Abstract
The nonequilibrium approach to correlated electron systems is often based on the paradigm that different degrees of freedom interact on different timescales. In this context, photoexcitation is treated as an impulsive injection of electronic energy that is transferred to other degrees of freedom only at later times. Here, by studying the ultrafast dynamics of quasiparticles in an archetypal strongly correlated chargetransfer insulator (La_{2}CuO_{4+δ}), we show that the interaction between electrons and bosons manifests itself directly in the photoexcitation processes of a correlated material. With the aid of a general theoretical framework (Hubbard–Holstein Hamiltonian), we reveal that subgap excitation pilots the formation of itinerant quasiparticles, which are suddenly dressed by an ultrafast reaction of the bosonic field.
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Introduction
The exotic electronic ground states of transitionmetal oxides (TMOs), such as hightemperature superconductivity and colossal magnetoresistance, arise from both the strong interactions among electrons and the complex interplay between the electronic and bosonic degrees of freedom such as lattice vibrations and magnetic excitations^{1,2,3,4}. A prime example of such complexity is represented by the cuprate family. Starting from parent compounds which are chargetransfer (CT) insulators^{5,6,7,8,9,10,11,12,13}, anomalous metallic and eventually hightemperature superconducting phases appear on electron or hole doping^{14,15}.
The properties of CT insulators are determined by Coulomb repulsion, which forces the electrons to localize, freezing charge fluctuations. In such a state, the coupling between localized fermions and bosonic excitations, which is mediated by charge dynamics, is naively expected to be strongly suppressed^{16,17}.
Although this scenario is seemingly met when we measure and study the properties of a steadystate Mott insulator, a different situation arises if the system is doped^{18,19,20} or driven out of equilibrium^{13,21,22}. In particular, the naive picture would suggest that if we try to excite the system by means of a pulse with photon energy smaller than the CT gap, because of strong electron correlations, we would observe little effect on the material as the localized carriers have no chance to change their state and, therefore, they have no means to couple to bosonic excitations. In our study, a completely different scenario emerges, in which a coupled bosoncharge mode is excited leading to an effective renormalization of the CT gap.
Note that in the following we use the general term ‘boson’. This is due to the fact that although the Hubbard–Holstein Hamiltonian (HHH) was developed to treat lattice vibrations, the same model can be used to address other kinds of bosonic excitations such as collective electronic modes. The punctual identification of these bosonic modes remains controversial in the cuprates and is beyond the scope of this paper. Further information can be found in refs 23, 24, 25. Here, we show that a model including both local Coulomb interaction and coupling with a boson mode (HHH) accounts for our main experimental observations. The effects of the electron–boson coupling are highlighted by comparing the dynamical optical responses of La_{2}CuO_{4+δ} driven by pulses with photon energy respectively larger and smaller than the gap, which corresponds to the energy needed to create free charge excitations.
Results
Timeresolved measurements on La_{2}CuO_{4+δ}
Transient reflectivity experiments on La_{2}CuO_{4+δ} were performed with abovegap (~3.1 eV) and subgap (~0.95 eV) ~100 fs long pump pulses and broadband probes as detailed in the Methods section. From the measured ΔR(ω, τ)/R(ω)=R(ω, τ)/R(ω)−1 with R(ω, τ), the timedependent pumpperturbed reflectivity, and R(ω), the equilibrium reflectivity, we calculate the pumpinduced changes of the optical conductivity Δσ_{1}(ω, τ)=σ_{1}(ω, τ)σ_{1}(ω) as a function of probe frequency (ω) and pump–probe delay time (τ). Here, σ_{1}(ω, τ) and σ_{1}(ω) are the timedependent lightperturbed and equilibrium optical conductivity, respectively. The CT peak (CTP) in σ_{1}(ω) is at ~2.3 eV, while the gap, estimated from a simple linear interpolation (red dashed line in Fig. 1), is at about Δ=1.8 eV for T=130 K and is consistent with previous reports^{6,7,8,9,26}. The nonvanishing optical conductivity below the gap is due to the presence of tail states. Such tail states originate from the strong interaction between the free charges in excess, coming from oxygen nonstoichiometry^{23}, and the bosonic degree of freedom^{13}. The excess oxygen content δ~5 × 10^{−3} is estimated from the measured Néel temperature of 229 K (ref. 27). Δσ_{1}(ω, τ) is obtained from ΔR(ω, τ)/R(ω) with a novel data analysis methodology based on the KramersKronig (KK) relations. In short, by knowing the equilibrium reflectivity R(ω) and the nonequilibrium ΔR(ω, τ)/R(ω), it is possible to calculate the pumpperturbed reflectivity R(ω, τ) and, from it, all the transient optical properties (see Methods).
