Witnessing the formation and relaxation of dressed quasi-particles in a strongly correlated electron system

Abstract

The non-equilibrium approach to correlated electron systems is often based on the paradigm that different degrees of freedom interact on different timescales. In this context, photo-excitation 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 quasi-particles in an archetypal strongly correlated charge-transfer insulator (La₂ CuO_4+δ ), we show that the interaction between electrons and bosons manifests itself directly in the photo-excitation processes of a correlated material. With the aid of a general theoretical framework (Hubbard-Holstein Hamiltonian), we reveal that sub-gap excitation pilots the formation of itinerant quasi-particles, which are suddenly dressed by an ultrafast reaction of the bosonic field.

ARTICLE

Received 9 Jun 2014 | Accepted 2 Sep 2014 | Published 7 Oct 2014

DOI: 10.1038/ncomms6112

Witnessing the formation and relaxation of dressed quasi-particles in a strongly correlated electron system

Fabio Novelli1,2, Giulio De Filippis3, Vittorio Cataudella3, Martina Esposito1, Ignacio Vergara4, Federico Cilento2, Enrico Sindici1, Adriano Amaricci5, Claudio Giannetti6,7, Dharmalingam Prabhakaran8, Simon Wall9,Andrea Perucchi2,10, Stefano Dal Conte11, Giulio Cerullo11, Massimo Capone5, Andrey Mishchenko12,13, Markus Grninger4, Naoto Nagaosa12,14, Fulvio Parmigiani1,2,4 & Daniele Fausti1,2

The non-equilibrium approach to correlated electron systems is often based on the paradigm that different degrees of freedom interact on different timescales. In this context, photo-excitation 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 quasi-particles in an archetypal strongly correlated charge-transfer insulator (La2CuO4 d), we show that the interaction between electrons and bosons manifests itself directly in the photo-excitation processes of a correlated material. With the aid of a general theoretical framework (HubbardHolstein Hamiltonian), we reveal that sub-gap excitation pilots the formation of itinerant quasi-particles, which are suddenly dressed by an ultrafast reaction of the bosonic eld.

1 Dipartimento di Fisica, Universit degli Studi di Trieste, Via Valerio 2, 34127 Trieste, Italy. 2 ElettraSincrotrone Trieste S.C.p.A., 34149 Basovizza, Italy.

3 SPIN-CNR and Dipartimento di Fisica, Universit di Napoli Federico II, I-80126 Napoli, Italy. 4 II. Physikalisches Institut, Universitat zu Kln, 50937 Kln, Germany. 5 CNR-IOM Democritos National Simulation Center and Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea 265,34136 Trieste, Italy. 6 I-LAMP (Interdisciplinary Laboratories for Advanced Materials Physics), Universit Cattolica del Sacro Cuore, I-25121 Brescia, Italy.

7 Department of Physics, Universit Cattolica del Sacro Cuore, I-25121 Brescia, Italy. 8 Department of Physics, University of Oxford, Oxford OX1 3PU, UK.

9 ICFO-Institut de Ciencies Fotoniques, Av. Carl Friedrich Gauss, 3, Castelldefels, 08860 Barcelona, Spain. 10 INSTM UdR Trieste-ST, 34149 Basovizza, Italy.

11 IFN-CNR, Dipartimento di Fisica, Politecnico di Milano, Piazza L. da Vinci 32, 20133 Milano, Italy. 12 RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan. 13 RRC Kurchatov Institute, 123182 Moscow, Russia. 14 Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan. Correspondence and requests for materials should be addressed to G.F. (email: mailto:[email protected]

Web End [email protected] ) or to D.F.(email: mailto:[email protected]

Web End [email protected] ).

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The exotic electronic ground states of transition-metal oxides (TMOs), such as high-temperature 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 excitations14. A prime example of such complexity is represented by the cuprate family. Starting from parent compounds which are charge-transfer (CT) insulators513, anomalous metallic and eventually high-temperature superconducting phases appear on electron or hole doping14,15.

The properties of CT insulators are determined by Coulomb repulsion, which forces the electrons to localize, freezing charge uctuations. In such a state, the coupling between localized fermions and bosonic excitations, which is mediated by charge dynamics, is naively expected to be strongly suppressed16,17.

