an isolated dimer and adopt a three-level model, which consists of the ground state |GS〉 and two excited states |IMa〉 and |IMf〉, which are the one-photon allowed and one-photon forbidden intramolecular excited states, respectively. The electronic states in a dimer are expressed using the following formula.

$$\left|\begin{array}{cc}LS & RS\\ LH & RH\end{array}\right\rangle$$
(1)

L and R show left and right site in a dimer, respectively. \(S\) and \(H\) represent the higher-energy and lower-energy orbital, which correspond to SOMO and HOMO in the original state, respectively (see Fig. 1d). An electron is expressed by an arrow, which shows the direction of each spin. For example, \(\left|\begin{array}{cc}\uparrow & \downarrow \\ \uparrow \downarrow & \uparrow \downarrow \end{array}\right\rangle\) represents the lowest-energy state with an up-spin electron in LS and a down-spin electron in RS. By assuming that the intradimer transfer integrals are negligibly small compared to intramolecular Coulomb repulsion energy, the ground state is expressed as follows:

$$|{{{{{\rm{GS}}}}}}\rangle =\frac{1}{\sqrt{2}}\left(\bigg|\begin{array}{cc}\uparrow & \downarrow \\ \uparrow \downarrow & \uparrow \downarrow \end{array}\bigg\rangle \,{-}\,\bigg|\begin{array}{cc}\downarrow & \uparrow \\ \uparrow \downarrow & \uparrow \downarrow \end{array}\bigg\rangle \right)$$
(2)

In the intramolecular excitations, we consider the excited states, \(\left|\begin{array}{cc}\uparrow & \uparrow \downarrow \\ \uparrow \downarrow & \downarrow \end{array}\right\rangle\) and \(\left|\begin{array}{cc}\uparrow \downarrow & \uparrow \\ \downarrow & \uparrow \downarrow \end{array}\right\rangle\), in which an up-spin electron is excited from HOMO to SOMO. The eigenstates can be expressed by the following two states, which should be nearly degenerate.

$$\begin{array}{c}|{{{{{{\rm{IM}}}}}}}_{{{{{{\rm{a}}}}}}}\rangle =\dfrac{1}{\sqrt{2}}\left(\bigg|\begin{array}{cc}\uparrow & \uparrow \downarrow \\ \uparrow \downarrow & \downarrow \end{array}\bigg\rangle \,+\,\bigg|\begin{array}{cc}\uparrow \downarrow & \uparrow \\ \downarrow & \uparrow \downarrow \end{array}\bigg\rangle \right)\\ |{{{{{{\rm{IM}}}}}}}_{{{{{{\rm{f}}}}}}}\rangle =\dfrac{1}{\sqrt{2}}\left(\bigg|\begin{array}{cc}\uparrow & \uparrow \downarrow \\ \uparrow \downarrow & \downarrow \end{array}\bigg\rangle \,-\,\bigg|\begin{array}{cc}\uparrow \downarrow & \uparrow \\ \downarrow & \uparrow \downarrow \end{array}\bigg\rangle \right)\end{array}$$
(3)

|IMa〉 and |IMf〉 are one-photon allowed and forbidden for \(E\perp a\), respectively.

When the EMV coupling exists, radiations from the dressed states, \(\left|{{{{{{\rm{IM}}}}}}}_{{{{{{\rm{a}}}}}}},\,\pm \!2n\right\rangle\) and \(\left|{{{{{{\rm{IM}}}}}}}_{{{{{{\rm{f}}}}}}},\pm \left(2n-1\right)\right\rangle\) (\(n\): natural number), can occur. The energy of |IMa〉 is approximately 2.0 eV, and |IMf〉 is almost degenerate with |IMa〉. Considering that the energy of mode 1 (mode 2) is 0.146 eV (0.165 eV), the probe pulse ranging from 1.75 to 2.25 eV can cover only the dressed states |IMf, ±1〉 associated with the originally one-photon forbidden excited state |IMf〉, which will be responsible for the oscillatory signals in \(\Delta {I}_{{{{{{\rm{R}}}}}}}\left({t}_{{{{{{\rm{d}}}}}}}\right)/{I}_{{{{{{\rm{R}}}}}}}\).

In this case, the radiation from the dressed states due to the nonlinear polarization with the frequency \(\omega\) is considered to occur by the third-order nonlinear polarization \({P}^{\left(3\right)}\), which is proportional to \({\left[{g}_{{{{{{\rm{S}}}}}}1}{Q}_{1}\,{{{+}}}\,{g}_{{{{{{\rm{S}}}}}}2}{Q}_{2}\right]}^{2}{E}_{{{{{{\rm{VIS}}}}}}}\). Here, \({E}_{{{{{{\rm{VIS}}}}}}}\) is the electric field of the probe pulse. \({Q}_{1}\) and \({Q}_{2}\) are generated by the forced oscillations of modes 1 and 2 by the mid-infrared pulse, respectively, so that the magnitude of \({P}^{\left(3\right)}\) is proportional to the product of \({E}_{{{{{{\rm{VIS}}}}}}}\) and the square of the mid-infrared electric field.