Temporal response
The results reported in Fig. 2a,b are for pump pulses with photon energy at E_{pump}=3.1 eV and E_{pump}=0.95 eV, respectively, with ~0.04 eV pulse bandwidth each (see Methods for details). In the first hundred femtoseconds, we observe striking differences between the photoexcitation with photons at energy E_{pump}>Δ (Fig. 2a) and E_{pump}<Δ (Fig. 2b). A smooth rise time (>100 fs) is detected only for E_{pump}>Δ, while subgap excitation leads to a pulsewidthlimited rise time of the signal followed by an ultrafast decay. This difference is made evident in Fig. 2c, where the optical response at the CTP is shown as a function of the pump–probe delay. The results in Fig. 2c are normalized to the energy absorbed by the sample for E_{pump}=3.1 eV (solid blue curve) and E_{pump}=0.95 eV (solid red curve), respectively (see Methods for details). Although in a different physical context, it is worth mentioning here that ultrafast responses to an optical pump below the gap in multiple quantum wells^{28} and below the electronic excitation energy in molecular systems^{29} has been previously reported.
In Fig. 2c, we show that the modifications of the optical response at the CTP induced by the two stimuli are different in the initial dynamics while they become similar for longer delays between the pump and the probe pulses. As the data are normalized to the absorbed energy, the good agreement for large delays indicates that this ‘slow’ response is directly related to absorption, that is, to real electronic excitations. Such ‘slow’ response is consistent with previous timeresolved measurements on La_{2}CuO_{4} (refs 30, 31), Nd_{2}CuO_{4} (refs 30, 31, 32), YBa_{2}Cu_{3}O_{y} (ref. 32) and Sr_{2}CuO_{2}Cl_{2} (ref. 21).
Sub20 fs pulse experiments
With the aim to resolve the ultrafast rise time observed for E_{pump}<Δ, we performed room temperature ~10 fs experiments with E_{pump}<Δ (1.4 eV with 0.1 eV bandwidth) and E_{pump}>Δ (2.25 eV with 0.25 eV bandwidth). In spite of the different excitation parameters used for the ~10 and ~100 fs measurements, we expect the outcome of these experiments to be meaningful for addressing the question concerning the nonthermal ultrafast rise time.
The ΔR(ω, τ)/R(ω) at E_{probe}~2.1 eV is shown in Fig. 2d, while the transient reflectivity spectra at two pump–probe delays are reported in Supplementary Fig. 1 and Fig. 2d. It is evident that also for ~10 fs E_{pump}<Δ pulses the ultrafast shift of the optical response is limited by the temporal resolution of the experiments (Supplementary Fig. 1). Such result sets the upper limit of the timescale of the nonthermal response in La_{2}CuO_{4+δ} to 20 fs. Unfortunately, under the present experimental conditions, a direct quantitative comparison of the thermalization times between the ~100 and the ~10 fs excitations is not meaningful. In particular, the thermalization time is significantly shorter for ~10 fs excitations (see Fig. 2c,d). We attribute this effect to the remarkably different conditions in terms of excitation density and spectral content of the pulses used in the experiments (see Methods).
In the following, we will discuss only the spectral responses of the pump–probe data obtained with ~100 fs pump pulses where the linear response regime has been proven (see Methods).
Energydependent response
We now take advantage of the broadband probe and describe the spectral dependence of the transient response. The energydependent Δσ_{1}(ω, τ) at a delay time τ=0.05 ps averaged over ±0.05 ps (τ=0.3 ps, averaged over ±0.1 ps) is reported in Fig. 3a,b for the two pump energies (blue for 3.1 eV and red for 0.95 eV pulses, respectively).
For E_{pump}>Δ at both pump–probe delays, we observe a depletion of the optical absorption centred at E_{probe}~2.3 eV and an enhanced absorption around E_{probe}~2.1 eV. This behaviour can be ascribed to a ‘thermal response’, that is, it is compatible with the shift towards lower energies of the CTP on heating^{27,30,31} as seen in Fig. 1. This is confirmed by comparing the measured Δσ_{1}(ω, τ) at long pump–probe delays with the expected variation of the equilibrium optical conductivity for a temperature increase resulting from the pumpinduced heating of the sample (black line in Fig. 3b, see Methods for details).
