Dynamical aspects of excitonic Floquet states generated by a phase-locked mid-infrared pulse in a one-dimensional Mott insulator

A periodic electric field of light applied on a solid is predicted to generate coupled states of the light electric fields and electronic system called photon-dressed Floquet states. Previous studies of those Floquet states have focused on time-averaged energy-level structures. Here, we report time-dependent responses of Floquet states of excitons generated by a mid-infrared (MIR) pulse excitation in a prototypical one-dimensional (1D) Mott insulator, a chlorine-bridged nickel-chain compound, [Ni(chxn)2Cl](NO3)2 (chxn = cyclohexanediamine). Sub-cycle reflection spectroscopy on this compound using a phase-locked MIR pump pulse and an ultrashort visible probe pulse with the temporal width of ∼7 fs revealed that large and ultrafast reflectivity changes occur along the electric field of the MIR pulse; the reflectivity change reached approximately 50% of the original value around the exciton absorption peak. It comprised a high-frequency oscillation at twice the frequency of the MIR pulse and a low-frequency component following the intensity envelope of the MIR pulse, which showed different probe-energy dependences. Simulations considering one-photon-allowed and one-photon-forbidden excitons reproduced the temporal and spectral characteristics of both the high-frequency oscillation and low-frequency component. These simulations demonstrated that all responses originated from the quantum interferences of the linear reflection process and nonlinear light-scattering processes owing to the excitonic Floquet states characteristic of 1D Mott insulators. The present results lead to the developments of Floquet engineering, and demonstrate the possibility of rapidly controlling the intensity of visible or near-IR pulse by varying the phase of MIR electric fields, which will be utilized for ultrafast optical switching devices.


Introduction
When an electronic system is subjected to an electric field of light, it interacts strongly with photons and multiple sidebands appear around the original electronic levels [1][2][3][4][5][6][7][8].This property is derived from Floquet's theorem [9][10][11], which is a temporal version of Bloch's theorem.The stationary state under a periodic field is represented by a superposition of these sidebands that are known as photon-dressed states, which is a typical example of Floquet states [1][2][3][4][5][6][7][8][12][13][14][15][16].By adjusting the electric-field frequency and intensity of a laser pulse, we will be able to control the nature of photon-dressed states and even change the electronic structures of solids [13].This method of electronic-state control is called Floquet engineering and has been preceded by theoretical studies [1-8, 12, 17, 18].More recently, studies on angle-resolved photoelectron spectroscopy have reported the observation of photon-dressed states [14][15][16].However, previous studies have focused on time-averaged changes in electronic or energy-level structures.Sub-cycle spectroscopy [19][20][21][22][23][24][25][26][27][28][29][30][31], which detects time-dependent responses within one cycle of the external field, provides detailed information on the amplitude and phase of the wavefunctions of photon-dressed states.A schematic of the photon-dressed states formed by a pump light with a frequency hΩ (period T = 2π/Ω) is shown in figure 1.New sidebands |φ , n⟩ (n: an integer) are generated in the energy intervals of hΩ around the original state |φ ⟩.These photon-dressed states change the time-averaged nature of the system.An important feature in their optical response is the occurrence of inelastic scattering such that a photon absorbed in a sideband |φ , n⟩ is emitted from a different sideband |φ , n ′ ⟩ [21,22,26,[32][33][34][35].More specifically, when a broadband probe pulse covering several sidebands is incident on a sample in a reflection (transmission) configuration, an inelastically scattered probe pulse is emitted, which interferes with the probe pulse of the same frequency that is reflected (transmitted) via a linear optical response, thus changing the reflected (transmitted) probe-pulse intensity; the intensity is expected to oscillate at (n − n ′ ) Ω depending on the delay time of the probe pulse relative to the pump pulse.For example, a narrow-band terahertz wave with a duration of approximately 5 ps was used as a pump pulse to form photon-dressed states of 1s and 2p excitons in a semiconductor quantum well.Absorption changes oscillating at even multiples of the terahertz-pulse frequency with delay time were indeed observed [22,26].In exotic materials such as strongly correlated systems, the frequency width of an electronic state is at least one order of magnitude greater than 1 THz (∼4 meV).To investigate the optical responses of the photon-dressed states in those materials, the pump-pulse frequency Ω should be considerably larger than 10 THz.Therefore, it is effective to use a mid-infrared (MIR) pulse for the excitation and an ultrashort visible or near-IR pulse for the detection.
The target material of this study is a chlorine-bridged nickel-chain compound [Ni(chxn) 2 Cl](NO 3 ) 2 (chxn = cyclohexanediamine) [36], which is a prototypical one-dimensional (1D) Mott insulator [37,38].Hereafter, this compound is referred to as NCN.The crystal structure of NCN is shown in figure 2(a).Ni 3+ and Cl − ions are arranged alternately along the b-axis.The strong ligand field created by the four N atoms of two chxn molecules and two Cl − ions splits the Ni-3d orbitals, and each 3d z 2 orbital has one electron (figure 2(b)).Hybridizations of Ni-3d z 2 and Cl-3p z orbitals form half-filled Ni-3d and occupied Cl-3p bands.Because of the large on-site Coulomb repulsion energy U on the Ni sites, the Ni-3d band is split into upper and lower Hubbard bands with the Cl-3p band located between them.The charge-transfer (CT) transition from the Cl-3p band to Ni-3d upper Hubbard band corresponds to the optical gap (figure 2(c)).Thus, NCN is a CT insulator, while it has been established that the single-band Hubbard model can well explain the optical responses of this material [37][38][39].NCN and its derivatives of 1D Mott insulators are known to show a large third-order optical nonlinearity associated with band-edge excitons [37][38][39].It makes us expect to clearly observe photon-dressed Floquet states.For the generation and detection of photon-dressed states in the 1D Mott insulator, we adopted the MIR pump visible sub-cycle reflection probe spectroscopy using a phase-locked MIR pulse and an ultrashort visible pulse.Using this method, we indeed observed large and ultrafast responses of photon-dressed Floquet states of excitons caused by strong coupling of excitons with photons under strong electron correlations.

