Abstract
Competition between electron localization and delocalization in Mott insulators underpins the physics of strongly correlated electron systems. Photoexcitation, which redistributes charge, can control this manybody process on the ultrafast timescale^{1,2}. So far, timeresolved studies have been carried out in solids in which other degrees of freedom, such as lattice, spin or orbital excitations^{3,4,5}, dominate. However, the underlying quantum dynamics of ‘bare’ electronic excitations has remained out of reach. Quantum manybody dynamics are observed only in the controlled environment of optical lattices^{6,7} where the dynamics are slower and lattice excitations are absent. By using nearly singlecycle nearinfrared pulses, we have measured coherent electronic excitations in the organic salt ETF_{2}TCNQ, a prototypical onedimensional Mott insulator. After photoexcitation, a new resonance appears, which oscillates at 25 THz. Timedependent simulations of the Mott–Hubbard Hamiltonian reproduce the oscillations, showing that electronic delocalization occurs through quantum interference between bound and ionized holon–doublon pairs.
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Main
In Mott insulators, conductivity at low energies is prevented by the repulsion between electrons. This state is fundamentally different from that of conventional band insulators, in which Bragg scattering from the lattice opens gaps in the singleparticle density of states. The electronic structure of Mott insulators is, therefore, sensitive to doping. Photoexcitation, in analogy to static doping, can trigger large changes in the macroscopic properties^{8}. However, the coherent physics driving these transitions has not been fully observed because the manybody electronic dynamics are determined by hopping and correlation processes that persist for only a few femtoseconds.
We report measurements of coherent manybody dynamics with ultrafast optical spectroscopy in the onedimensional Mott insulator bis(ethylendithyo)tetrathiafulvalenedifluorotetracyanoquinodimethane (ETF_{2}TCNQ). Several factors make this possible: ETF_{2}TCNQ has a narrow bandwidth (∼100 meV), which corresponds to hopping times of tens of femtoseconds; the material has a weak electron–lattice interaction; we use a new optical device producing pulses of 9 fs at the 1.7 μm Mott gap; we study this physics in a onedimensional system, allowing the evolution of the manybody wavefunction to be calculated and compared with experimental data.
ETF_{2}TCNQ is a stacked molecular ionic solid^{9} in which ET molecules are donors and F_{2}TCNQ are acceptors, which form quasi onedimensional chains of ET molecules. Conducting electrons are localized on ET sites because the large onsite Coulomb repulsion (U∼1 eV) exceeds the singleelectron hopping amplitude (t∼0.1 eV; ref. 10).
Figure 1 shows the optical properties of ETF_{2}TCNQ. No Drude weight is found at low energies and light polarized parallel to the ET chains is strongly reflected at ∼0.7 eV. This feature corresponds to intersite charge transfer between neighbouring ET ions, resulting in a hole on one lattice site (a holon) and a neighbouring site with two electrons (a doublon). The charge transfer feature in ETF_{2}TCNQ is sharp, reflecting a bandwidth lower than the gap energy. Figure 1a shows the static reflectivity of ETF_{2}TCNQ fitted with a multiLorentzian dielectric function of the form , where A_{j} is the oscillator strength, ω_{j0} is the resonance frequency and γ_{j} is the damping rate. A single oscillator at ω_{Mott}=0.675 eV is sufficient to describe the Mott gap. Figure 1b shows the optical conductivity obtained from the imaginary part of the fitted dielectric function.
Photoexcitation across the Mott gap of ETF_{2}TCNQ transiently generates a metallic state^{11}, evidenced by the transfer of spectral weight from the Mott gap to a Drude response at low frequencies resulting from charge delocalization. To observe the initial delocalization dynamics, short light pulses are needed. Even for the low hopping amplitudes of this compound (t∼100 meV) we expect electron delocalization to require a time of order h/t∼40 fs, where h is Planck’s constant. Similarly, we expect correlated electrons to be dressed on timescales of the order of h/U. Furthermore, the pulses need to be resonant with the 0.7 eV charge transfer resonance, corresponding to a wavelength of 1.7 μm, where sub10fs pulses have not been previously achieved. Therefore, we developed a source at 1.7 μm with 9 fs duration^{12}.
Figure 2 reports timeresolved reflectivity measurements of the photoinduced response of ETF_{2}TCNQ in the paramagnetic, Mott insulating phase at room temperature. The timedependent reflectivity was probed with a replica of the pump pulse and spectrally resolved. Both pump and probe were polarized along the crystal’s a axis, to excite charges across the Mott gap. Figure 2a shows the spectrally integrated reflectivity change. After a prompt decrease, relaxation back to the ground state occurs with a biexponential decay with time constants of 130 fs and 840 fs. Excitation with light perpendicular to the chains produced no observable dynamics, confirming that the experiments address the onedimensional Mott physics of ETF_{2}TCNQ.