On the contrary, Δσ_{1}(ω, τ) for E_{pump}<Δ reveals a significant spectral evolution with time. The main feature remains an optical conductivity depletion around the CTP, but the energy dependence of the response depends on the pump–probe delay time τ. The Δσ_{1} (ω, τ) has a minimum at E_{probe}~2.2 eV within ~100 fs and drifts towards the thermal response only on longer times.
It is important to notice that for τ≤100 fs and E_{pump}=0.95 eV the optical conductivity displays mainly a negative variation around E_{probe}~2.2 eV, which consists of an overall nonthermal quench of the CTP. This ultrafast variation of the optical conductivity resembles the one induced by small doping at equilibrium (see the dashed line in Fig. 1b), that is, in both cases a broadband negative response at the CTP is present. To understand these experimental results, we now introduce the HHH.
Discussion
In the oneband Hubbard model, the optical gap for free charge excitations is related to the energy cost for unbound double occupancies (doublons). To explain the differences between the responses to subgap and abovegap excitations, one has to consider the two limiting cases predicted by timedependent perturbation theory: nonadiabatic and adiabatic regimes. The nonadiabatic regime describes real transitions between electronic quantum states where the response is present for some time after the pump pulse is over. On the contrary, in the adiabatic regime the response comes from the deformation of the wave functions by the electric field of the light and the system comes back to the ground state when the pump pulse is over. Taking into account that the system is still out of equilibrium for several picoseconds after the pump pulse is over, we argue that the nonadiabatic contribution with real electronic transitions dominates the experimental response for E_{pump}>Δ (Fig. 2a). Conversely, both contributions are essential for describing the dynamics for E_{pump}<Δ.
To describe the differences between the responses driven in La_{2}CuO_{4+δ} by subgap (E_{pump}=0.95 eV) and abovegap (E_{pump}=3.1 eV) ~100 fs long pump pulses (Fig. 2), we turn to the HHH and introduce the effects of the electromagnetic fields through the classical vector potential A(τ) (see Methods). The HHH represents a standard approach for underdoped cuprates where the insulating state arises from the strong repulsion between the electrons. By taking into account both onsite Coulomb repulsion U and an effective electron–boson interaction^{30}, the HHH can reproduce the equilibrium optical properties of cuprates both undoped and with tiny doping^{13} and rationalizes the presence of tail states responsible for the broad belowgap absorption^{13,27,33}. We argue that these tail states are responsible for the nonadiabatic contribution for subgap pumping and for the linearity of the optical response as a function of pump fluence (see Methods). Furthermore, we will show that the ultrafast transient optical response observed for E_{pump}<Δ unveils a mixed regime of light–matter interaction where the lightdriven gain of the electron kinetic energy leads to a sudden reaction of the bosonic field. As doping also drives a kinetic energy gain of the carriers, one naively expects an offequilibrium ultrafast response similar to the one associated to doping at equilibrium (see Figs 1b and 3).
The HHH is given by the sum of three terms:
which account for the kinetic energy of the electrons (equation (1)), the onsite Coulomb repulsion (equation (2)) and the electron–boson interaction (equation (3)), respectively. In equations (1, 2, 3), t is the hopping amplitude, is the fermionic creation operator, μ is a lattice vector, creates a boson at site i with frequency ω_{0} and n_{i} is the electron number operator. The units are such that ħ=1. We choose model parameters typical for the cuprates (t=0.3 eV, ω_{0}=0.2t, and the electron–boson coupling constant ). The value of the Hubbard repulsion U=10t yields lowenergy physics very similar to that of the t−J model with J=4t^{2}/U (see Methods). Although the exact solution for this Hamiltonian is not available, the time evolution of the manybody wavefunction Ψ(τ)› can be obtained in a small twodimensional lattice (eight sites with periodic boundary conditions) by using an exact diagonalization method (timedependent Lanczos approach) based on a smart truncation of the boson Hilbert space, which has been proven successful in different systems^{34,35}. Spin and charge degrees of freedom are treated exactly within numerical precision. The optical conductivity is then calculated for a given value of the pump–probe delay τ, within the linear response theory, starting from the wavefunction Ψ(τ)› (see Methods for more details).