Hereafter, we consider only the high-frequency oscillatory signals to simplify the discussion. In the frequency domain, the third-order nonlinear polarization \({P}^{\left(3\right)}\left(\omega \right)\) with frequency \(\omega\) originating from the higher dressed states |IMf, +1〉 associated with mode 1 and mode 2 is dominated mainly by the terms of \({\chi }_{{\varOmega }_{1},{\varOmega }_{1}}^{\left(3\right)}\left(\omega \right){Q}_{1}\left({\varOmega }_{1}\right){Q}_{1}\left({\varOmega }_{1}\right){E}_{{{{{{\rm{VIS}}}}}}}\left(\omega -2{\varOmega }_{1}\right)\), \({\chi }_{{\varOmega }_{2},{\varOmega }_{2}}^{\left(3\right)}\left(\omega \right){Q}_{2}\left({\varOmega }_{2}\right){Q}_{2}\left({\varOmega }_{2}\right){E}_{{{{{{\rm{VIS}}}}}}}\left(\omega -2{\varOmega }_{2}\right)\), and \({\chi }_{{\varOmega }_{1},{\varOmega }_{2}}^{\left(3\right)}\left(\omega \right){Q}_{1}\left({\varOmega }_{1}\right){Q}_{2}\left({\varOmega }_{2}\right){E}_{{{{{{\rm{VIS}}}}}}}\left(\omega -{\varOmega }_{1}-{\varOmega }_{2}\right)\). Here, \({\chi }_{{\varOmega }_{a},{\varOmega }_{b}}^{(3)}(a,b={{{{\mathrm{1,2}}}}})\) is third-order nonlinear susceptibility. \({Q}_{1}\left(\omega \right)\) and \({Q}_{2}\left(\omega \right)\) are the frequency dependences of the coordinates of mode 1 and 2, respectively, and \({E}_{{{{{{\rm{VIS}}}}}}}(\omega )\) is the spectrum of the probe pulse. The linear polarization \({P}^{\left(1\right)}\left(\omega \right)={\varepsilon }_{0}{\chi }^{(1)}{E}_{{{{{{\rm{VIS}}}}}}}(\omega )\) is also generated due to the transition from |GS〉 to the originally one-photon allowed state |IMa〉. Compared to the radiation from \({P}^{\left(1\right)}\left(\omega \right)\), the radiations from \({P}^{\left(3\right)}\left(\omega \right)\) related to \({\chi }_{{\varOmega }_{1},{\varOmega }_{1}}^{(3)}\), \({\chi }_{{\varOmega }_{2},{\varOmega }_{2}}^{(3)}(\omega )\), and \({\chi }_{{\varOmega }_{1},{\varOmega }_{2}}^{(3)}({\chi }_{{\varOmega }_{2},{\varOmega }_{1}}^{\left(3\right)})\) have the additional phase of \(2{\varOmega }_{1}{t}_{{{{{{\rm{d}}}}}}}\), \(2{\varOmega }_{2}{t}_{{{{{{\rm{d}}}}}}}\), and \(\left({\varOmega }_{1}+{\varOmega }_{2}\right){t}_{{{{{{\rm{d}}}}}}}\), respectively. It is because the polarizations of frequency \({\varOmega }_{1}\) and \({\varOmega }_{2}\) due to the mode 1 and mode 2 vibrations are generated at a time \({t}_{{{{{{\rm{d}}}}}}}\) before the probe pulse is incident to the sample. In this situation, the two types of the radiations with the frequency \(\omega\) from the polarizations of \({P}^{\left(1\right)}\left(\omega \right)\) and \({P}^{\left(3\right)}\left(\omega \right)\) interfere with each other, giving rise to the oscillations with the frequencies of \(2{\varOmega }_{1}\), \(2{\varOmega }_{2}\), and \(\left({\varOmega }_{1}+{\varOmega }_{2}\right)\) in \(\Delta {I}_{{{{{{\rm{R}}}}}}}\left({t}_{{{{{{\rm{d}}}}}}}\right)/{I}_{{{{{{\rm{R}}}}}}}\) as observed in the pump-probe signal. This mechanism of the oscillation generation with \(2{\varOmega }_{1}\) related to \({\chi }_{{\varOmega }_{1},{\varOmega }_{1}}^{(3)}\) is illustrated in Fig. 4a, in which each arrow indicates the excitation by the probe pulse or the mid-infrared pulse. The phase part of each optical process is written in the same figure. An oscillation of the frequency \(2{\varOmega }_{1}\) appears due to the interference between the radiation oscillating at \(\omega t\) from \({P}^{\left(1\right)}\left(\omega \right)\) and the radiation oscillating at \(\omega t+2{\varOmega }_{1}{t}_{{{{{{\rm{d}}}}}}}\) from \({P}^{\left(3\right)}\left(\omega \right)\). The oscillation related to the lower dressed state |IMf, –1〉 occurs due to the similar mechanism as illustrated in Fig. 4b. As a result, the high-frequency oscillations with the frequencies of \(2{\varOmega }_{1}\), \(2{\varOmega }_{2}\), and \(\left({\varOmega }_{1}+{\varOmega }_{2}\right)\) are expected to have two peaks separated by \(2{\varOmega }_{1}\), \(2{\varOmega }_{2}\), and \(\left({\varOmega }_{1}+{\varOmega }_{2}\right)\), respectively, in the probe photon-energy dependence, that is, in the spectrum. As for the low-frequency oscillation with the frequency of \(\left({\varOmega }_{2}-{\varOmega }_{1}\right)\), we can also consider the similar generation mechanism based upon the third-order optical nonlinearity.