Excitonic states in [Ni(chxn) 2 Cl](NO 3 ) 2
The spectra of the reflectivity R and imaginary part of the dielectric constant ε 2 along the b-axis at 77 K of NCN are shown in figure 2(d).A sharp peak structure appears at 1.95 eV in the ε 2 spectrum.According to the previous studies of electro-reflectance (ER) and third-harmonic generation (THG) spectroscopy and of the excitation spectrum of photoconductivity in this compound, it has been revealed that this peak is due to a one-photon-allowed exciton with odd-parity |φ odd ⟩ and that the second-lowest excited state is a one-photon-forbidden exciton with even-parity |φ even ⟩ which is located at 2.04 eV [38,39].The energy positions of the one-photon-allowed-and forbidden-excitons are indicated by the red and green bars, respectively, in figure 2(d).Those studies also revealed that the continuum starts at the higher energy, approximately 2.15 eV, while the oscillator strength is concentrated on the one-photon allowed exciton with odd-parity at 1.95 eV and that of the interband transition is much smaller, reflecting the pure 1D nature of the electronic state.Figure 2(e) schematically shows the envelope of the doublon wavefunction in an odd-parity (even-parity) exciton, |φ odd ⟩ (|φ even ⟩), when the hole is placed at the central Cl site.The spatial extensions of these excitons are almost equal except for their phases.As a result, the energy interval between the two excitons E s is very small (∼90 meV), and the transition dipole moment ⟨φ odd |x| φ even ⟩ between them is high (∼11 Å), which is responsible for the large third-order optical nonlinearity.Moreover, the spectral width of odd-parity (even-parity) exciton is small at 27 meV (53 meV) at 77 K [38,39].Considering that spectral widths are generally very broad in correlated electron systems, NCN is one of the exceptionally suitable materials for spectrally resolved observations of the photon-dressed states.

Sample preparations
Single crystals of [Ni(chxn) 2 Cl](NO 3 ) 2 were grown using an electrochemical method based on a previous report [36].The typical crystal size was approximately 1.5 × 0.3 × 0.2 mm 3 .These single crystals have good flat surfaces suitable for the optical measurements and fairly stable in air.Nearly similar data can be obtained with each crystal.

MIR pump sub-cycle reflection probe spectroscopy
Figure 3(a) shows a schematic of the MIR pump visible sub-cycle reflection probe spectroscopy.In this measurement, a Ti:sapphire regenerative amplifier (RA) with a central photon energy of 1.55 eV, repetition rate of 1 kHz, pulse duration of 35 fs, and pulse energy of 7.5 mJ was used as the light source.The RA output is divided into two beams.One of the outputs is fed as an input to a handmade noncollinear optical parametric amplifier (NOPA) to generate ultrashort visible pulses.The details of the NOPA have been previously reported [40,41].To stabilize the temporal profile of the probe pulses, this study adopts a method to control the chirp using a static setup with chirped mirrors and fused silica plates [41].The intensity spectrum of the probe pulses, which was measured using a calibrated spectrometer, is shown by the green line in figure 3(d).The phase spectrum of the electric fields of probe pulses, which was measured using a second-harmonic generation frequency-resolved optical gating (SHG-FROG) method, is shown by the grey line in figure 3(d).The temporal profile of the visible probe pulse is shown in figure 3(e).Its full temporal width at half maximum is 6.9 fs.Details of the SHG-FROG measurements and evaluation procedures of the temporal profile of the probe pulse are reported in appendix A.
The other output of RA is used to excite two optical parametric amplifiers (OPAs).A carrier envelope phase (CEP)-stable MIR pulse was obtained by a difference-frequency generation between the idler outputs of two OPAs [42,43].To stabilize the long-term CEP of the MIR pulses, we performed feedback control of the optical path lengths.The standard deviation of the CEP of MIR pulses was within 200 mrad (1 fs) over a few tens of hours at the sample position.The electric-field waveform of the MIR pulse, E MIR (τ ), was measured using the electro-optic sampling (EOS) method, in which a part of the probe pulse is used as the sampling pulse.The details of the generation and stabilization methods and the EOS of MIR pulses have been previously reported [44].The electric-field waveform and Fourier power spectrum of the MIR pulse are depicted by the red lines in figures 3(b) and (c), respectively.The central frequency Ω (period T) of the MIR pulse is 31.5 THz (31.7 fs), and its carrier-envelope phase is kept constant.The temporal width of the probe pulse, 6.9 fs, is much shorter than the value T = 31.7 fs of the MIR pulse, due to which implementation of sub-cycle spectroscopy is possible.The photon energy of the MIR pulse (130 meV) is much smaller than the lowest one-photon allowed excitonic state of NCN at 1.95 eV, and the material is almost transparent to this MIR pulse (see appendix B).Therefore, we can investigate pure electric field effects on the electronic states in NCN.
In the actual MIR pump visible sub-cycle reflection probe measurements on NCN, a single-crystal sample was placed on the cold sample holder of a conduction-type optical cryostat.The measurement temperature was set to 77 K, at which the energy level structures of the excited states were previously evaluated based on the analyses of the ER spectra [38,39].The electric fields of both the MIR pump pulse and visible probe pulse were set parallel to the Ni-Cl chain axis b in NCN.The MIR pump and visible probe pulses were focused almost coaxially toward the sample such that the angle between them was smaller than 20 mrad.This was to avoid a decrease in the time resolution owing to non-overlapping of the wavefront of the MIR pulse and pulse front of the visible pulse.The spot diameter (full width at half maximum) of the MIR pump pulse and ultrashort visible probe pulse on the sample surface is 55 µm and 30 µm, respectively.The fluence of the MIR pulse at an electric field amplitude of 0.9 MV cm −1 is approximately 0.1 mJ cm −2 .The intensity of reflected light, including emission from the photon-dressed states, was detected using a silicon photodiode.To extract the reflected light at a specific photon energy hω p , a bandpass filter (BPF) was inserted in front of the detector.For all BPFs used, transmission spectra were measured using a well-calibrated spectrometer.As a transmission photon energy of each BPF, the central value of the transmission band obtained in the experiment was used.The bandwidths of the BPFs were approximately 30 meV (10 nm).Unless mentioned otherwise, the BPFs having those bandwidths were used.By changing the transmissive photon energy of BPFs, the responses were measured at 17 different photon energies ranging from 1.74 eV to 2.26 eV.The time origin was set to the peak of the envelope of the MIR pulse.The electric field amplitude E MIR of the MIR pulse was also defined by its envelope value at the time origin.The delay time τ of the probe pulse from the time origin was controlled by a mechanical delay stage.