Figure 2b shows the spectrally resolved reflectivity. A prompt shift of spectral weight towards lower energies is followed by a drop in the reflectivity at the Mott gap. The dashed lines in Fig. 2 show the fitted reflectivity at each delay. Whereas the unperturbed reflectivity was accurately described by a single oscillator at the Mott gap, the transient reflectivity of the photoexcited system requires an additional, lower energy, oscillator. The appearance of a new resonance, which is clearly visible in the reflectivity lineouts of Fig. 2, suggests that belowgap bound states are formed, reminiscent of the bound holon–doublon pairs observed in other Mott insulators^{13}.
The characteristics of this new peak are timedependent, as shown in Fig. 2c, where we have normalized the reflectivity at each time step. Two contours are shown in Fig. 2c. On the blue side, a prompt redshift and recovery of the resonance is observed, whereas the red side shows a longerlived component, containing a damped oscillatory response at 25 THz. Static Raman data on ETF_{2}TCNQ (see Supplementary Information) do not show any equivalent features, strongly suggesting that the oscillation is not due to coherent phonons, but of an electronic origin.
To investigate such dynamics, we used a onedimensional Mott–Hubbard Hamiltonian for a halffilled chain, with N=10 sites, with electron hopping, t, and onsite and nearestneighbour Coulomb repulsion U and V,
where c_{l,σ}^{†} and c_{l,σ} are the creation and annihilation operators for an electron at site l with spin σ,n_{l,σ} is the number operator and n_{l}=n_{l,↑}+n_{l,↓}. We described the initial state as , where represents a manybody wavefunction with one electron per site and total spinvector . This reflects the fact that, at room temperature, charges are localized, but possess no magnetic ordering.
We calculate the static optical conductivity (see the Methods section) to find values of U,V and t that provide the best fit to the static measurements. The best fit, shown in Fig. 3c (t=−200 fs), gave U=820 meV, V =100 meV and t=50 meV. It was not possible to fit the optical conductivity using U and t alone and the intersite correlation energy, V, was needed^{14}.
These static parameters were used to fit to the timedependent optical properties. We considered states created by the laser, ρ_{e,L}, which consist of a neighbouring holon–doublon pair, delocalized over L lattice sites, with an optical conductivity σ_{e,L}(ω). To model the timedependence, the system was described by an incoherent mixture of two states: ρ(T)=p_{g}(T)ρ_{g}+p_{e}(T)ρ_{e,1}, where T is pump–probe time delay and p_{s}(T) is the fraction of the sample in state s. The timedependent optical conductivity was described as σ(ω,T)=p_{g}(T)σ_{g}(ω)+p_{e}(T)σ_{e,1}(ω), p_{g}(T) and p_{e}(T) were varied, under the constraint p_{g}(T)+p_{e}(T)=1, to obtain the best fit.
The result is shown in Fig. 3b, where the calculated timedependent optical conductivity is compared to that obtained from the transient reflectivity by fitting the dielectric function with two timedependent oscillators. The model provides a good fit at long time delays (>200 fs), indicating that the longterm dynamics are dictated by incoherent holon–doublon decay. However, the model fails at early times when coherent processes are primarily responsible for the dynamics.
To simulate the coherent dynamics, we calculate the time evolution of the state ρ_{e,10} under the Hamiltonian. The temporal evolution has no free parameters, and the optical conductivity was calculated after the state evolved for a time T. In Fig. 4 the normalized result of this simulation is compared to the normalized experimental optical conductivity. The simulation reproduces the 25 THz oscillations observed experimentally, as shown by the Fourier transforms in Fig. 4c. The numerical simulation contains no damping or dephasing, resulting in a narrower Fourier transform, yet, it is remarkable that this model reproduces the frequency with this accuracy.
The origin of these oscillations can be understood by considering interference between the different photoexcitation paths of the Mott insulator. Optical excitation acts on the ground state …111111…〉 to create bound states of the form …110211…〉. In this case, the holon–doublon pair is bound by an energy V. This state can evolve into a superposition of bound states and ionized states of the form …110121…〉 by electron hopping. These excitations interfere in the time domain, giving rise to the observed oscillations. This conclusion is validated by simulations with V =0, in which these oscillations are not observed (see Supplementary Information).