Fig. 4: Schematics of generation mechanism of high-frequency oscillations in reflectivity changes.
figure 4

a The case that only the mode 1 of the frequency of \({\varOmega }_{1}\) is excited by the mid-infrared pulse, and the state |IMf, +1〉 is resonantly excited by the combination of three processes indicated by the arrows. The probe pulse covers the energies of the states |IMa〉, |IMf, + 1〉, and |IMf, −1〉. The linear absorption spectrum is broad and therefore the state |IMa〉 is also excited. The radiation from the linear polarization of |IMa〉 and that from the nonlinear polarization of |IMf, + 1〉 have the phase of \(\omega t\) and \((\omega t+2{\varOmega }_{1}{t}_{{{{{{\rm{d}}}}}}})\), respectively. \({t}_{{{{{{\rm{d}}}}}}}\) is the delay time of the probe pulse relative to the pump pulse. Those two radiations interfere with each other, giving rise to the oscillation of the frequency \(2{\varOmega }_{1}\) in the pump-probe signal. Phase part of each excitation and radiation is shown by colored characters. b The case that the state |IMf, +1〉 is resonantly excited by the probe pulse. The radiation from the linear polarization of |IMa〉 and that from the nonlinear polarization of |IMf, –1〉 have the phase of \(\omega t\) and \((\omega t-2{\varOmega }_{1}{t}_{{{{{{\rm{d}}}}}}})\) and interfere with each other, giving rise to the oscillation of the frequency \(2{\varOmega }_{1}\).

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Next, we show the relation between the radiation from the phonon-dressed states |IMf, ±1〉 and the third-order nonlinear susceptibility \({\chi }_{{\varOmega }_{a},{\varOmega }_{b}}^{(3)}\). The change in the intensity of the radiated light, \(\Delta {I}_{{{{{{\rm{R}}}}}}}\left(\omega ,{t}_{{{{{{\rm{d}}}}}}}\right)\), as a function of the delay time \({t}_{{{{{{\rm{d}}}}}}}\) is expressed as

$$\Delta {I}_{{{{{{\rm{R}}}}}}}\left(\omega ,{t}_{{{{{{\rm{d}}}}}}}\right)= \,{\left|{\widetilde{r}}_{y}\left(\omega \right){E}_{{{{{{\rm{VIS}}}}}}}\left(\omega ,{t}_{{{{{{\rm{d}}}}}}}\right)+\Delta {E}_{{{{{{\rm{R}}}}}}}\left(\omega ,{t}_{{{{{{\rm{d}}}}}}}\right)\right|}^{2}\\ -{\left|{\widetilde{r}}_{y}\left(\omega \right){E}_{{{{{{\rm{VIS}}}}}}}\left(\omega ,{t}_{{{{{{\rm{d}}}}}}}\right)\right|}^{2}.$$
(4)

Here, \({\widetilde{r}}_{y}\left(\omega \right)\) is the complex amplitude reflectivity and \(\Delta {E}_{{{{{{\rm{R}}}}}}}\left(\omega ,{t}_{{{{{{\rm{d}}}}}}}\right)\) is the change in the radiated electric field. The oscillatory component \(\Delta {I}_{{\varOmega }_{a}+{\varOmega }_{b}}\left(\omega \right)\) of the oscillation frequency \({\varOmega }_{a}+{\varOmega }_{b}\) (\({\varOmega }_{a},{\varOmega }_{b}={\varOmega }_{1},{\varOmega }_{2}\)) in \(\Delta {I}_{{{{{{\rm{R}}}}}}}\left(\omega ,{t}_{{{{{{\rm{d}}}}}}}\right)\) is expressed as follows:

$$\Delta {I}_{{\varOmega }_{a}+{\varOmega }_{b}}\left(\omega \right)\propto \; {\xi }_{{\varOmega }_{a},{\varOmega }_{b}}\left(\omega \right){\chi }_{{\varOmega }_{a},{\varOmega }_{b}}^{(3)}\left(\omega \right){E}_{{{{{{\rm{VIS}}}}}}}\left(\omega -{\varOmega }_{a}-{\varOmega }_{b}\right){E}_{{{{{{\rm{VIS}}}}}}}^{* }\left(\omega \right)\\ +{\left[{\xi }_{-{\varOmega }_{a},-{\varOmega }_{b}}\left(\omega \right){\chi }_{{\varOmega }_{-a},{\varOmega }_{-b}}^{(3)}\left(\omega \right){E}_{{{{{{\rm{VIS}}}}}}}\left(\omega +{\varOmega }_{a}+{\varOmega }_{b}\right){E}_{{{{{{\rm{VIS}}}}}}}^{* }\left(\omega \right)\right]}^{* }$$
(5)