Although this scenario is seemingly met when we measure and study the properties of a steady-state Mott insulator, a different situation arises if the system is doped1820 or driven out of equilibrium13,21,22. 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 boson-charge 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 HubbardHolstein 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 identication of these bosonic modes remains controversial in the cuprates and is beyond the scope of this paper. Further information can be found in refs 2325. 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 electronboson coupling are highlighted by comparing the dynamical optical responses of La2CuO4 d driven by pulses with photon energy respectively

larger and smaller than the gap, which corresponds to the energy needed to create free charge excitations.

ResultsTime-resolved measurements on La2CuO4 d. Transient

reectivity experiments on La2CuO4 d were performed with

above-gap (B3.1 eV) and sub-gap (B0.95 eV) B100 fs long pump pulses and broadband probes as detailed in the Methods section. From the measured DR(o, t)/R(o) R(o, t)/R(o) 1

with R(o, t), the time-dependent pump-perturbed reectivity, and R(o), the equilibrium reectivity, we calculate the pump-induced changes of the optical conductivity Ds1(o, t)

s1(o, t)-s1(o) as a function of probe frequency (o) and pump probe delay time (t). Here, s1(o, t) and s1(o) are the time-dependent light-perturbed and equilibrium optical conductivity, respectively. The CT peak (CTP) in s1(o) is at B2.3 eV, while the gap, estimated from a simple linear interpolation (red dashed line in Fig. 1), is at about D 1.8 eV for T 130 K and is con

sistent with previous reports69,26. The non-vanishing 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 non-stoichiometry23, and the bosonic degree of freedom13. The excess oxygen content dB5 10 3 is estimated from the measured Nel temperature of

229 K (ref. 27). Ds1(o, t) is obtained from DR(o, t)/R(o) with a novel data analysis methodology based on the Kramers-Kronig (KK) relations. In short, by knowing the equilibrium reectivity R(o) and the non-equilibrium DR(o, t)/R(o), it is possible to calculate the pump-perturbed reectivity R(o, t) 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 Epump 3.1 eV and

Epump 0.95 eV, respectively, with B0.04 eV pulse bandwidth

each (see Methods for details). In the rst hundred femtoseconds, we observe striking differences between the photo-excitation with photons at energy Epump4D (Fig. 2a) and EpumpoD (Fig. 2b).

A smooth rise time (4100 fs) is detected only for Epump4D,

while sub-gap excitation leads to a pulse-width-limited 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 pumpprobe delay. The results in Fig. 2c are normalized to the energy absorbed by the sample for Epump 3.1 eV (solid blue curve) and Epump 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 wells28 and below the electronic excitation energy in molecular systems29 has been previously reported.

In Fig. 2c, we show that the modications 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 time-resolved measurements on La2CuO4 (refs 30,31), Nd2CuO4 (refs 3032), YBa2Cu3Oy (ref. 32) and Sr2CuO2Cl2 (ref. 21).

Sub-20 fs pulse experiments. With the aim to resolve the ultra-fast rise time observed for EpumpoD, we performed room

temperature B10 fs experiments with EpumpoD (1.4 eV with

0.1 eV bandwidth) and Epump4D (2.25 eV with 0.25 eV band

width). In spite of the different excitation parameters used for the B10 and B100 fs measurements, we expect the outcome of these experiments to be meaningful for addressing the question concerning the non-thermal ultrafast rise time.

The DR(o, t)/R(o) at EprobeB2.1 eV is shown in Fig. 2d, while

the transient reectivity spectra at two pumpprobe delays are reported in Supplementary Fig. 1 and Fig. 2d. It is evident that also for B10 fs EpumpoD 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 non-thermal response in La2CuO4 d to 20 fs.

Unfortunately, under the present experimental conditions, a direct quantitative comparison of the thermalization times between the B100 and the B10 fs excitations is not meaningful. In particular, the thermalization time is signicantly shorter for B10 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 pumpprobe data obtained with B100 fs pump pulses where the linear response regime has been proven (see Methods).