Results
Figure 3(f) shows the time characteristic of the reflectivity changes ∆R (τ ) /R caused by the MIR pulse with an electric-field amplitude E MIR = 0.9 MV cm −1 as a function of delay time τ of the visible probe pulse relative to the MIR pulse.The detection photon energy of the probe pulse is hω p = 2.18 eV.∆R (τ ) /R reaches 46%, which surprisingly indicates a reflectivity increase of approximately 1.5 times.More specifically, ∆R (τ ) /R consists of a low-frequency component I low (τ ) and a high-frequency oscillation I high (τ ).I high (τ ) was extracted by a high pass filter and shown in figure 3(g).I low (τ ) was obtained by subtracting I high (τ ) from ∆R (τ ) /R and shown by the green line in figure 3(f), that varies along the intensity envelope of the MIR pulse (the red broken line in figure 3(b)).The oscillation amplitude in I high (τ ) is also very large, reaching approximately 15% of the original reflectivity.Its Fourier power spectrum is indicated by the blue line in figure 3(c).The central frequency is 63 THz, which is exactly 2Ω, that is, twice the central frequency Ω of the MIR pulse.The ∆R (τ ) /R signal diminishes instantaneously when the MIR electric field disappears.
In figure 4(a), we show the magnitudes of I low (τ ) defined by their values at 0 fs, I low (0 fs), and the oscillation amplitudes of I high (τ ) defined by the envelope value of I high (τ ) at the time origin, as a function of the electric-field amplitude E MIR of the MIR pulse.Both are proportional to indicating that the observed responses were dominated by third-order optical nonlinearity in this electric-field range.In figure 4(b), we show the same data as figure 4(a) up to E MIR of 4.5 MV cm −1 in linear scale.At E MIR > 1 MV cm −1 , I low (0 fs) and the amplitude of I high (τ ) tend to saturate and eventually start to decrease, which is attributable to the appearance of the higher order nonlinear optical responses.This decreasing behavior is discussed later in the Discussion section referring to the electric-field dependence of wavefunctions of photon-dressed Floquet states theoretically calculated.Next, we measured ∆R (τ ) /R by changing the detection photon energy hω p .The results are shown in figure 5(a).The magnitude and sign of the low-frequency components I low (τ ) and the amplitude and phase of the high-frequency oscillatory components I high (τ ) show different detection photon-energy dependences, while the temporal profiles of I low (τ ) or I high (τ ) are approximately independent of hω p .A two-dimensional (2D) plot of ∆R (τ ) /R as a function of τ and hω p is shown in figure 5(b).The magnitude of I low (0 fs) for each hω p is indicated by circles in figure 5(c).The amplitude and phase of the high-frequency oscillation I high (τ ) for each hω p are indicated by circles in figures 5(d) and (e), respectively.For the oscillation phase, the value of the Fourier component of ∆R (τ ) /R at 63 THz was used.With the increase in hω p from the low energy side, the sign of I low (0 fs) changes from positive to negative around 1.85 eV and then changes to  positive again above 2 eV (figure 5(c)).The oscillation amplitude has two peaks at approximately 1.9 and 2.2 eV (figure 5(d)).With the increase in hω p from the low energy side, the oscillation phase first decreases from 1.8 eV to 2 eV and then increases above 2 eV, forming a V shape (figure 5(e)).To understand the mechanism of the detection photon energy dependence of the optical responses, as observed in ∆R (τ ) /R, we performed a simulation of the results in figures 5(c)-(e), in which the MIR electric field was incorporated as a perturbation, and the changes in reflectivity were calculated by considering nonlinear polarization to the third order.The reflectivity change at frequency ω, ∆R (ω, τ ) /R (ω) , is expressed as a function of τ as follows: E p (ω) and r (ω) are the electric field of the incident probe pulse and complex amplitude reflectivity, respectively.The change in the electric field of the detected light ∆E p (ω, τ ), which originates from the third-order nonlinear polarization created by the MIR pump pulse and the visible probe pulse, is expressed as follows (see appendix C): ( ñ (ω) is the complex refractive index, tsv (ω) ( tvs (ω)) is the amplitude transmittance of light from the sample (vacuum) to vacuum (sample), E MIR (ω) is the electric field of the pump pulse, and χ (3) is the third-order nonlinear susceptibility that dominates the nonlinear polarization generated by E MIR (ω 1 ), E MIR (ω 2 ), and E p (ω − ω 1 − ω 2 ).∆k denotes the phase mismatch in the nonlinear process.Among the physical quantities in equations ( 1), (2), the linear optical constants, namely ñ (ω), r (ω), tsv (ω), and tvs (ω) , were derived from the reflectivity spectrum (figure 2(d)) using the Kramers-Kronig transformation.χ (3) was calculated as follows.In NCN, the spectral weight of linear absorption is concentrated at the odd-parity exciton |φ odd ⟩.This state and the even-parity exciton |φ even ⟩ dominate third-order optical nonlinearities as mentioned above.To reproduce the nonlinear optical spectra precisely, it is necessary to consider another odd-parity state |φ odd2 ⟩, which is located near the edge of the continuum that begins from a level slightly above |φ even ⟩.The previous study reported that the ER and THG spectra were well reproduced by the four-level model consisting of these three excited states and the ground state φ g [39].χ (3)  2) can be calculated using the four-level model.The derivation of χ (3) For the parameters concerning the three excited states, that is, the energy positions, widths, and related transition dipole moments, values deduced from the ER spectrum were used and listed in table 1.
Figure 5(f) shows the simulated time and spectral characteristics of ∆R (ω, τ ) /R (ω).In this simulation, the Fourier components of ∆R (ω, τ ) /R (ω) with frequencies above 30 THz were multiplied by 0.5 to reproduce the amplitudes of the experimental high-frequency oscillation I high (τ ).This is probably because the experimentally obtained amplitudes were suppressed owing to the short-time phase fluctuations of the MIR pulses and resultant decrease in the effective temporal resolution.The overall features of the simulated ∆R (ω, τ ) /R (ω) values are in good agreement with those of the experimental results in figure 5(b).The solid lines in figures 5(c)-(e) show the magnitude of I low (0 fs) and the amplitude and phase of I high (τ ) obtained by the present simulation, which reproduce both their experimental spectral shapes and absolute values (circles) without any parameter adjustments other than a constant multiplication of the oscillation amplitude mentioned above.
It is important to consider the effects of the spectral shape and chirp of the probe pulse on the simulations of reflectivity changes as indicated by the solid lines in figures 5(c)-(e).The high-frequency oscillation I high (τ ) in the reflectivity changes is affected by the spectral shape and chirp of the probe pulse, especially below 1.75 eV and above 2.2 eV, where the intensity and phase largely decrease as seen in figure 3(d).This is because I high (τ ) originates from the process in which the frequency of emitted light shifts by ±2Ω from that of the input probe pulse.Since the change in the electric field ∆E p (ω, τ ) of the light reflected (or emitted) from the sample induced by the MIR electric field in such a process is proportional to the electric field of the incident probe light E (ω ∓ 2Ω), the amplitude and phase of I high (τ ) are affected by the ratio E (ω ∓ 2Ω) /E (ω) of the electric fields of the incident probe pulse.To exclude the effects of the spectral shape and chirp of the probe pulse, we simulated the magnitude of I low (0 fs) and the amplitude and phase of I high (τ ) assuming that the temporal width of the incident probe pulse was infinitesimal and its electric field had no frequency dependence, that is, E (ω) = constant.The results are indicated by the broken lines in figures 5(c)-(e).The simulation results shown by the solid and broken lines for the low-frequency component I low (0 fs) (figure 5(c)) are in perfect agreement with each other.However, the solid and broken lines in the amplitude and phase of I high (τ ) are different below 1.80 eV and between 1.95 and 2.05 eV.These differences are attributable to the chirping of the probe pulse below 1.8 eV as seen in figure 3(d).Since the probe pulse has a finite intensity above 1.7 eV, the phase in the region of 1.96-2.06eV corresponding to the region of 1.7-1.8eV shifted by 2Ω (= 0.26 eV) should be affected by the chirp as well as in the region of 1.7-1.8eV.Except for these two regions, the amplitude or phase spectra shown by the broken and solid lines are nearly identical as seen in figures 5(d) and (e).From these analyses, we can consider that the broken lines in figures 5(c)-(e) reflect the actual response of the photon-dressed states in NCN.