In Fig. 3c,d we show the calculated Δσ_{1}(ω, τ) at two different delay times both for abovegap (~3.2 eV) and subgap (~1.9 eV) ~40 fs long pump pulses. Please note that the slightly different pulse parameters used in the model compared with the experiment are required by the small lattice adopted in the calculations (see Methods). The major photoinduced feature calculated for E_{pump}>Δ (Fig. 3c,d) is a reduction of the optical conductivity, which is consistent with the experiments (Fig. 2a and Fig. 3a,b). However, we observe quantitative discrepancies between the calculated linewidths and the measured ones. In particular, the calculated reduction of the optical conductivity extends over a broader energy range. We argue that this is due to the fact that the model is well suited to address doping phenomena^{13} but lacks both the detailed crystal structure and a proper thermal reservoir. This interpretation is supported by comparing the expected variation of the optical conductivity on doping^{6} (black dashed curve in Fig. 1b) with the model results (Fig. 3b). Hence, for long pump–probe delays or E_{pump}>Δ, the single boson mode contained in the HHH acts as an effective but quantitatively inaccurate thermal bath.
Nonetheless, we stress that the theory accounts for the main experimental features and that a comparison between theory and experiment is fair, as both lie in the linear response regime (see Methods). In particular, both theory and experiments for E_{pump}<Δ display a spectral response exhibiting an evolution on ultrafast timescales (Fig. 3a–d). We argue that such ultrafast changes of the spectral response are described by the model exclusively in the presence of strong electron–boson interaction, as can be seen by inspection of Fig. 3e,f that are obtained for electron–boson coupling λ=0. The ultrafast shift of the optical response for E_{pump}<Δ may be rationalized as the result of a distortion of the bosonic field around holons and doublons^{36,37}, leading to an energy renormalization close to the experimentally observed value^{13}.
In the following we discuss the theoretical results in detail. We separate the wave function into the ground state Ψ_{GS}› and the photoexcited component φ(τ)›, that is, the part of the wavefunction different from the ground state. Here b(τ)^{2} represents the weight of φ(τ)› in the total wavefunction Ψ(τ)›. As φ(τ)› includes a doublon–holon pair (see Methods), b(τ)^{2} is also a rough estimate of the double occupancy density. Indeed, the number of doublons left in the system at τ>0 is an order of magnitude smaller for E_{pump}<Δ than for E_{pump}>Δ (Fig. 4a), confirming that the nonadiabatic processes are less relevant for subgap pumping.
To highlight the modifications driven in the system by the pump, we calculate the expectation values of different physical observables in the photoexcited state, that is, ‹φ(τ)Aφ(τ)›. The operators of interest are the kinetic energy, the number of bosons and the interaction energy. These correspond to the expectation values of H_{t} (equation (1)), of the first part of H_{ebi} divided by ω_{0}, and of the second part of H_{ebi} (equation (3)), respectively. These expectation values are related to the ones calculated on the total wavefunction ‹Ψ(τ)AΨ(τ)› through the weight b(τ)^{2} (see Methods for details). The interaction energy carries direct information on the interplay of the bosonic field with the excited charge carriers. Namely, this quantity is proportional to the number of bosons that are accumulated just on the site where doublons or holons are situated.
For E_{pump}>Δ, itinerant doublon–holon pairs are created (Fig. 4a) with a significant kinetic energy during the laser pulse (Fig. 4b). These pairs subsequently relax to lower energies emitting bosons on a longer timescale. The decrease of the average kinetic energy is related to the increase of the average boson number (Fig. 4b). The physics emerging for E_{pump}>Δ is thus consistent with a thermalization scenario of the photoexcited doublons and holons.
In the case of E_{pump}<Δ, a completely different response is observed. During the pulse, the kinetic energy is negative both for λ=0 and for finite λ, but most important is the behaviour immediately after the pulse (Fig. 4c). In case of finite λ, we observe an ultrafast reaction of the bosons, which is apparent both from the maximum of the boson number at τ_{1}=35 fs as well as from the minimum of the interaction energy at the same value of τ (inset of Fig. 4a). Owing to this ultrafast reaction, the kinetic energy increases immediately after the pulse. Altogether, this is a clear signature of the ultrafast formation of strongly dressed quasiparticles. This transient state, leading to the observed optical conductivity drop around the CTP, persists for few tens of femtoseconds and only at longer times evolves into a thermal incoherent state (τ_{2}>50 fs). The time dependence of the kinetic energy and boson number (Fig. 4c) shows that for subgap pumping the energy is transferred nearly ‘instantaneously’ (on a timescale related to the inverse of the boson frequency) from the electromagnetic field into the boson subsystem through the electrons. The comparison of the kinetic energies calculated with and without the electron–boson coupling (solid and dashed lines in Fig. 4c) indicates that the persistence of some dynamics after the end of the pump pulse for E_{pump}<Δ is possible only with nonvanishing electron–boson interaction. In Fig. 4d we redraw for clarity the data of Fig. 2c, and cartoons of the physical mechanisms described above are shown in Fig. 4e,f for E_{pump}>Δ and for E_{pump}<Δ, respectively.