Here, \({\xi }_{\pm {\varOmega }_{a},\pm {\varOmega }_{b}}\left(\omega \right)\) is a parameter related to the linear optical constants. \(\Delta {I}_{{\varOmega }_{a}+{\varOmega }_{b}}\left(\omega \right)\) depends not only on \({E}_{{{{{{\rm{VIS}}}}}}}\left(\omega \right)\) but also on \({E}_{{{{{{\rm{VIS}}}}}}}\left(\omega \pm \left({\varOmega }_{a}+{\varOmega }_{b}\right)\right)\). In Fig. 5a, we show the intensity spectrum of the probe pulse \({\left|{E}_{{{{{{\rm{VIS}}}}}}}\left(\omega \right)\right|}^{2}\) and the spectra shifted by \(\pm 2{\varOmega }_{1}\), \({\left|{E}_{{{{{{\rm{VIS}}}}}}}\left(\omega \pm 2{\varOmega }_{1}\right)\right|}^{2}\). \({\left|{E}_{{{{{{\rm{VIS}}}}}}}\left(\omega \right)\right|}^{2}\) ranges from 1.75 to 2.25 eV. \(2\hslash {\varOmega }_{1}\) is approximately 0.3 eV, and thus \({\left|{E}_{{{{{{\rm{VIS}}}}}}}\left(\omega -2{\varOmega }_{1}\right)\right|}^{2}\) and \({\left|{E}_{{{{{{\rm{VIS}}}}}}}\left(\omega +2{\varOmega }_{1}\right)\right|}^{2}\) cover 2.05–2.55 and 1.45–1.95 eV, respectively. Therefore, for \({\hslash} \omega \, > \, 2{{{{{\rm{eV}}}}}}\) \(({{\hslash }}\omega \le 2{{{{{\rm{eV}}}}}})\), the first (second) term in Eq. (5) dominates the high-frequency oscillations. Accounting for this experimental condition, we derive the spectrum of \({\left|{\chi }_{{\pm \varOmega }_{a},\pm {\varOmega }_{b}}^{\left(3\right)}(\omega )\right|}^{2}\) as follows:

$${\left|{\chi }_{{\pm \varOmega }_{a},\pm {\varOmega }_{b}}^{\left(3\right)}(\omega )\right|}^{2}\propto \frac{{\left|\Delta {I}_{{\varOmega }_{a}+{\varOmega }_{b}}\left(\omega \right)\right|}^{2}}{{\left|{\xi }_{\pm {\varOmega }_{a}\pm ,{\varOmega }_{b}}\left(\omega \right){E}_{{{{{{\rm{VIS}}}}}}}\left(\omega \mp {\varOmega }_{a}\mp {\varOmega }_{b}\right){E}_{{{{{{\rm{VIS}}}}}}}^{* }\left(\omega \right)\right|}^{2}}\\ (+\,{{{{{\rm{for}}}}}}\,{{\hslash }}\omega \, > \, 2\;{{{{{\rm{eV}}}}}},\,-\,{{{{{\rm{for}}}}}}\,{{\hslash }}\omega \,\le\, 2\;{{{{{\rm{eV}}}}}})$$
(6)

The derivations of Eqs. (5) and (6) and the formula of \({\xi }_{\pm {\varOmega }_{a},\pm {\varOmega }_{b}}\left(\omega \right)\) are detailed in Supplementary Note 4.

Fig. 5: Probe photon-energy dependence of high-frequency oscillations in optical responses associated with phonon-dressed states in K-TCNQ.
figure 5

a The power spectrum of the visible-probe pulse \({\left|{E}_{{{{{{\rm{VIS}}}}}}}\left(\omega \right)\right|}^{2}\) and those shifted by \(\pm 2{\varOmega }_{1}\). b–d The corrected intensities of high-frequency oscillations A–C expressed by Eq. (6) for \({\hslash} \omega > 2\) eV (blue circles) and for \({\hslash} \omega \le 2\) eV (red circles). Blue and red lines show simulated spectra of \({|{\chi }_{{\pm \varOmega }_{1},\pm {\varOmega }_{1}}^{(3)}(\omega )|}^{2}\) (b), \({|{\chi }_{{\pm \varOmega }_{1},\pm {\varOmega }_{2}}^{(3)}(\omega )|}^{2}\) (c), and \({|{\chi }_{{\pm \varOmega }_{2},\pm {\varOmega }_{2}}^{(3)}(\omega )|}^{2}\) (d) based upon the three-level model, which correspond to the intensity spectra for oscillations A, B, and C, respectively. The error bars indicate the standard deviation of the least-squares fit. e The optical process corresponding to \({\chi }_{{\pm \varOmega }_{1},\pm {\varOmega }_{1}}^{\left(3\right)}(\omega )\) (oscillation A). The probe pulse produces a transition from the ground state |GS〉 to the \(\mp\)1-phonon-dressed state, |IMf, ∓1〉, and the light emission associated with a transition from |IMf, ± 1〉 to |GS〉 occurs. It generates the oscillatory component whose frequency equals twice the frequency of the phonon.

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Probe photon-energy dependence of oscillatory signals in optical radiation from phonon-dressed states

By monochromizing the change of the radiation, \(\Delta {I}_{{{{{{\rm{R}}}}}}}\left({t}_{{{{{{\rm{d}}}}}}}\right)\), and obtaining the spectra of the high-frequency oscillations, we can verify the interpretation of the oscillatory signals mentioned in the previous sub-section. For this measurement, we also use a mid-infrared pulse exciting both ag modes 1 and 2 as shown in Fig. 2a, although its electric-field waveform is slightly different from that shown in Fig. 2b (see Supplementary Note 5). Here, we focus only on the \({\hslash} \omega\)-dependence of high-frequency oscillations A–C in \(\Delta {I}_{{{{{{\rm{HF}}}}}}}\left(t\right)\). Via the fitting procedure detailed in the Method section and Supplementary Note 2, we evaluate the time profile of each oscillatory component for various probe photon energies. From those data, we obtained each Fourier power spectrum and its integrated intensity \({I}_{{{{{{\rm{A}}}}}}}\), \({I}_{{{{{{\rm{B}}}}}}}\), and \({I}_{{{{{{\rm{C}}}}}}}\) as a function of \({\hslash} \omega\), which correspond to \({|\Delta {I}_{{\varOmega }_{a}+{\varOmega }_{b}}(\omega )|}^{2}\) in Eq. (6).