Discussion
This section provides an interpretation of the characteristic spectral shapes of the magnitude of the low-frequency component, I low (0 fs), and the amplitude and phase of the high-frequency oscillation I high (τ ) shown in figures 5(c)-(e) and the mechanism of the dynamical responses of photon-dressed excitonic Floquet states.For this purpose, we simply consider two original excitons |φ odd ⟩ and |φ even ⟩ and their photon-dressed Floquet states under the MIR electric field, |φ odd , n⟩ and |φ even , n⟩, and neglect the higher-energy odd-parity state |φ odd2 ⟩.This is because the nonlinear optical responses in NCN are dominated by these two exciton states, |φ odd ⟩ and |φ even ⟩, and the contributions of |φ odd2 ⟩ are small [38,39].
First, we calculate the Floquet quasi-energies of photon-dressed states of the two excitons |φ odd ⟩ and |φ even ⟩, and the ground state φ g subjected to a monochromatic MIR electric field.In the case where the Hamiltonian is periodic with respect to time, the solution of the Schrödinger equation is deduced from Floquet's theorem as shown below [4,9,10]: where |Φ α (t) ⟩ is a wavefunction with the same periodicity as that of the Hamiltonian and ε α is the Floquet quasi-energy.The Hamiltonian and |Φ α (t) ⟩ can be expanded into a Fourier series as follows: Substituting these expressions in the Schrödinger equation yields the following time-independent form of the eigenvalue problem [4,9]: This can be represented using a matrix as follows: Figure 6.Electric field amplitude E0 dependences of (a), (b) quasi-energies and (c)-(e) magnitudes of the photon-dressed exciton states and the ground state in NCN calculated using equation ( 6).(a) Red and green solid lines show quasi-energies of the photon-dressed odd-and even-parity excitons, respectively.The horizontal red and green broken lines show the original energies of hΩ odd and hΩeven, respectively, at E0 = 0 MV cm −1 .(b) Quasi-energy of the photon-dressed ground state.The horizontal broken line represents the original energy of the ground state at E0 = 0 MV cm −1 , which is zero.The vertical broken line indicates E0 = 0.9 MV cm −1 , which is equal to the electric-field amplitude of the MIR pulse, EMIR, used in the experiments.(c)-(e) Magnitudes of (c) and (e) photon-dressed excitonic states, ⟨φ odd , n|φ odd , n⟩ and ⟨φ even, n|φ even, n⟩ (n = 0, ±1, and ±2), and (e) the ground state, ⟨φ g, 0|φ g, 0⟩.
Ĥ0 is the Hamiltonian in the absence of an external electric field of light.As mentioned above, we consider only two exciton states |φ odd ⟩ and |φ even ⟩ and the dipole moment between them ⟨φ odd |x| φ even ⟩ is the major factor in the off-diagonal part of ĤF [38].In this case, each part of the Hamiltonian can be expressed as follows Here, E 0 is the amplitude of the electric field of light.
To calculate the Floquet quasi-energies of |φ odd } , |φ even } , and |φ g } of NCN, we solve the eigenvalue problem in equations ( 6)-( 9) by using the parameters listed in table 1 and changing the size of the matrix in equation (6), that is, the value range of m,−N c ⩽ m ⩽ N c , in equation (5).When N c is larger than 20, all the quasi-energies of |φ odd } , |φ even } , and |φ g } are constant with increasing N c for electric-field amplitudes E 0 < 7.5 MV cm −1 .In figures 6(a) and (b), we plot the quasi-energies of |φ odd }, |φ even } , and |φ g } corresponding to the energies ε odd , ε even , and ε g of |φ odd , 0⟩, |φ even , 0⟩, and the ground state, respectively, for N c = 40 as a function of E 0 .
The energy splitting between |φ odd ⟩ and |φ even ⟩, E s , which is 90 meV under no electric fields, is decreased as E 0 increases.This is related to the fact that the photon energy (frequency) of the MIR pump pulse, hΩ = 130 meV (Ω = 31.5THz), is larger than the original energy splitting of 90 meV.In this case, the energies of |φ odd , 0⟩ and |φ even , 0⟩ are expected to approach each other under the MIR electric field, similarly to the case of AC Stark effect.In figure 6(a), this feature appears up to approximately 1.2 MV cm −1 .As E 0 further increases, the energies of the odd-and even-parity excitons cross each other and oscillate.This is considered to be characteristic of the formation of photon-dressed excitonic states by the irradiation of the MIR light.Those behaviors at the high electric-field amplitudes are discussed later again.The reason for such large changes in the energies of these excitons is that the original energy difference between the two excitons (approximately 90 meV) is close to the photon energy of the MIR light (130 meV) and the transition dipole moment between them is very large (⟨φ odd |x| φ even ⟩ = 11.5 Å) (table 1).On the other hand, the calculation result in figure 6(b) shows that the MIR light hardly any changes in energy in the ground state.It is because the ground state is approximately 2 eV away in energy from these exciton levels, and the transition dipole moment between the ground state and the lowest odd-parity exciton is relatively small (⟨φ g |x| φ odd ⟩ = 1.08 Å) (table 1).Thus, for the ground state, the formation of the photon-dressed state can be neglected.
As seen in figures 6(a) and (b), at E 0 = 0.9 MV cm −1 , which corresponds to the electric field amplitude of the MIR pulse, E MIR , used in our main spectroscopic experiments shown in figure 5, the higher (lower) energy shift of |φ odd , 0⟩ (|φ even , 0⟩) is approximately 30 meV, whereas the energy of the ground state shifts only by approximately −0.025 meV.Taking these results into account, we schematically show the expected energy positions of the excitonic states and their photon-dressed Floquet states in figures 7(a)-(c).Based on the spatial inversion symmetry of NCN, the even-order sidebands |φ odd , n⟩ (n = 0, ±2, ±4, . ..) of |φ odd ⟩ and odd-order sidebands |φ even , n⟩ (n = ±1, ±3, . ..) of |φ even ⟩ are optically allowed and shown by thick solid lines in figures 7(b) and (c).The positions of the optically allowed sidebands located in the range of the probe-pulse spectrum are represented by the vertical red and green lines in figures 5(c)-(e).
On the basis of these theoretical expectations, we can discuss the spectrum of the magnitude of I low (0 fs) in figure 5(c).Using the MIR electric field, the original transition from φ g to |φ odd ⟩ shown by blue arrows in figure 7(a) was converted into transitions from φ g to |φ odd , 0⟩ and |φ even , ± 1⟩ shown by the blue arrows in figures 7(b) and (c), respectively.I low (0 fs) is negative at 1.9-2.05eV (figure 5(c)) because of the decrease in intensity and higher energy shift of the transition from φ g to |φ odd , 0⟩ (figure 7(b)) compared to those of the original transition from φ g to |φ odd ⟩ (blue arrows in figure 7(a)).The increases of I low (0 fs) at ∼1.8 and ∼2.2 eV can be ascribed to the appearance of transitions from φ g to |φ even , −1⟩ and |φ even , +1⟩, respectively (blue arrows in figure 7(c)).In addition, transitions from φ g to photon-dressed states |φ odd , ±2⟩ can exist in the spectral region of the probe pulse (the green line in figure 3(d)).However, they correspond to fifth-order optical nonlinearities, and their intensities should be proportional to (E MIR ) 4 .
Because ∆R (ω, τ ) /R (ω) signals in the electric-field (E MIR ) region where the data in figure 5 are measured are proportional to (E MIR ) 2 as mentioned above, |φ odd , ±2⟩ does not contribute to I low (0 fs).
Next, we focus on the characteristic spectral shapes of the amplitude and phase of the high-frequency oscillation I high (τ ) shown in figures 5(d) and (e), respectively.For this purpose, we detail the reason why the high frequency oscillation was observed at the wide photon energy region as seen in figures 5(a) and (d).
When the probe pulse is incident under the MIR electric field, the reflected light at the detection photon energy includes the emissions via inelastic scattering processes as shown by purple arrows in figures 7(d)-(h).These processes can be formulated in the Floquet framework [22,32].Here, we consider the optically allowed states covered by the probe pulse, |φ odd , 0⟩, |φ odd , ± 2⟩, and |φ even , ±1⟩, and second order scattering processes related to these states.Scattering caused by the MIR electric field induces a frequency change between the incident and emitted light by ±2Ω.We illustrate this scattering process using figure 7(f) as an example.When the transition from φ g to |φ odd , 0⟩ is generated by the incident probe pulse under an MIR electric field of frequency Ω, emission from |φ odd , +2⟩ to φ g can occur via the scattering process, accompanied by a frequency shift of +2Ω.The phase difference between the incident and emitted light varies with τ as +2Ωτ .The emitted light, represented by the purple downward arrow with the open circle in the right part of figure 7(f), interferes with light at the same frequency reflected by a linear optical process, shown by the blue downward arrow with the open circle in the left part.Consequently, the high-frequency oscillation I high (τ ) with a frequency of 2Ω appears as τ increases.Note that the broadband probe pulse can simultaneously generate two excitations with different frequencies shown by the upward blue and purple arrows in figure 7(f).In the other cases depicted in figures 7(d), (e), (g) and (h), similar scattering processes are expected to occur, as represented by the purple arrows with the phase difference, ±2Ωτ , and the light emitted by the scattering process (purple downward arrows with circles) and linear optical process (blue downward arrows with circles) interfere with each other.As a result, high-frequency oscillations I high (τ ) with a frequency 2Ω can appear in all the cases in figures 7(d)-(h) and therefore are indeed observed in the wide photon energy region as seen in figure 5(a).Now, we can discuss the amplitude and phase spectra of the high-frequency oscillation I high (τ ) (figures 5(d) and (e)).Because the amplitude of I high (τ ) is proportional to the electric field amplitude of the scattered light (purple downward arrows in figures 7(d)-(h)), it should be enhanced at energies of |φ even , ±1⟩, |φ odd , 0⟩ and |φ odd , ±2⟩.In the case of figure 7(d), however, the oscillation amplitude at |φ odd , 0⟩ is not enhanced significantly because the phase changes of the two processes denoted by the purple arrows in the middle and right parts, −2Ωτ and +2Ωτ , respectively, cancel each other.In addition, in the region of |φ odd , −2⟩ around 1.7 eV, the probe pulse is relatively weak (see the green line in figure 3(d)), making it difficult to detect the oscillation signals due to the scattering process shown in figure 7(e).Thus, the enhancements of the oscillation amplitudes are observed only around |φ even , ±1⟩ and |φ odd , +2⟩ as seen in figure 5(d).