The scenario stemming from the exact diagonalization of the Hubbard–Holstein model explains the different response measured for abovegap and subgap excitations. In spite of the fact that the model includes only one bosonic field, such field can be excited in a different way by the two excitation schemes. Although for E_{pump}>Δ the boson field acts as a bath where the excess kinetic electronic energy is deposited after the excitation, subgap pumping triggers a quasiinstantaneous distortion of the bosonic field. It is interesting to note that for both theory and experiment the ultrafast response in the optical conductivity for E_{pump}<Δ gives a transient feature, which is qualitatively analogous to the changes associated to a small doping (Fig. 1c, black dashed curve). This observation leads us towards the speculation that a perturbation with subgap photon energy drives a nonthermal tendency to create delocalized strongly dressed quasiparticles. This discloses several possible scenarios where coherent electromagnetic fields can be used to manipulate quantum coherent phases of matter^{38,39}.
Methods
Experimental details
We performed timeresolved experiments on La_{2}CuO_{4+δ} by combining broadband probes with ~100 fs pump pulses at different wavelengths. To address the pulsewidthlimited rise time observed for E_{pump}<Δ, we performed additional measurements with sub20 fs pulses.
The ~100 fs pump pulse measurements were performed with a Ti:Sa amplified laser at 250 kHz repetition rate. The pump energies were tuned to 0.95 eV (E_{pump}<Δ) and to 3.1 eV (E_{pump}>Δ) with durations corresponding to ~100 and ~140 fs, respectively, and about 0.04 eV bandwidth each. The 0.95 eV pulses are generated with a commercial optical parametric amplifier, the 3.1 eV pulses by second harmonic generation of the fundamental laser light in a 200μmthick Beta Barium Borate (BBO) crystal. The broadband probe was generated in a sapphire crystal. As expected for whitelight generation^{40}, the chirp of these probes is mainly linear. Thus, it has been simply corrected with the same straight line for both abovegap (~3.1 eV) and subgap (~0.95 eV) pump pulses. Freshly polished a–b oriented samples were mounted on the cold finger of a helium flow cryostat. The reflectivity changes ΔR(ω, τ)=R(ω, τ)/R(ω)−1 were measured in the 1.5–3.1 eV range at several temperatures between 50 and 330 K. The pump fluences of the measurements shown in Figs 2a–c and 3 are equal to 330 and 170 μJ cm^{−2} for the 3.1 and 0.95 eV pump pulses, respectively. These fluences correspond to about 4 × 10^{−3} and 1 × 10^{−3} Ph u^{−1} for E_{pump}>Δ and E_{pump}<Δ, respectively, where ph u^{−1} stands for photons (ph) absorbed by a chemical unit (u) of La_{2}CuO_{4}. The amplitude of the measured photoinduced variation of the reflectivity is linear with the pump intensity up to two times the excitation densities used in the reported experiments (see Supplementary Fig. 2).
The sub20 fs measurements were performed with a Ti:Sa amplified source at 1 kHz. The pump energies were tuned to 1.4 eV with 0.15 eV bandwidth (E_{pump}<Δ) and to 2.25 eV with 0.25 eV bandwidth (E_{pump}>Δ) with duration corresponding to ~12 and ~7 fs, respectively. The short pulses are obtained by compressing the output of a tunable noncollinear optical parametric amplifier with chirped mirrors or with two prisms. The measurements shown in Fig. 2d and in Supplementary Fig.1 have been performed at room temperature with pump fluences corresponding to 1.3 and 2.8 mJ cm^{−2} for abovegap or subgap pump pulses, respectively. The higher noise conditions of this set of measurements do not allow to investigate the linear response regime for the system, while the high intensities in both pump and probe pulses (≥0.2 mJ cm^{−2}) are responsible for unavoidable artefacts in the transient responses. For these reasons, we limited the KKbased analysis of the optical conductivity to the long pulse measurements, which are safely in the linear response limit.