Using Eq. (6) and the photon-energy dependence of \({|\Delta {I}_{{\varOmega }_{a}+{\varOmega }_{b}}(\omega )|}^{2}\) experimentally obtained, we deduce the \({|{\chi }_{{\pm \varOmega }_{a},\pm {\varOmega }_{b}}^{(3)}(\omega )|}^{2}\) spectra of oscillations A−C, which are shown by red and blue circles in Fig. 5b–d. The three spectra all show two peaks around 1.8 and 2.1 eV, although the data points vary to some extent. The energy difference between the two peaks, ∼0.3 eV, is almost equal to double the intramolecular vibration frequency (\(2{\hslash } {\varOmega }_{1}=0.29\) eV and \(2{\hslash} {\varOmega }_{2}=0.33\) eV). In addition, the energy gravity of the two peaks, 1.95 eV, is close to the intramolecular transition energy (Fig. 2a). These are consistent with the situation shown in Figs. 1a and 4. Thus, the two obtained peaks are attributed to \(\pm\)1-phonon-dressed states of the one-photon-forbidden intramolecular excited state |IMf〉, and |IMf〉 is located at almost the same energy with the one-photon allowed state |IMa〉 as expected.

Simulations of probe photon-energy dependence of oscillatory signals

To confirm the interpretations proposed in the preceding section, we formulated the nonlinear optical susceptibility \({\chi }_{{\pm \varOmega }_{a},\pm {\varOmega }_{b}}^{\left(3\right)}(\omega )\) by using the perturbation treatment in the framework of the third-order nonlinear optics. In this formulation, we adopted the three-level model mentioned above, which includes the intramolecular excited states \(\left|{{{{{{\rm{IM}}}}}}}_{{{{{{\rm{a}}}}}}}\right\rangle\) and \(|{{{{{{\rm{IM}}}}}}}_{{{{{{\rm{f}}}}}}}\rangle\), and the ground state \({{{{{\rm{|GS}}}}}}{\rangle }\) in each dimer and considered that the phonon-dressed states of these excited states are formed by periodic perturbation to electrons via the EMV coupling. The specific expression of \({\chi }_{{\pm \varOmega }_{a},\pm {\varOmega }_{b}}^{\left(3\right)}(\omega )\) is shown in the Methods section.

Using the experimentally obtained energy \(({\hslash }{\omega }_{{{{{{\rm{IMa}}}}}}}=1.99{{{{{\rm{eV}}}}}})\) and damping constant (\(\hslash {\gamma }_{{{{{{\rm{IMa}}}}}}}=0.111\) eV) of \(\left|{{{{{{\rm{IM}}}}}}}_{{{{{{\rm{a}}}}}}}\right\rangle\) (Fig. 2a), and assuming the energy \(({\hslash}{\omega }_{{{{{{\rm{IMf}}}}}}}=1.97{{{{{\rm{eV}}}}}})\) and damping constant (\(\hslash {\gamma }_{{{{{{\rm{IMf}}}}}}}=0.08\) eV) of \(|{{{{{{\rm{IM}}}}}}}_{{{{{{\rm{f}}}}}}}\rangle\), we simulate the \({\left|{\chi }_{{\pm \varOmega }_{a},\pm {\varOmega }_{b}}^{\left(3\right)}(\omega )\right|}^{2}\) spectra for the three kinds of oscillations A–C, which are shown by the solid red and blue lines in Fig. 5b–d. The procedures of the simulations are detailed in Supplementary Note 5. The energy differences between two peaks for \({\left|{\chi }_{{\pm \varOmega }_{1},\pm {\varOmega }_{1}}^{\left(3\right)}(\omega )\right|}^{2}\) (Fig. 5b), for \({\left|{\chi }_{{\pm \varOmega }_{1},\pm {\varOmega }_{2}}^{\left(3\right)}(\omega )\right|}^{2}\) (Fig. 5c), and for \({\left|{\chi }_{{\pm \varOmega }_{2},\pm {\varOmega }_{2}}^{\left(3\right)}(\omega )\right|}^{2}\) (Fig. 5d) are equal to \(2{\hslash} {\varOmega }_{1}\), \({\hslash} ({\varOmega }_{1}+{\varOmega }_{2})\), and \(2{\hslash}{\varOmega }_{2}\), respectively, which are shown by the double-headed arrows in each panel. These \({\left|{\chi }_{{\pm \varOmega }_{a},\pm {\varOmega }_{b}}^{\left(3\right)}(\omega )\right|}^{2}\) spectra almost reproduce the experimental intensity spectra of oscillations A-C. This demonstrates that \(\pm\)1-phonon-dressed states are responsible for the observed oscillatory signals.