The characteristic V-shape observed in the phase of I high (τ ) (figure 5(e)) originates from the complex phase of χ (3) .That is, the generation of third-order nonlinear polarizations and the resultant light emissions are reflected by the phase of I high (τ ).The probe energy dependences of the important factors determining the phase and amplitude of I high (τ ) are detailed in appendix E. In the simulation, the energy shifts of the electronic states were not considered in processes involving sidebands with n ̸ = 0 because they correspond to fifth-order optical nonlinearities.
It should be emphasized that the high-frequency oscillation reported here could be observed only with the subcycle spectroscopy.If the temporal width of the probe pulse is almost the same as that of MIR pump pulse (approximately 100 fs), its spectral width is shortened to ∼20 meV.In this case, the measurement is an ordinary pump-probe reflection spectroscopy, in which the response of the probe pulse depending on the oscillating electric field of the MIR pump pulse, can be detected only as a time-averaged response.To detect inelastic scattering processes in this time-averaged measurement, it is necessary to detect light emitted at a photon energy different from that of the incoming probe pulse.However, the intensity of this light is proportional to the square ( ∆E p 2 ) of the change in the detected light, ∆E p , as seen in equation ( 1), and therefore is extremely low and difficult to detect.On the other hand, the subcycle spectroscopy uses probe pulses with a short temporal width and wide spectral width, making it possible to measure the interference between light due to the linear response and light due to an inelastic scattering process, as represented in figures 7(d)-(h).In this case, the amplitude of the oscillatory structure due to this interference is, therefore, proportional to the change in the electric field of the emitted light, ∆E p , as seen in equation ( 1).As a result, the subcycle spectroscopy can detect the inelastic scattering processes with higher sensitivity than ordinary pump-probe spectroscopy which measures the time-averaged signal.Furthermore, only the subcycle spectroscopy can obtain the phase information of the inelastic scattering processes.So far, the spectral features of the magnitude of I low (0 fs), and the amplitude and phase of I high (τ ) shown in figure 5  As mentioned above, these responses can be treated in the framework of third-order optical nonlinearity.If E MIR is further increased, higher-order responses involving the Floquet side bands are expected to be detected, and their signatures should appear in the response to the MIR pulse with E MIR > 1 MV cm −1 shown in figure 4(b).To obtain information of those responses to the higher electric fields, the magnitudes of the sidebands, ⟨φ odd , n|φ odd , n⟩, ⟨φ even , n|φ even , n⟩, and ⟨φ g , n|φ g , n⟩ were also calculated using equation ( 6) and those of several low-lying sidebands are plotted as a function of E 0 in figures 6(c)-(e).At the low electric fields of E MIR ⩽ 0.5 MV cm −1 , the magnitudes of the first-order sideband states, ⟨φ odd , ±1|φ odd , ±1⟩ and ⟨φ even , ±1|φ even , ±1⟩, increase in proportion to (E MIR ) 2 .When E 0 exceeds 1.5 (2.3) MV cm −1 , the magnitudes of ⟨φ odd , +1|φ odd , +1⟩ and ⟨φ even , −1|φ even , −1⟩ (⟨φ odd , −1|φ odd , −1⟩ and ⟨φ even , +1|φ even , +1⟩) decrease, and the magnitudes of the second-order sideband states ⟨φ odd , −2|φ odd , −2⟩ and ⟨φ even , +2|φ even , +2⟩ (⟨φ odd , +2|φ odd , +2⟩ and ⟨φ even , −2|φ even , −2⟩) increase, being dominant above ∼3 MV cm −1 .Keeping these calculation results in our mind, we discuss the behaviors of the amplitude of I high (τ ) and the magnitude of I low (0 fs) at the higher electric field for E MIR > 1 MV cm −1 shown in figure 4(b).The central energy of the BPF used in the measurement in figure 4(b) is 1.89 eV, which corresponds to the energy position of |φ even , −1⟩ (figure 5(c)).Therefore, the high-frequency oscillatory component characterized by I high (τ ) is caused by the process as shown in figure 7(g), i.e.The radiation from |φ even , −1⟩ after the probe light is absorbed at |φ even , +1⟩.When the magnitudes of ⟨φ odd , ±1|φ odd , ±1⟩ and ⟨φ even , ±1|φ even , ±1⟩ decrease, the dipole moments between the sideband states and the ground state, ⟨φ g |x|φ even , ±1⟩ should be suppressed.Since the intensity of the light radiated from |φ even , −1⟩ is dominated by the product of these dipole moments, ⟨φ g |x| φ even , −1⟩ × ⟨φ g |x| φ even , +1⟩, it should be proportional to (⟨φ even , −1|φ even , −1⟩ × ⟨φ even , +1|φ even , +1⟩) 1 2 ≡ I even,±1 .The electric-field (E 0 ) dependence of the calculated I even,±1 values is shown by the red broken line in figure 4(b), which roughly reproduces the E MIR dependence of I high (τ ) (the purple squares) up to approximately 2 MV cm −1 .Strictly, the maximum of the amplitude of I high (τ ) is located at E MIR ∼ 2.4 MV cm −1 (figure 4(b)), which is higher than E 0 ∼ 2 MV cm −1 at which I even,±1 takes the maximum (figure 4(b)), and the deviation of I high (τ ) from I even,±1 is enhanced with the increase of E 0 .This may be attributed to several possible reasons.The first possible reason is that this analysis only considers the process in figure 7(g).Under the high electric fields, the processes shown in figures 7(d) and (e) in which the radiation photon energies are relatively close to the probe energy (1.89 eV), could also contribute to the responses.The second possible reason is that the higher order nonlinear processes, which are not included in the simple theoretical framework presented above, appear.For example, the nonlinearly scattered probe light is so large at such electric field amplitudes, comparable in amplitude to the incident probe light, that cascaded nonlinear processes may take place i.e. light radiated from a sideband may be absorbed by a sideband again, repeating the processes multiple times inside the crystal.The resulting reflectivity change should no longer be proportional to the magnitude of the response due to photon-dressed states, and it cannot be described precisely by a single scattering event.The other possibility is that the E MIR values obtained in the experiments are slightly overestimated since it is determined from the MIR electric-field waveform and the total fluence of the MIR pulse.As for the low-frequency component at the time origin, I low (0 fs), its spectrum (figure 5(c)) is considerably broadened, and the response related to |φ odd , 0⟩ overlaps it.Therefore, the magnitude of |I low (0 fs)| does not show a clear relation with the magnitudes of the sideband states characterized by I even,±1 as compared to the amplitude of I high (τ ).A more accurate reproduction of these experimentally observed responses to the strong electric fields is a future subject.
Finally, we compare the response to a periodic electric field of light in NCN with those previously reported in other insulating materials.When band insulators of diamond and MgF 2 are excited with a near-IR pulse having a photon energy of hΩ below their band gaps, an oscillation with a frequency 2Ω is observed in the reflectivity change in the extreme ultraviolet (XUV) region far above their band gap energies [20,28].In those near-IR pump XUV sub-cycle probe spectroscopic studies, not only the 2Ω oscillation appears, but also its oscillation phase shows a V-shaped dependence on the photon energy of the XUV probe pulse.These phenomena are interpreted as the dynamics of conduction electrons due to the electric field of the near-IR pulse corresponding to the dynamical Franz Keldysh effect (DFKE), which is reflected in the interband transition in diamond [20] or in the interband and exciton transitions in MgF 2 [28].The V-shaped probe energy dependence of the phase of the 2Ω oscillations observed in these studies resembles the result in figure 5(e) observed in NCN, but the mechanism is quite different.As mentioned above, the V-shaped phase shift in figure 5(e) is not related to the dynamics of band electrons but can be explained in the Floquet framework that takes only the odd-and even-parity excitons, and ground state into account.Namely, this V-shaped phase shift is essentially different from the previously reported ones due to DFKE and is characteristic of the response of well-defined excitons to the oscillating electric field.Hence, the present result does not contradict with the previous studies.It is interesting that the responses in these two cases possess some similarities although they originate from different mechanisms.