Furthermore, we notice that the observed thermalization times are different for the two excitation schemes with ~100 and sub20 fs pulses. This might be partially due to the fact that excitations produced by light pulses at certain wavelengths undergo a statespecific thermalization pathway. The longer times detected with ~100 fs pulses partially support this scenario. However, we stress that the two excitation schemes differ not only in terms of the excited states involved, but also regarding the pulse energies used, and this can affect the observed timescales. Further studies are needed to clarify this issue. Nonetheless, it is remarkable that both results confirm a pulsewidthlimited nonthermal reaction of the bosonic field for E_{pump}<Δ.
Finally, we can exclude the slow rise time measured for E_{pump}>Δ to be related to diffusion of the photoexcitation energy along the surface normal. In such case, one would expect slower rise times for shorter probe wavelengths. This effect is not observed in our experiments (see Supplementary Fig. 3).
Thermal effects
The optical response of the CTP as a function of pump–probe delay is reported in Fig. 2c for E_{pump}>Δ (solid blue curve) and E_{pump}<Δ (solid red curve), respectively. The transient conductivity probed at 2.3 eV for E_{pump}=0.95 eV is properly normalized by the factor N, calculated as follows. The pump energy density absorbed by the sample and detected by a singlecolour probe with energy E_{probe} is equal to , where Ф(E) is the fluence, R(E) is the reflectivity and x(E) is the penetration depth for photons with energy E. Considering the bandwidth of the pump pulse and averaging over the probe energy range ΔE_{probe}, the effective energy density is . Making use of the reflectivity and the penetration depth as determined from the ellipsometry data (see Fig. 1 and Supplementary Fig. 4b), the ratio turns out to be ~14 at T=130 K, which is the normalization factor used in Fig. 2c.
The absorbed pump energy density can be directly related to the increase of the sample temperature ΔT detected by the probe pulses. In particular, where δ denotes the mass density and C the specific heat. Being at T=130 K (ref. 41) and , the expected temperature reached by the sample when all the degrees of freedom thermalize is T+ΔT≈139 K. Thus, we expect that the laserdriven transient response at long pump–probe delays should be comparable to the difference between the equilibrium optical quantities measured at 130 K and at 139 K. A comparison among the offequilibrium Δσ_{1}(ω, τ=5 ps) and the difference between the equilibrium conductivities σ_{1}(T=139 K) and σ_{1}(T=130 K) reveals a very good agreement, see the black dashed curves in Fig. 1c and in Fig. 3b and Supplementary Fig. 4. The dashed line in Fig. 1c is obtained by a linear interpolation of the data in ref. 6.
Data analysis
To extract the timedependent optical quantities such as the dielectric function (ε) or the optical conductivity (σ) out of pump–probe reflectivity ΔR(ω, τ)/R(ω)=R(ω, τ)/R(ω)−1, we proceed as follows. We obtain the equilibrium reflectivity R(ω) from the measured ellipsometry data between 1 and 3.2 eV and from the literature outside of this range^{6,42}. We perform a Drude–Lorentz fit^{43,44} of the broadband equilibrium reflectivity over a wide energy range, from a few meV up to several tens of eV. We calculate the real and imaginary parts of the equilibrium dielectric function, ε_{1}(ω) and ε_{2}(ω), in the entire energy range through the KK relations^{45}.
The timeresolved ΔR(ω, τ)/R(ω) is measured in the 1.5–3.1 eV energy range. To obtain the timedomain changes of the optical functions, we consider only the regime where the variations of the reflectivity ΔR(ω, τ) (up to 10^{−2} as shown in Supplementary Fig. 5) is linear with the pump fluence. We make the assumption that the variations of the pumpperturbed reflectivity outside the probed energy range are either small or distant on the energy scale so that they do not affect significantly the optical response in the probed range of 1.5–3.1 eV. Starting from these assumptions, both the measured reflectivity ΔR(ω, τ)/R(ω) and the equilibrium reflectivity R(ω) are known over a broad energy range, that is, from a few meV up to several tens of eV. Hence, the time evolution of the optical quantities such as ε_{1}(ω, τ) and ε_{2}(ω, τ) can be obtained by the KK transformations. The results of this analysis performed on the data measured at 130 K are reported in Fig. 2. The photoinduced variation of the real part of the optical conductivity Δσ_{1}(ω, τ)=σ_{1}(ω, τ)σ_{1}(ω) is shown for E_{pump}=3.1 eV (Fig. 2a) and E_{pump}=0.95 eV (Fig. 2b). We include for clarity the reflectivity data in Supplementary Fig. 5 and Supplementary Fig. 6. We emphasize that our analysis is consistent because the same normalization factor is used in Fig. 2c (that displays Δσ_{1}) and in Supplementary Fig. 7 (where ΔR/R is shown).