Energy-dependent response. We now take advantage of the broadband probe and describe the spectral dependence of the transient response. The energy-dependent Ds1(o, t) at a

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1,000

130 K

CTP 200 K

300 K

[afii9846] 1(1 cm1)

[afii9846] (139 K)-[afii9846] (130 K)

[afii9846] (x=0.01)-[afii9846] (x=0) (/15)

5

500

[afii9846] 1([afii9853]) (1 cm1)

a

b

c

5

0

1.0 1.5 2.5

2.0 Energy (eV)

gap 3.0

0

2.0

2.5

Energy (eV)

Figure 1 | Equilibrium optical conductivity of lanthanum-copper oxide for polarization in the oxygen plane. (a) La2CuO4 unit cell with copper in purple, oxygen in red and, lanthanum in yellow. (b) In plane optical conductivity at 130 K, 200 K, and room temperature as determined via ellipsometry. The charge-transfer peak (CTP) is indicated. The red dashed straight line allows to dene the gap at B1.8 eV. (c) Expected variation of s1(o) on doping (dashed line) or heating (solid line). The solid line is obtained from our ellipsometry data, the dashed line is the difference between the optical conductivity of

La1.99Sr0.01CuO4 (x 0.01) and La2CuO4 (x 0) at room temperature. This last curve has been calculated starting from the data reported in ref. 6.

Specically, to obtain the x 0.01 doping reported in c, we performed a linear interpolation between the x 0 and the x 0.02 spectra shown in ref. 6.

1.8

0

Energy (eV)

2.4

2.6 [afii9846]1 (1cm1)

Epump=3.1 eV

20

20

40

60

0

2.2

[afii9846] 1(E=2.30.1eV) (1cm1)

50

Pulse3.1 eV

0.95 eV (14)

1.0 1.5

2.0

100

0.0 0.5Delay (ps)

2.6

0.00

Epump=0.95 eV

2

2

4

8

6

[afii9846]1 (1cm1)

Energy (eV)

2.4

R/R(E=2.10.1eV)

0.10

0.05

0

2.2

2.0

0.15

1.8

0.0 0.5

Delay (ps)

1.0 1.5

0.0

Delay (ps)

2.25 eV1.4 eV (norm) Pulse

0.1

Figure 2 | Time-domain evolution of optical conductivity. The measurements performed at 130 K are reported as a function of probe energy for pump energy (a) larger (3.1 eV with 0.04 eV bandwidth) and (b) smaller (0.95 eV with 0.04 eV bandwidth) than D. (c) The transient optical conductivity at

Eprobe 2.3 eV for both pump energies is shown (3.1 eV in blue, 0.95 eV in red). The response for sub-gap excitation is multiplied by

the ratio of the absorbed energy densities (see Methods). The black curve depicts the 3.1 eV pump autocorrelation. (d) Normalized pumpprobe reectivity measurements performed at room temperature with B10 fs pulses (in black the pulse duration, see Methods for details).

delay time t 0.05 ps averaged over 0.05 ps (t 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 Epump4D at both pumpprobe delays, we observe a depletion of the optical absorption centred at EprobeB2.3 eV and

an enhanced absorption around EprobeB2.1 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 heating27,30,31 as seen in Fig. 1. This is conrmed by comparing the measured Ds1(o, t) at long pumpprobe delays with the expected variation of the equilibrium optical conductivity for a

temperature increase resulting from the pump-induced heating of the sample (black line in Fig. 3b, see Methods for details).

On the contrary, Ds1(o, t) for EpumpoD reveals a signicant 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 pumpprobe delay time t. The Ds1 (o, t) has a minimum at EprobeB2.2 eV within

B100 fs and drifts towards the thermal response only on longer times.

It is important to notice that for tr100 fs and Epump

0.95 eV the optical conductivity displays mainly a negative variation around EprobeB2.2 eV, which consists of an overall

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0.0

0.0

[afii9846] 1([afii9853]) (a.u.)