Discussions

Here, we discuss the nature of observed phonon-dressed states based on the Floquet theory. According to the theory, the periodic potential of an infrared-active phonon creates new one-photon allowed states, \(\left|{{{{{{\rm{IM}}}}}}}_{{{{{{\rm{a}}}}}}},\pm 2n\right\rangle\) and \(\left|{{{{{{\rm{IM}}}}}}}_{{{{{{\rm{f}}}}}}},\pm \left(2n-1\right)\right\rangle\) (\(n\): natural number), around the original states, \(\left|{{{{{{\rm{IM}}}}}}}_{{{{{{\rm{a}}}}}}}\right\rangle\) and \(\left|{{{{{{\rm{IM}}}}}}}_{{{{{{\rm{f}}}}}}}\right\rangle\), respectively. In our case, \(2{\varOmega }_{1}\) and \(2{\varOmega }_{2}\), which are twice the intramolecular vibration frequencies, are comparable to a half of the spectral width of the probe pulse, ∼0.25 eV, and thus we can observe only \(\left|{{{{{{\rm{IM}}}}}}}_{{{{{{\rm{f}}}}}}},\pm 1\right\rangle\) through light emission. The two optical processes shown in Fig. 5e are the origins of \({\chi }_{{\varOmega }_{1},{\varOmega }_{1}}^{(3)}\left(\omega \right)\) and \({\chi }_{-{\varOmega }_{1},{-\varOmega }_{1}}^{(3)}\left(\omega \right)\), which generates the peaks of oscillation A located at \({\hslash }{\omega }_{{{{{{\rm{IMf}}}}}}}+{\hslash}{\varOmega }_{1}\) and \({\hslash }{\omega }_{{{{{{\rm{IMf}}}}}}}-{\hslash} {\varOmega }_{1}\), respectively. This is the reason why all high-frequency components, associated with phonon-dressed states, have peaks situated at energy of \(\left|{{{{{{\rm{IM}}}}}}}_{{{{{{\rm{f}}}}}}},\pm 1\right\rangle\). More strictly, the peaks of oscillation B should appear at \({\hslash }{\omega }_{{{{{{\rm{IMf}}}}}}}\pm {\hslash}{\varOmega }_{1}\) and \({\hslash}{\omega }_{{{{{{\rm{IMf}}}}}}}\pm {\hslash}{\varOmega }_{2}\), whereas only two peaks are observed in Fig. 5c. This is because the intramolecular excited state has a broad spectral width (∼0.2 eV) (Fig. 2a). The detailed formulation of phonon-dressed states based on the Floquet theory is reported in Supplementary Notes 6 and 7.

In summary, we investigated the phonon-dressed states in a half-filled organic Mott insulator. Under molecular dimerization, the ag modes of intramolecular vibrations became infrared-active and modulated the energy levels of π-molecular orbitals via EMV couplings. Using this feature, we added strong periodic potentials to the \(\pi\)-molecular orbitals by exciting a couple of ag modes. Through sub-cycle spectroscopy with a mid-infrared pump pulse and an ultrashort visible-probe pulse, we observed optical radiation with the second-order oscillatory components of two ag modes. From the analyses of their probe-energy dependences, we demonstrated that phonon-dressed states of the one-photon-forbidden intramolecular excited state were formed and responsible for the characteristic optical radiation. Our method of combining coherent phonon excitations and ultrafast sub-cycle spectroscopy can be used for advanced studies of Floquet states and further phase controls in various types of molecular materials.

Methods

Sample preparations

Single crystals of K-TCNQ were grown from KI and TCNQ through slow diffusion in acetonitrile according to the literature31. The typical crystal size was approximately 5 mm × 0.5 mm × 0.5 mm. Almost the same data are obtained in distinct samples.

Steady-state optical spectroscopy measurements

The polarized reflectivity \((R)\) spectrum in a single crystal of K-TCNQ was measured using a specially designed spectrometer with a 25-cm-grating monochromator in the visible and near-infrared region (0.5–5.0 eV) and a Fourier-transform infrared spectrometer in the mid-infrared region (0.08–1.2 eV). Both of these spectrometers were equipped with an optical microscope. The spectra of the real \(({\varepsilon }_{1})\) and imaginary \(({\varepsilon }_{2})\) parts of the complex dielectric constant were obtained from the polarized \(R\) spectra using the Kramers–Kronig transformation.

Mid-infrared-pump visible-probe reflectivity measurements

To perform the mid-infrared-pump visible-probe reflectivity measurements, we employed a Ti:sapphire regenerative amplifier (RA) with a central photon energy of 1.55 eV, a repetition rate of 1 kHz, a pulse duration of 35 fs, and a pulse energy of 7.5 mJ as the light source. The output of the RA is divided into two beams, which were used to generate ultrashort visible pulses and carrier-envelope-phase (CEP)-stable mid-infrared pulses.

To generate an ultrashort probe pulse, we used a hand-made non-collinear optical parametric amplifier (NOPA)32, which was excited by one beam from the RA (pulse energy 3.5 mJ). In the NOPA, we further divided the output of RA into two beams. One was focused on a sapphire crystal, from which broadband white light was generated. The other was converted to second-harmonic light in a second-order nonlinear crystal, β-BaB2O4. Another β-BaB2O4 crystal was used to enhance the white light via non-collinear optical parametric amplification, in which the second-harmonic light and white light were used as pump light and seed light, respectively. The pulse width was approximately 8 fs and the spectrum range was from 1.75 to 2.25 eV. The temporal profile of the pulse was measured by second-harmonic-generation frequency-resolved-optical-gating using the retrieval algorithm33,34. The fundamental and second-harmonic spectra of the pulse were measured by a spectroscope, the sensitivity of which was calibrated using a well-calibrated deuterium-halogen lamp.