We implement a transfermatrix method to correct the effect of the mismatch between the penetration depths of the pump and the probe pulses on the amplitude of the timedomain signal. We assume a pumpperturbed exponentially graded index of refraction at the material surface by means of multiple planes, each with a thickness of 1 nm. The total reflection is calculated exploiting the continuity conditions for the electric fields across these boundaries. Further details can be found in refs 46, 47 Being the penetration depth at ~0.95 eV larger than the one of any probe in the 1.5–3 eV range, one expects significant corrections only for E_{pump} ~3.1 eV. The detailed numerical analysis confirms this simple picture. This is made evident in Supplementary Figs 8 and 9 for pump–probe delay time τ=0.1 ps averaged over ±0.1 ps. The ΔR(ω, τ)/R(ω) obtained in this way for E_{pump} ~3.1 eV has been used as starting point for the KK analysis described in the previous paragraph. The corrections to the transient reflectivity are obtained as summarized in Supplementary Fig. 9.
HHH details
The timedependent potential vector adopted is^{48}
and the optical conductivity at time τ, after the end of the pump, is given by^{49,50}
Here , T is the timeordering operator, j (t) is the current operator in the Heisenberg representation along one of the lattice axes, η is a broadening factor taking into account additional dissipative processes and M is the number of lattice sites. The Ψ(τ)› state is obtained through the Lanczos timepropagation method^{51}. In the following, we use periodic boundary conditions on twodimensional lattices^{52} with M=8.
The local nature of the interactions involved in the HHH allows us to extract the relevant physics of the model (at least at a qualitative level) also by using a small lattice, which, on the other hand, reproduces most of the discrete symmetries of the square lattice, and contains the relevant momenta: , (π, 0), (0, π), (0, 0) and (π, π). It is worth noting that the chosen lattice does not frustrate the antiferromagnetic ground state^{52}. These important properties are a consequence of the ‘special’ eight site lattice introduced first in ref. 53.
Within the used approach, the charge and spin degrees of freedom are exactly treated: all the possible states are considered in subspaces with a welldefined value of the z component of the total spin. The real bottleneck comes from the infinite dimensional Hilbert space required by the boson excitations. To solve this problem, we use a generalization of the method recently introduced in the Holstein^{35} and SuSchriefferHeeger model^{34}: the doubleboson approach. It is based on a smart truncation of the boson Hilbert space including two distinct sets of states. In the first set of basis elements, there are up to four bosons distributed in all possible ways. Of course, in this way the scattering processes among charge carriers and boson excitations up to four quanta are treated exactly also in q space. This set is able to recover the selfconsistent Born approximation and goes beyond it, including vertex corrections. In the other set of states, local excitations of any strength (up to 15 boson excitations per site) are distributed at most on two sites, which are located at arbitrary positions with respect to the charge carriers. The doubleboson approach used in this study is based on the idea that the physical properties for any value of the electron–boson interaction can be described by diagonalizing the Hamiltonian in the bosonic Hilbert subspace generated by the two sets of states introduced above (excluding obvious double counting).
The pump pulses adopted in the model have slightly different temporal and energetic content with respect to the ones used in the experiments. This is mainly due to finite size effects, explained as follows.
Concerning the frequency of the pump pulses, we have chosen E_{pump}=1.9 eV and E_{pump}=3.2 eV for subgap and abovegap pump pulses, respectively. In particular, for the subgap case, we were forced to use a larger value than that one used in the experiments for two reasons. First, solving the HHH on a small lattice implies few sparse discrete eigenstates at low energy, resulting in the lowest optical transition at ~1.9 eV that is absent in the bare Hubbard model. Second, our model does not include the free charge in excess present in the experimental sample. In addition to this, we highlight that the model is provided at T=0 K while it is well known that the CT gap in La_{2}CuO_{4} shifts to higher energies on cooling^{30,31,32}.
Moreover, the theory adopted ~40 fs long pulses, while the experiments made use of ~100 fs pulses. In the case of a small lattice, as the one used in our model, the allowed eigenvalues are restricted to a discrete set with a finite separation between them. However, due to the uncertainty principle, the more the duration of the pump pulse increases, the narrower is the energy. Hence, our model cannot use pump pulses exceeding ~40 fs, as the energy uncertainty of the pump pulses would be smaller than the typical distance between two adjacent eigenvalues. In this condition, a light pulse would excite only one eigenstate of the Hamiltonian resulting in a state not evolving in time.