Summary
In summary, we performed sub-cycle reflection spectroscopy using a phase-stable MIR pulse and an ultrashort visible pulse on a chlorine-bridged nickel-chain compound to investigate the photon-dressed Floquet states of a 1D Mott insulator.An MIR pulse with an amplitude of 0.9 MV cm −1 induced a reflectivity change of approximately 50%, which included the low-frequency component and high-frequency oscillation.The simulation considering nonlinear polarizations of excitonic states to the third order reproduced well the amplitude and phase spectra of the high-frequency oscillations as well as the spectrum of the low-frequency component.This demonstrated the formation of photon-dressed excitonic Floquet states.The observation of the high-frequency oscillation with a large amplitude reaching 15% of the original reflectivity suggests that the intensity of a visible or near-IR pulse can be controlled by changing the electric field phase of the MIR pulse.This phenomenon can never be obtained in the studies of the time-averaged responses of photon-dressed Floquet states but can be used as an advanced mechanism for ultrafast optical switching devices with high repetition rates.algorithm.The intensity spectrum of the probe pulse directly measured by a spectrometer and that retrieved from the FROG trace are shown in figure 9, in which the intensities of the two spectra are normalized so that their integrated intensities are the same.The two spectra were in good agreement with each other, thus confirming that the retrieval was performed successfully.The retrieved phase spectrum the probe pulses is shown by the grey line in figure 3(d).The phase changes are sufficiently small in the range of 1.75-2.3eV.The temporal profile of the probe pulses obtained by the retrieval is shown by the solid green line in figure 3(e).The absorption peak in the range of 155-180 meV observed in NCN does not exist in NCC.Therefore, this peak can be assigned to the vibrational mode related to NO 3 − ions.The red line in figure 1(a) shows the spectrum of the pump pulse in MIR pump visible-reflectivity probe spectroscopy reported in the main As can be seen in figure 10, there are several sharp peaks in the spectral range of the MIR pump light.Therefore, it might be possible that an inelastic scattering of the probe light by these phonon modes causes the reflectivity to oscillate with respect to the probe light at twice its frequency, and a response like that shown in figure 3(f) could be observed.If this is the case, such a response due to phonons should persist as long as the phonons continue to oscillate after the MIR pump light disappears.As seen in figure 3, however, the reflectivity change observed in the experiment quickly returns to zero after the disappearance of the MIR pump light, demonstrating that the signals are not due to phonons.This feature of the signals also rules out the possibility that the reflectivity is modulated by a Raman-active phonon excited by two-photon absorption of the MIR pulse.In fact, it was reported from a previous Raman scattering measurement on NCN that no Raman-active modes exist in the region corresponding to twice the frequency of the MIR pulses [47].
where the superscript signs in f ± (ω), A ± (ω), B ± (ω), C ± (ω), and D ± (ω) correspond to the signs of the frequency shifts.A ± (ω), B ± (ω), C ± (ω), and D ± (ω) are the terms expressing light emission originating from the nonlinear polarization, factor caused by the amplitude transmittance and reflectivity at the sample surface, factor due to the phase mismatches inside the sample, and factor due to the spectral shape of the incident probe pulse, respectively.
The phase and amplitude of I high (ω, τ ) calculated using equations (A15)-(A17) are shown in figures 11(a) and (b), respectively.To determine the contributions of A ± (ω), B ± (ω), C ± (ω), and D ± (ω) individually in equation (A17), the phase of I high (ω, τ ) for each term is calculated by setting the complex phases of other terms in f ± (ω) * + f − (ω) in equation (A15) to zero.For example, to evaluate the contribution of A ± to the phase of f ± (ω), f ± (ω) = A ± |B ± C ± D ± | is substituted in equation (A15), and the phase of I high (ω, τ ) is calculated.The effective phase values of the four terms A ± (ω), B ± (ω), C ± (ω), and D ± (ω) thus obtained are shown in figures 11(c)-(f), respectively.Their amplitudes are shown in figures 11(g)-(j), respectively.It can be seen that both the V-shape in the phase spectrum and peak structures in I high (ω, τ ) are owing to the term A ± (ω).