0

0.5

1.9 eV3.2 eV

0.5

1.9 eV3.2 eV

[afii9848]=0.050.05 ps

0.95 eV3.1 eV

[afii9848]=0.035 ps

[afii9838]>0

[afii9848]=0.035 ps

[afii9838]=0

1

1.0

1.0

1

0.0

0.0

[afii9846] 1([afii9853]) (a.u.)

0

0.5

1.9 eV3.2 eV

0.5

1.9 eV3.2 eV

[afii9848]=0.30.1 ps

0.95 eV3.1 eV[afii9846]1(139 K)-[afii9846]1(130 K)

[afii9848]=0.055 ps

[afii9838]>0

[afii9848]=0.055 ps

[afii9838]=0

1

1.5 2.0 Energy (eV)

1.0

1.0

2.5 1.5 2.0 Energy (eV)

2.5 1.5 2.0 Energy (eV)

2.5

Figure 3 | Experimental and theoretical energy dependence of Dr1 at signicant pumpprobe delays. In a and b, the experimental data are shown (the pulse bandwidth of about 0.04 eV is omitted in the labels); in cf, the model calculations are reported. In e and f, the zero-coupling results are reported for comparison. All the graphs share the same energy axis. The data are normalized for the sake of comparison.

non-thermal 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.

DiscussionIn the one-band 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 sub-gap and above-gap excitations, one has to consider the two limiting cases predicted by time-dependent perturbation theory: non-adiabatic and adiabatic regimes. The non-adiabatic 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 eld 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 non-adiabatic contribution with real electronic transitions dominates the experimental response for Epump4D (Fig. 2a). Conversely, both

contributions are essential for describing the dynamics for EpumpoD.

To describe the differences between the responses driven in La2CuO4 d by sub-gap (Epump 0.95 eV) and above-gap

(Epump 3.1 eV) B100 fs long pump pulses (Fig. 2), we turn to

the HHH and introduce the effects of the electromagnetic elds through the classical vector potential A(t) (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 electronboson interaction30, the HHH can reproduce the equilibrium optical properties of cuprates both undoped and with tiny doping13 and rationalizes the presence of tail states responsible for the broad below-gap absorption13,27,33. We argue that these tail states are responsible for the non-adiabatic contribution for sub-gap pumping and for

the linearity of the optical response as a function of pump uence (see Methods). Furthermore, we will show that the ultrafast transient optical response observed for EpumpoD unveils a mixed

regime of lightmatter interaction where the light-driven gain of the electron kinetic energy leads to a sudden reaction of the bosonic eld. As doping also drives a kinetic energy gain of the carriers, one naively expects an off-equilibrium 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:

Ht t X

i;m;s

i m;sci;s H:c:

eiAtcy

; 1

HU U X

1

2

ni;#

ni;"

; 2

1

2

i

Hebi o0 X

ay

i ai go0 X

i

ai

; 3

which account for the kinetic energy of the electrons (equation (1)), the onsite Coulomb repulsion (equation (2)) and the electronboson interaction (equation (3)), respectively. In equations (13), t is the hopping amplitude, cyi;s is the fermionic creation operator, m is a lattice vector, ayi creates a boson at site i with frequency o0 and ni is the electron number operator. The units are such that : 1. We choose model

parameters typical for the cuprates (t 0.3 eV, o0 0.2t, and the

electronboson coupling constant l g

2o0

4t

ay

i

1 ni

i

0:5). The value of

the Hubbard repulsion U 10t yields low-energy physics very

similar to that of the t J model with J 4t2/U (see Methods).

Although the exact solution for this Hamiltonian is not available, the time evolution of the many-body wavefunction |C(t)i can be

obtained in a small two-dimensional lattice (eight sites with periodic boundary conditions) by using an exact diagonalization method (time-dependent Lanczos approach) based on a smart truncation of the boson Hilbert space, which has been proven successful in different systems34,35. Spin and charge degrees of freedom are treated exactly within numerical precision. The

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[afii9846] 1(E=2.30.1eV) (1 cm1)

a

d

0

Interaction energy

0.5

Thermal response

1.9 eV (10)3.2 eV Pulse

10

1.0

0 20

Delay (fs)

40 60

40 20

|b()|2

0.2

50

Ultrafast boson response

3.1 eV0.95 eV (14)