This physics is reminiscent of singleparticle excitations in multiple quantum wells. An electron is transferred from a well of depth U into a neighbouring well with depth V, representing the bound holon–doublon pair. This ‘exciton’ can tunnel, with an energy t, into a third well, representing the ionized pair (see Fig. 4d). Such a system will oscillate at , in close agreement with the experimental and numerical observations. Thus, our experiments highlight, in a roomtemperature solid, coherent manybody physics that has, so far, been accessible only in ultracold gases^{15}. Our experiments captured dynamics on the timescale associated with hopping and intersite correlations. However, higher temporal resolution could observe dressing resulting from onsite Coulomb correlations, and the coherent formation of the holon–doublon pairs. These measurements would require temporal resolutions approaching the attosecond regime^{16} and, therefore, we anticipate that timeresolved photoelectron spectroscopy^{17} and ultrafast soft Xray techniques^{18} will play major roles in these studies.
Methods
Sample preparation.
Single crystals were grown by first purifying commercially available ET molecules and synthesizing F_{2}TCNQ (ref. 19) by repeated cycles of recrystallization and sublimation. Single crystals of approximately 3 mm×10 mm×0.5 mm were achieved by slowly cooling a hot chlorobenzene solution of purified ET and F_{2}TCNQ.
Experimental setup.
A nearinfrared optical parametric amplifier is driven by an amplified Ti:sapphire laser (100 μJ, 150 fs, 1 kHz pulses at 800 nm). A whitelight seed, generated in a sapphire plate, is amplified in a 3 mm TypeI βbariumborate crystal to ≈2 μJ energy. The ultrabroadband optical parametric amplifier pulses, with a spectrum covering the 1,200–2,200 nm range, are compressed using a deformable mirror to a nearly transformlimited 9 fs duration. The pump fluence at the sample surface was 3.5 mJ cm^{−2}. Transient reflectivity is measured in a degenerate pump–probe configuration at a nearnormal angle of incidence and the probe spectrum is detected by an InGaAs optical multichannel analyser.
Numerical simulation.
The initial state of the system is assumed to be described by the density matrix, . The laser creates excited states of the form ρ_{e,L}∝X_{L}ρ_{g}X_{L}^{†}, where the excitation operator corresponds to the creation of a single holon–doublon pair at neighbouring sites, delocalized over L lattice sites.
The optical conductivity of a given state, ρ, is calculated using the unequal time current–current correlation function c_{j j}(τ,τ′)=tr[ρ j(τ)j(τ′)]θ(τ−τ′), where τ>τ′, is the current density operator, τ is time, θ(τ) is the Heaviside function and j(τ)=exp(i H τ)jexp(−i H τ) is the current operator in the Heisenberg picture. The definition of H is given in the main text.
For the incoherent model we take c_{j j}(τ,τ′)=c_{j j}(τ,0)=c_{j j}(τ) and the regular finitefrequency optical conductivity then follows from c_{j j}(ω), the Fourier transform taken with respect to τ, as σ(ω>0)∝(Re{c_{j j}(ω)})/ω. The total evolution time over which c_{j j}(τ) is computed was limited to τ_{max}=5h/t. The Fourier transform was carried out with a Gaussian windowing function exp[−4(τ/τ_{max})^{2}], leading to a broadening and smoothing of c_{j j}(ω) compared with its exact limit, which is instead composed of numerous δfunctions. This is justified because we focus on features in the high angular frequency range 8t<ω<32t and because the probe pulse used in the experiment has similar spectral limitations.
The timedependent optical conductivity for the state ρ_{e,10} was calculated using the full twotime current–current correlation function c_{j j}(τ,T), where T is the pump–probe time delay and the Fourier transform is taken with respect to τ−T.
Change history
09 December 2010
In the version of this Letter originally published online, the x axis of Fig. 4a was incorrect. This has now been corrected in all versions of the Letter.
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Acknowledgements
This work was supported by the European Community Access to Research Infrastructure Action, Contract RII3CT2003506350 (Centre for Ultrafast Science and Biomedical Optics, LASERLABEUROPE). S.R.C and D.J thank the National Research Foundation and the Ministry of Education of Singapore for support. A.A. is supported by the Royal Society. H.O. is grateful for support by a GrantinAid for Scientific Research (No. 20110005) from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
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S.W., D.B., H.P.E. and G.C. carried out the pump–probe experiments. D.B., S.B. and G.C. designed and built the experimental apparatus. H.U., Y.T., T.H. and H.O. provided samples. S.W. analysed the experimental data. S.R.C. and D.J. carried out the numerical simulations. S.W., A.C., A.A., S.R.C. and D.J. interpreted the data and the simulations. A.C and S.W. wrote the manuscript. A.C. conceived and coordinated the project.
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Wall, S., Brida, D., Clark, S. et al. Quantum interference between charge excitation paths in a solidstate Mott insulator. Nature Phys 7, 114–118 (2011). https://doi.org/10.1038/nphys1831
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DOI: https://doi.org/10.1038/nphys1831
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