Figure 2 .
Figure 2. Electronic structure and excitonic states of [Ni(chxn)2Cl](NO3)2.(a) Crystal structure.(b) Electron configuration in the 3d orbitals of Ni 3+ ion.(c) Band structure consisting of the Ni-3d upper-Hubbard and lower-Hubbard bands along with the occupied Cl-3p valence band.(d) Spectra of reflectivity R and imaginary part of dielectric constant ε2 with the electric field of light parallel to the Ni-Cl chain axis b.(e) Wavefunctions of odd-parity exciton |φ odd ⟩ (red line) and even-parity exciton |φ even⟩ (green line).

Figure 3 .
Figure 3. MIR pump visible sub-cycle reflection probe spectroscopy on [Ni(chxn)2Cl](NO3)2.(a) Schematic of the measurement system.Polarizations of the MIR pump and visible probe pulses are both parallel to the b-axis.(b) Typical electric-field waveform of the MIR pump pulse with a frequency of 31.5 THz.(c) Fourier power spectra of the MIR pulse and oscillation I high (τ ) in (g).(d) Intensity and phase spectra of the visible probe pulse.The red line shows the ε2 spectrum.(e) Temporal profile of the probe pulse.(f) Time characteristic of ∆R (τ ) /R at 2.18 eV.The green line is the low-frequency component I low (τ ).(g) Time characteristic of the high frequency oscillation I high (τ ) extracted from ∆R (τ ) /R in (f).