0.1

0.0

100

b

[afii9848]2 1.0

1.5

Kinetic energy Epump=3.2 eV

Boson number

0.0 0.5Delay (ps)

e

Kinetic energy

1.0 10

Boson number

Epump >

[afii9848] > 0

0.8

5

e

Thermal dressing

Coherent dressing

0

c

0.0

Oxygen Copper

Epump=1.9 eV

Kinetic energy ([afii9838]=0) Kinetic energy Boson number

f

10

Boson number

Epump <

[afii9848] << [afii9848]2 [afii9848] > [afii9848]2

Kinetic energy

0.1

0.2

5

0

40 20 20 Delay (fs)

40

0

Figure 4 | HubbardHolstein calculations. (a) The weight, |b(t)|2, of the photo-excited component of the wave function as a function of pumpprobe delay for excitations below (red) and above (blue) D (the sub-gap weight is multiplied by 10.2, that is, it is normalized to that above gap). In the inset of a, the interaction energy in eV is shown. The average number of bosons (thick line) and the electron kinetic energy in eV (thin line) for the two excitation wavelengths are reported in b and c. In a, b and c, the differences between the pump-perturbed quantities and the ground state ones are shown. In d, Fig. 2c is reproduced for clarity, with orange and blue shaded areas to underline the slow and fast responses of the bosonic eld associated to Epump4D and

EpumpoD, respectively. In e and f, cartoons of the physical mechanism are sketched. For Epump4D (e), the dynamics is consistent with a simple thermalization or thermal dressing scenario; for EpumpoD (f), the ultrafast drop of kinetic energy goes in pair with the reaction of the bosonic eld and a coherent dressing mechanism is in action at very short time delays.

optical conductivity is then calculated for a given value of the pumpprobe delay t, within the linear response theory, starting from the wave-function |C(t)i (see Methods for more details).

In Fig. 3c,d we show the calculated Ds1(o, t) at two different

delay times both for above-gap (B3.2 eV) and sub-gap (B1.9 eV) B40 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 photo-induced feature calculated for Epump4D (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 phenomena13 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 doping6 (black dashed curve in Fig. 1b) with the model results (Fig. 3b). Hence, for long pumpprobe delays or Epump4D, 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 EpumpoD display a spectral response exhibiting an evolution on ultrafast timescales (Fig. 3ad). We argue that such ultrafast changes of the spectral response are described by the model exclusively in the presence of strong electronboson interaction, as can be seen by inspection of Fig. 3e,f that are obtained for electronboson coupling l 0. The

ultrafast shift of the optical response for EpumpoD may be

rationalized as the result of a distortion of the bosonic eld around holons and doublons36,37, leading to an energy renormalization close to the experimentally observed value13.

In the following we discuss the theoretical results in detail. We separate the wave function Cti

j

p

1 j bt j 2

CGS

j i

bt ft

j i into the ground state |CGSi and the photo-

excited component |f(t)i, that is, the part of the wavefunction

different from the ground state. Here |b(t)|2 represents the weight of |f(t)i in the total wavefunction |C(t)i. As |f(t)i includes

a doublonholon pair (see Methods), |b(t)|2 is also a rough estimate of the double occupancy density. Indeed, the number of doublons left in the system at t40 is an order of magnitude smaller for EpumpoD than for Epump4D (Fig. 4a), conrming

that the non-adiabatic processes are less relevant for sub-gap pumping.

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To highlight the modications driven in the system by the pump, we calculate the expectation values of different physical observables in the photo-excited state, that is, hf(t)|A|f(t)i.

The operators of interest are the kinetic energy, the number of bosons and the interaction energy. These correspond to the expectation values of Ht (equation (1)), of the rst part of Hebi divided by o0, and of the second part of Hebi (equation (3)), respectively. These expectation values are related to the ones calculated on the total wavefunction hC(t)|A|C(t)i through the

weight |b(t)|2 (see Methods for details). The interaction energy carries direct information on the interplay of the bosonic eld 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 Epump4D, itinerant doublonholon pairs are created (Fig. 4a) with a signicant 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 Epump4D is

thus consistent with a thermalization scenario of the photo-excited doublons and holons.