A CEP-stable mid-infrared pulse was obtained by difference-frequency generation (DFG) between the idler outputs of two optical parametric amplifiers (OPA1 and OPA2) excited by another output of the RA (pulse energy 4.0 mJ). OPA1 delivered 1640 nm pulses with 390 µJ energy, and OPA2 delivered pulses tuneable from 1900 to 2080 nm with 120 µJ energy. We obtained mid-infrared pulses with Type I DFG in a GaSe crystal of thickness 500 µm between two pulses generated from OPA1 and OPA2. The GaSe crystal was placed at an external phase-matching angle of approximately 34°35,36. By tuning the wavelength of the output from OPA2, we obtained two types of mid-infrared pulses for pumps I and II, as shown in Fig. 2b. To stabilize the long-term CEP of the mid-infrared pulses, we performed feedback controls of the optical path lengths by using a pair of CaF2 wedge plates inserted after OPA2 and adjusting two delay lines37. The CEP stability of the controlled mid-infrared pulses was within ±1 fs over a few tens of hours at the sample position.

The electric-field waveform of the mid-infrared pulse was measured using an electro-optic sampling (EOS) method. For this measurement, a 20-µm-thick LiGaS2 (LGS) crystal was used as a second-order nonlinear crystal, and the output of NOPA was used as a sampling pulse. The detection wavelength range of EOS was limited due to the absorptions of LGS and water vapor; electric fields of pulses with wavelengths longer than 12 µm or of 5.5–7.7 µm cannot be detected. The maximum electric-field amplitude of the mid-infrared pulse, \({E}_{{{{{{\rm{MIR}}}}}}}\), was evaluated from a spatial intensity profile obtained by a knife-edge method, and an electric-field profile obtained from EOS and the fluence of the pulse. The maximum electric field \({E}_{{{{{{\rm{MIR}}}}}}}\) (fluence) of the pulses for pumps I and II were 8.07 MV cm−1 (500 nJ) and 8.79 MV cm−1 (1000 nJ), respectively. The amplitude of the mid-infrared pulse was weakened by a pair of wire-grid polarizers.

For the mid-infrared-pump visible-probe reflectivity measurements, the polarization of the electric field \({E}_{{{{{{\rm{MIR}}}}}}}\,({E}_{{{{{{\rm{VIS}}}}}}})\) of the mid-infrared pump pulse (visible-probe pulse) was set parallel (perpendicular) to the \(a\)-axis. The intensity of the reflection light, including the emission from the phonon-dressed states, was detected by a silicon photodiode. We inserted a bandpass filter with a full width at half maximum of approximately 40 meV in front of the detector to select the probe photon energy \(\hslash \omega\). The delay time of the probe pulse relative to the mid-infrared pulse was controlled by a delay stage.

Fitting analyses of high-frequency components of reflectivity changes

To obtain detailed information about the oscillatory signals, we perform a fitting analysis of the oscillatory components, \(\Delta {I}_{{{{{{\rm{HF}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\) and \(\Delta {I}_{{{{{{\rm{LF}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\), with the following formula:

$$\frac{\Delta {I}_{{{{{{\rm{HF}}}}}}}\left({t}_{{{{{{\rm{d}}}}}}}\right)}{{I}_{{{{{{\rm{R}}}}}}}}= \; {{{{{\rm{HPF}}}}}}\left\{\mathop{\sum }\limits_{\lambda =1}^{2}{\alpha }_{\lambda \lambda }{\left[{e}_{{{{{{\rm{MIR}}}}}}}\left({t}_{{{{{{\rm{d}}}}}}}\right)\otimes {f}_{\lambda }\left({t}_{{{{{{\rm{d}}}}}}}\right)\right]}^{2}\right.\\ +2{\alpha }_{12}\left[{e}_{{{{{{\rm{MIR}}}}}}}({t}_{{{{{{\rm{d}}}}}}})\otimes {f}_{1}({t}_{{{{{{\rm{d}}}}}}})\right]\left[{e}_{{{{{{\rm{MIR}}}}}}}({t}_{{{{{{\rm{d}}}}}}})\otimes {f}_{2}({t}_{{{{{{\rm{d}}}}}}})\right]\bigg\}$$
(7)
$$ \frac{\Delta {I}_{{{{{{\rm{LF}}}}}}}({t}_{{{{{{\rm{d}}}}}}})}{{I}_{{{{{{\rm{R}}}}}}}} ={{{{{\rm{LPF}}}}}}\left\{2{\beta }_{12}\left[{e}_{{{{{{\rm{MIR}}}}}}}({t}_{{{{{{\rm{d}}}}}}})\otimes {f}_{1}({t}_{{{{{{\rm{d}}}}}}})\right]\left[{e}_{{{{{{\rm{MIR}}}}}}}({t}_{{{{{{\rm{d}}}}}}})\otimes {f}_{2}({t}_{{{{{{\rm{d}}}}}}})\right]\right\}\\ {f}_{\lambda }\left({t}_{{{{{{\rm{d}}}}}}}\right) ={{\exp }}\left(-\frac{{t}_{{{{{{\rm{d}}}}}}}}{{T}_{\lambda }}\right){{\sin }}\left({\varOmega }_{\lambda }{t}_{{{{{{\rm{d}}}}}}}+{\phi }_{\lambda }\right)\theta ({t}_{{{{{{\rm{d}}}}}}})$$
(8)