Figure 4 .
Figure 4. MIR electric-field amplitude dependence of low-frequency component and high-frequency oscillation in reflectivity changes with (a) logarithmic scales and (b) linear scales.The blue squares show the magnitudes of the low-frequency component I low (τ ) of the reflectivity changes at the time origin |I low (0 fs)|, and the purple squares show the amplitudes of the high-frequency oscillation I high (τ ).The probe energy was 1.89 eV.The BPF used to obtain these data had a full width at half-maximum of approximately 110 meV.The solid lines in (a) show the linear relation between each signal and (EMIR) 2 .The red broken line in (b) shows the calculated I even,±1 , which is a crude measure of I high (τ ).

Figure 5 .
Figure 5. Detection-energy dependences of transient reflectivity changes and their analyses in [Ni(chxn)2Cl](NO3)2.(a) Detection-energy dependences of ∆R (τ ) /R induced by the MIR pulse shown in figure 3(b).(b) 2D plot of the experimentally measured ∆R (τ ) /R as a function of delay time τ and detection photon energy.(c)-(e), Detection photon-energy dependences of (c) the magnitude of low-frequency component I low (0 fs), and (d) the amplitude and (e) phase of the oscillation I high (τ ) (circles) and their simulation curves (solid and broken lines).The vertical red and green lines show the energy positions of photon-dressed states associated with the odd-parity and even-parity excitons, respectively, that are deduced from the simulation.(f) 2D plot of the simulated ∆R (τ ) /R as a function of delay time τ and detection photon energy.

Figure 7 .
Figure 7. Energy-level structures of photon-dressed excitonic Floquet states and related optical processes in [Ni(chxn)2Cl](NO3)2 under the MIR electric field.(a) Original excitonic states |φ odd ⟩ and |φ even⟩.(b) and (c) Photon-dressed Floquet states of (b) odd-parity exciton |φ odd , n⟩ and (c) even-parity exciton |φ even, n⟩ under the MIR electric field of frequency Ω. Blue lines show linear optical processes involving the incident and emitted light of same frequency.(d)-(h) Light emission via inelastic scattering processes (purple arrows) associated with excitonic Floquet states shown by thick solid lines.In the inelastic scattering processes, the frequencies of the incident and emitted light differ by 2Ω.In each figure, a linear optical process is depicted by the blue arrows in the left part.The light reflected via a linear optical process and the inelastically scattered light, whose phase differs by 2Ωτ , interfere with each other, resulting in an oscillatory response I high (τ ) with frequency 2Ω.
figures 7(b)-(h) are observed among the responses related to the Floquet side bands appearing in succession.As mentioned above, these responses can be treated in the framework of third-order optical nonlinearity.If E MIR is further increased, higher-order responses involving the Floquet side bands are expected to be detected, and their signatures should appear in the response to the MIR pulse with E MIR > 1 MV cm −1 shown in figure4(b).To obtain information of those responses to the higher electric fields, the magnitudes of the sidebands, ⟨φ odd , n|φ odd , n⟩, ⟨φ even , n|φ even , n⟩, and ⟨φ g , n|φ g , n⟩ were also calculated using equation (6) and those of several low-lying sidebands are plotted as a function of E 0 in figures 6(c)-(e).At the low electric fields of E MIR ⩽ 0.5 MV cm −1 , the magnitudes of the first-order sideband states, ⟨φ odd , ±1|φ odd , ±1⟩ and ⟨φ even , ±1|φ even , ±1⟩, increase in proportion to (E MIR )2 .When E 0 exceeds 1.5(2.3)

Figure 9 .
Figure 9. Spectrum of the visible probe pulse.The green line shows the spectrum measured directly using a spectrometer.The blue line represents the spectrum retrieved from the FROG trace.The intensities of the two spectra were normalized such that their integrated intensities were the same.

Figures 10 (
Figures 10(a) and (b) shows the absorption spectra measured in the transmission configuration of single crystals of [Ni(chxn) 2 Cl](NO 3 ) 2 (NCN) and [Ni(chxn) 2 Cl]Cl 2 (NCC) with thicknesses of 48 and 56 µm, respectively.NCC is another Ni-Cl chain compound in which the counter ion NO 3 − of NCN is changed to Cl − and has the same crystal structure as NCN.The electric field of light E in the absorption spectra is parallel or perpendicular to the Ni-Cl chain axis b.The absorption peak in the range of 155-180 meV observed in NCN does not exist in NCC.Therefore, this peak can be assigned to the vibrational mode related to NO 3 − ions.The red line in figure1(a) shows the spectrum of the pump pulse in MIR pump visible-reflectivity probe spectroscopy reported in the main