In the case of EpumpoD, a completely different response is observed. During the pulse, the kinetic energy is negative both for l 0 and for nite l, but most important is the behaviour

immediately after the pulse (Fig. 4c). In case of nite l, we observe an ultrafast reaction of the bosons, which is apparent both from the maximum of the boson number at t1 35 fs as well

as from the minimum of the interaction energy at the same value of t (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 quasi-particles. 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 (t2450 fs). The time dependence of the kinetic energy and boson number (Fig. 4c) shows that for sub-gap pumping the energy is transferred nearly instantaneously (on a timescale related to the inverse of the boson frequency) from the electromagnetic eld into the boson subsystem through the electrons. The comparison of the kinetic energies calculated with and without the electronboson coupling (solid and dashed lines in Fig. 4c) indicates that the persistence of some dynamics after the end of the pump pulse for EpumpoD is possible only with

non-vanishing electronboson 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 Epump4D

and for EpumpoD, respectively.

The scenario stemming from the exact diagonalization of the HubbardHolstein model explains the different response measured for above-gap and sub-gap excitations. In spite of the fact that the model includes only one bosonic eld, such eld can be excited in a different way by the two excitation schemes. Although for Epump4D the boson eld acts as a bath where the

excess kinetic electronic energy is deposited after the excitation, sub-gap pumping triggers a quasi-instantaneous distortion of the bosonic eld. It is interesting to note that for both theory and experiment the ultrafast response in the optical conductivity for EpumpoD 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 sub-gap photon energy drives a non-thermal tendency to create delocalized strongly dressed quasi-particles. This discloses several possible scenarios where coherent electromagnetic elds can be used to manipulate quantum coherent phases of matter38,39.

Methods

Experimental details. We performed time-resolved experiments on La2CuO4d

by combining broadband probes with B100 fs pump pulses at different wavelengths. To address the pulse-width-limited rise time observed for EpumpoD,

we performed additional measurements with sub-20 fs pulses.

The B100 fs pump pulse measurements were performed with a Ti:Sa amplied laser at 250 kHz repetition rate. The pump energies were tuned to 0.95 eV (EpumpoD) and to 3.1 eV (Epump4D) with durations corresponding to B100 and

B140 fs, respectively, and about 0.04 eV bandwidth each. The 0.95 eV pulses are generated with a commercial optical parametric amplier, the 3.1 eV pulses by second harmonic generation of the fundamental laser light in a 200-mm-thick Beta

Barium Borate (BBO) crystal. The broadband probe was generated in a sapphire crystal. As expected for white-light generation40, the chirp of these probes is mainly linear. Thus, it has been simply corrected with the same straight line for both above-gap (B3.1 eV) and sub-gap (B0.95 eV) pump pulses. Freshly polished ab oriented samples were mounted on the cold nger of a helium ow cryostat. The reectivity changes DR(o, t) R(o, t)/R(o) 1 were measured in the 1.53.1 eV

range at several temperatures between 50 and 330 K. The pump uences of the measurements shown in Figs 2ac and 3 are equal to 330 and 170 mJ cm 2 for the3.1 and 0.95 eV pump pulses, respectively. These uences correspond to about

4 10 3 and 1 10 3 Ph u 1 for Epump4D and EpumpoD, respectively, where

ph u 1 stands for photons (ph) absorbed by a chemical unit (u) of La2CuO4. The amplitude of the measured photo-induced variation of the reectivity is linear with the pump intensity up to two times the excitation densities used in the reported experiments (see Supplementary Fig. 2).

The sub-20 fs measurements were performed with a Ti:Sa amplied source at 1 kHz. The pump energies were tuned to 1.4 eV with 0.15 eV bandwidth (EpumpoD) and to 2.25 eV with 0.25 eV bandwidth (Epump4D) with duration

corresponding to B12 and B7 fs, respectively. The short pulses are obtained by compressing the output of a tunable non-collinear optical parametric amplier 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 uences corresponding to 1.3 and 2.8 mJ cm 2 for above-gap or sub-gap 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 (Z0.2 mJ cm 2) are responsible for unavoidable artefacts in the transient responses. For these reasons, we limited the

KK-based 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 B100 and sub-20 fs pulses. This might be partially due to the fact that excitations produced by light pulses at certain wavelengths undergo a state-specic thermalization pathway. The longer times detected with B100 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 conrm a pulse-width-limited non-thermal reaction of the bosonic eld for EpumpoD.