Here, \({e}_{{{{{{\rm{MIR}}}}}}}({t}_{{{{{{\rm{d}}}}}}})\) is a normalized electric-field waveform of a mid-infrared pulse. \({f}_{\lambda }\left({t}_{{{{{{\rm{d}}}}}}}\right)\) expresses a damped oscillator \(\lambda\) with frequency \({\varOmega }_{\lambda }\). \({\phi }_{\lambda }\) and \({T}_{\lambda }\) are the initial phase and relaxation time of oscillator \(\lambda\), respectively. \(\theta ({t}_{{{{{{\rm{d}}}}}}})\) is a unit step function. HPF (LPF) is an operator that extracts a high- (low-)frequency component from the \(\Delta {I}_{{{{{{\rm{R}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\) signals. \(\otimes\) shows a convolution integral. \({e}_{{{{{{\rm{MIR}}}}}}}({t}_{{{{{{\rm{d}}}}}}})\otimes {f}_{\lambda }({t}_{{{{{{\rm{d}}}}}}})\) expresses the dynamics of a normal mode of the oscillator \(\lambda\), \({Q}_{\lambda }({t}_{{{{{{\rm{d}}}}}}})\). \({\alpha }_{11}\), \({\alpha }_{12}\), \({\alpha }_{22}\), and \({\beta }_{12}\) show the amplitudes of oscillations A, B, C, and D, with frequencies of \({\varOmega }_{{{{{{\rm{A}}}}}}}\), \({\varOmega }_{{{{{{\rm{B}}}}}}}\), \({\varOmega }_{{{{{{\rm{C}}}}}}}\), and \({\varOmega }_{{{{{{\rm{D}}}}}}}\), respectively. Using Eqs. (7) and (8), the experimental high-frequency oscillation (red line) is well reproduced (black line in Fig. 2j). Furthermore, power spectra (red circles) of the low- and high-frequency oscillations in Fig. 3a, b are also reproduced as shown by the black solid lines, which validates the fitting using Eqs. (7) and (8). This fitting can clearly separate the high-frequency components into oscillations A–C (colored lines in Fig. 3c). Full analyses, including the low-frequency and background components of \(\Delta {I}_{{{{{{\rm{R}}}}}}}({t}_{{{{{{\rm{d}}}}}}})/{I}_{{{{{{\rm{R}}}}}}}\), are reported in Supplementary Note 2.

Expression of third-order nonlinear susceptibility

In the three-level model in which the intramolecular excited states \(\left|{{{{{{\rm{IM}}}}}}}_{{{{{{\rm{a}}}}}}}\right\rangle\) and \(|{{{{{{\rm{IM}}}}}}}_{{{{{{\rm{f}}}}}}}\rangle\), and the ground state \({{{{{\rm{|GS}}}}}}{\rangle }\) are taken into account, \({\chi }_{{\pm \varOmega }_{a},\pm {\varOmega }_{b}}^{\left(3\right)}(\omega )\) is obtained from the perturbation theory as follows:

$${\chi }_{{\pm \varOmega }_{a},\pm {\varOmega }_{b}}^{(3)}\left(\omega \right)\equiv {\chi }^{\left(3\right)}\left(\!\pm \!{\varOmega }_{a},\pm \!{\varOmega }_{b},\omega \;{ \mp}\; {\varOmega }_{a}\;{ \mp}\; {\varOmega }_{b}\right)$$
(9a)
$$ {\chi }^{\left(3\right)}\left({\omega }_{1},{\omega }_{2},{\omega }_{3}\right)\\ \quad\propto \bigg[\frac{1}{\left({\omega }_{{{{{{\rm{IMa}}}}}}}-{\omega }_{1}-{\omega }_{2}-{\omega }_{3}-i{\gamma }_{{{{{{\rm{IMa}}}}}}}\right)\left({\omega }_{{{{{{\rm{IMf}}}}}}}-{\omega }_{2}-{\omega }_{3}-i{\gamma }_{{{{{{\rm{IMf}}}}}}}\right)\left({\omega }_{{{{{{\rm{IMa}}}}}}}-{\omega }_{3}-i{\gamma }_{{{{{{\rm{IMa}}}}}}}\right)}\\ \qquad+\frac{1}{\left({\omega }_{{{{{{\rm{IMa}}}}}}}-{\omega }_{1}-{\omega }_{2}-{\omega }_{3}-i{\gamma }_{{{{{{\rm{IMa}}}}}}}\right)\left({\omega }_{{{{{{\rm{IMf}}}}}}}-{\omega }_{1}-{\omega }_{3}-i{\gamma }_{{{{{{\rm{IMf}}}}}}}\right)\left({\omega }_{{{{{{\rm{IMa}}}}}}}-{\omega }_{3}-i{\gamma }_{{{{{{\rm{IMa}}}}}}}\right)}\bigg]$$
(9b)

Here, \({\hslash }{\omega }_{{{{{{\rm{IMa}}}}}}}\) (\({\hslash}{\omega }_{{{{{{\rm{IMf}}}}}}}\)) and \({\hslash }{\gamma }_{{{{{{\rm{IMa}}}}}}}\) (\(\hslash {\gamma }_{{{{{{\rm{IMf}}}}}}}\)) are the energy and the damping constant of the state \(\left|{{{{{{\rm{IM}}}}}}}_{{{{{{\rm{a}}}}}}}\right\rangle\) (\(|{{{{{{\rm{IM}}}}}}}_{{{{{{\rm{f}}}}}}}\rangle\)), respectively. The details of the derivations are reported in Supplementary Note 4.