Nonetheless, we stress that this limit of the theory is intrinsic in the finite size of our lattice and a comparison between experiment and theory is possible as the experimental pulse will always excite a superposition of eigenstates of the system. The linearity with the pump intensity of the observed response for both experiment and theory further consolidates the validity of the comparison.
In the main text, we focused our attention on the properties of φ(τ)›, the photoexcited component of the wavefunction Ψ(τ)›. It is defined in terms of the wave function at time τ and Ψ_{GS}›, the normalized ground state of the HHH in absence of the electromagnetic field: .
The expectation value of any observable A, measured with respect to the value in the ground state wavefunction, is:, where ΔA is proportional to the matrix element of the observable between the ground state and photoexcited component. As the last term is negligible (we checked it for all the quantities investigated in the paper), the ratio between the values of any fixed physical quantity, measured in the wavefunctions corresponding to above and subgap pulses, is always proportional to the ratio between the weights associated to the photoexcited components of the two wavefunctions. This ratio is about 10 for the chosen parameter values and delay larger than the pump pulse duration. Taking into account the smaller excitation fluence for the subgap case in the experimental data, we find a good agreement between theory and experiments.
Finally, we address the role of the pump pulse duration in the Hubbard–Holstein calculations (Supplementary Fig. 10). In general, one expects that for longer pump pulses the nonthermal part of the dynamical response of a system to be diminished in favour of the thermal dynamics. As the observed ultrafast shift of the transient optical conductivity is related to the nonthermal response in La_{2}CuO_{4+δ} for E_{pump}<Δ, one expects that the shorter the pump pulse, the larger the shift in Δσ_{1}(ω, τ). This is confirmed by the calculations displayed in Supplementary Fig. 10 corresponding to ~20 fs long pump pulses, and by the experiments shown in Supplementary Fig. 1. We stress that all the calculations predict an ultrafast transient negative change of Δσ_{1}(ω, τ) for E_{pump}<Δ, that evolves towards the thermal response at long pump–probe delays. This is confirmed by both ~10 and ~100 fs resolution experiments.
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How to cite this article: Novelli, F. et al. Witnessing the formation and relaxation of dressed quasiparticles in a strongly correlated electron system. Nat. Commun. 5:5112 doi: 10.1038/ncomms6112 (2014).
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Acknowledgements
F.N. and D.F. are grateful to Dr Marco Malvestuto and Francesco Randi for insightful discussions and critical reading of the manuscript. The research leading to these results has received funding from the European Union, Seventh Framework Programme (FP7 2007–2013), under grant number 280555 (GO FAST). N.N. was supported by GrantinAids for Scientific Research (number 24224009) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, Strategic International Cooperative Program (Joint Research Type) from Japan Science and Technology Agency, and by Funding Program for WorldLeading Innovative R&D on Science and Technology (FIRST Program). M.C. and A.A. are financed by EU/FP7 through ERC Starting Grant SUPERBAD, Grant Agreement 240524. A.P. acknowledges support from Italian Ministry of Research (MIUR) through the FIRB ‘Futuro in Ricerca’ grant no. RBFR10PSK4. G.C. acknowledges support from the EU Graphene Flagship (contract no. CNECTICT604391).
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F.N., D.F. and E.S. performed the broadband probe experiments. F.C. and D.F. developed the whitelight optical setup. F.N., M.E., S.C. and G.C. performed the hightemporalresolution measurements. F.N., D.F. and F.P. analysed the data and coordinated the project. M.G. and I.V. performed and analysed magnetic and broadband ellipsometry measurements at several temperatures of interest. G.F., V.C., N.N. and A.M. developed the theoretical framework. G.F. and V.C. performed the calculations. F.N., D.F., G.F. and V.C. wrote the manuscript after discussing with all the coauthors. S.W., C.G., A.A. and M.C. have given particularly important comments. A.P. performed equilibrium reflectivity measurements in the visible range. D.P. and S.W. made and furnished highquality oriented and polished single crystals.
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Novelli, F., De Filippis, G., Cataudella, V. et al. Witnessing the formation and relaxation of dressed quasiparticles in a strongly correlated electron system. Nat Commun 5, 5112 (2014). https://doi.org/10.1038/ncomms6112
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DOI: https://doi.org/10.1038/ncomms6112
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