Finally, we can exclude the slow rise time measured for Epump4D to be related

to diffusion of the photo-excitation 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 pumpprobe delay is reported in Fig. 2c for Epump4D (solid blue curve) and EpumpoD(solid red curve), respectively. The transient conductivity probed at 2.3 eV for Epump 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 single-colour probe

with energy Eprobe is equal to F E

xE 1 REpump

1 e xE

=xE

,

where ^(E) is the uence, R(E) is the reectivity 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 DEprobe, the effective energy density is

QEpump; E

probe

=xE

h i dEprobedEpump. Making use of the reectivity and the penetration depth as determined from the ellipsometry data (see Fig. 1 and Supplementary Fig. 4b), the ratio N

QE 3:1eV;1:9eVoE o2:5eV

QE 0:95eV;1:9eVoE o2:5eV turns out to be B14 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

Q of the sample temperature DT detected by the probe pulses. In particular, DT

E ;E

d C where d denotes the mass density and C the specic heat. Being

C 0:2J=K g at T 130 K (ref. 41) and d 7g cm 3, the expected temperature

reached by the sample when all the degrees of freedom thermalize isT DTE139 K. Thus, we expect that the laser-driven transient response at long

pumpprobe delays should be comparable to the difference between the equilibrium optical quantities measured at 130 K and at 139 K. A comparison among the off-equilibrium Ds1(o, t 5 ps) and the difference between the

1 DEprobe

ZZ

F Epump

x Eprobe

1 REpump

1 e xE

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equilibrium conductivities s1(T 139 K) and s1(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 time-dependent optical quantities such as the dielectric function (e) or the optical conductivity (s) out of pumpprobe reectivity DR(o, t)/R(o) R(o, t)/R(o) 1, we proceed as follows. We obtain the equili

brium reectivity R(o) from the measured ellipsometry data between 1 and 3.2 eV and from the literature outside of this range6,42. We perform a DrudeLorentz t43,44 of the broadband equilibrium reectivity 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, e1(o) and e2(o), in the entire energy range through the KK relations45.

The time-resolved DR(o, t)/R(o) is measured in the 1.53.1 eV energy range. To obtain the time-domain changes of the optical functions, we consider only the regime where the variations of the reectivity DR(o, t) (up to 10 2 as shown in

Supplementary Fig. 5) is linear with the pump uence. We make the assumption that the variations of the pump-perturbed reectivity outside the probed energy range are either small or distant on the energy scale so that they do not affect signicantly the optical response in the probed range of 1.53.1 eV. Starting from these assumptions, both the measured reectivity DR(o, t)/R(o) and the equilibrium reectivity R(o) 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 e1(o, t) and e2(o, t) 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 photo-induced variation of the real part of the optical conductivity Ds1(o, t) s1(o, t)-s1(o) is shown for Epump 3.1 eV (Fig. 2a) and

Epump 0.95 eV (Fig. 2b). We include for clarity the reectivity 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 Ds1) and in Supplementary Fig. 7 (where DR/R is shown).

We implement a transfer-matrix method to correct the effect of the mismatch between the penetration depths of the pump and the probe pulses on the amplitude of the time-domain signal. We assume a pump-perturbed exponentially graded index of refraction at the material surface by means of multiple planes, each with a thickness of 1 nm. The total reection is calculated exploiting the continuity conditions for the electric elds across these boundaries. Further details can be found in refs 46,47 Being the penetration depth at B0.95 eV larger than the one of any probe in the 1.53 eV range, one expects signicant corrections only for Epump

B3.1 eV. The detailed numerical analysis conrms this simple picture. This is made evident in Supplementary Figs 8 and 9 for pumpprobe delay time t 0.1 ps

averaged over 0.1 ps. The DR(o, t)/R(o) obtained in this way for Epump B3.1 eV

has been used as starting point for the KK analysis described in the previous paragraph. The corrections to the transient reectivity are obtained as summarized in Supplementary Fig. 9.

HHH details. The time-dependent potential vector adopted is48

A A0e

cos opump t t0

;

4

and the optical conductivity at time t, after the end of the pump, is given by49,50

s o; t

1Mo I Z

p

1 bt

j j

j i bt ft

2 CGS

j i.

The expectation value of any observable A, measured with respect to the value in the ground state wavefunction, is:hC t

A Cti hCGS A

j jCGSi b t

j j2

hf t

A

j jf t

i hCGS A

j jCGSi DA, where DA is proportional to the matrix

element of the observable between the ground state and photo-excited 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 xed physical quantity, measured in the wavefunctions corresponding to above- and sub-gap pulses, is always proportional to the ratio between the weights associated to the photo-excited 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 uence for the sub-gap case in the experimental data, we nd 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 non-thermal 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 non-thermal response in La2CuO4d for EpumpoD, one expects that the shorter the pump pulse, the larger

the shift in Ds1(o, t). This is conrmed by the calculations displayed in Supplementary Fig. 10 corresponding to B20 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 Ds1(o, t) for EpumpoD, that evolves towards the thermal response at long pumpprobe delays. This is conrmed by both B10 and B100 fs resolution experiments.

10 iei oiZthCt j j t

; j0

j Ctidt: 5

Here Ct

j i Te iR

Ht dt Ct

j 0i, T is the time-ordering operator, j (t)

is the current operator in the Heisenberg representation along one of the lattice axes, Z is a broadening factor taking into account additional dissipative processes and M is the number of lattice sites. The |C(t)i state is obtained through the

Lanczos time-propagation method51. In the following, we use periodic boundary conditions on two-dimensional lattices52 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: p2 ; p2

, (p,

0), (0, p), (0, 0)

and (p, p). It is worth noting that the chosen lattice does not frustrate the anti-ferromagnetic ground state52. These important properties are a consequence of the special eight site lattice introduced rst 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 well-dened value of the z component of the total spin. The real bottleneck comes from the innite dimensional Hilbert space required by the boson excitations. To solve this problem, we use a generalization of the method recently introduced in the Holstein35 and Su-Schrieffer-Heeger model34: the double-boson approach. It is based on a smart truncation of the boson Hilbert space including two distinct sets of states. In the rst 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 self-consistent 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 double-boson 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 nite size effects, explained as follows.

Concerning the frequency of the pump pulses, we have chosen Epump 1.9 eV

and Epump 3.2 eV for sub-gap and above-gap pump pulses, respectively. In

particular, for the sub-gap 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 B1.9 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 La2CuO4 shifts to higher energies on cooling3032.

Moreover, the theory adopted B40 fs long pulses, while the experiments made use of B100 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 nite 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 B40 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 nite 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 |f(t)i, the

photo-excited component of the wave-function |C(t)i. It is dened in terms of the

wave function at time t and |CGSi, the normalized ground state of the HHH in

absence of the electromagnetic eld: Ct

j i

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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 20072013), under grant number 280555 (GO FAST). N.N. was supported by Grant-in-Aids for Scientic 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 World-Leading Innovative R&D on Science and Technology (FIRST Program). M.C. and A.A. are nanced 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. CNECT-ICT-604391).

Author contributions

F.N., D.F. and E.S. performed the broadband probe experiments. F.C. and D.F. developed the white-light optical setup. F.N., M.E., S.C. and G.C. performed the high-temporal-resolution 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. andV.C. wrote the manuscript after discussing with all the co-authors. S.W., C.G., A.A. and M.C. have given particularly important comments. A.P. performed equilibrium reectivity measurements in the visible range. D.P. and S.W. made and furnished high-quality oriented and polished single crystals.

Additional information

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How to cite this article: Novelli, F. et al. Witnessing the formation and relaxation of dressed quasi-particles in a strongly correlated electron system. Nat. Commun. 5:5112 doi: 10.1038/ncomms6112 (2014).

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