Cooling quasiparticles in A$_3$C$_{60}$ fullerides by excitonic mid-infrared absorption

TL;DR

This paper proposes a new mechanism involving excitonic mid-infrared absorption in A$_3$C$_{60}$ fullerides, explaining how IR-induced excitons can cool quasiparticles and enable transient superconductivity at higher temperatures.

Contribution

It introduces a novel excitonic mechanism involving a super-exciton in A$_3$C$_{60}$, providing a new explanation for IR-induced transient superconductivity.

Findings

Identification of a broad IR absorption peak as a super-exciton involving $t_{1u}$ to $t_{1g}$ electronic transition.

Proposal that IR-induced excitons act as a cooling mechanism, extending superconductivity to higher temperatures.

Explanation of the broad absorption feature's origin, challenging previous phonon-based interpretations.

Abstract

Long after its discovery superconductivity in alkali fullerides A$_3$C$_{60}$ still challenges conventional wisdom. The freshest inroad in such ever-surprising physics is the behaviour under intense infrared (IR) excitation. Signatures attributable to a transient superconducting state extending up to temperatures ten times higher than the equilibrium $T_c\sim$ 20 K have been discovered in K$_3$C$_{60}$ after ultra-short pulsed IR irradiation -- an effect which still appears as remarkable as mysterious. Motivated by the observation that the phenomenon is observed in a broad pumping frequency range that coincides with the mid-infrared electronic absorption peak still of unclear origin, rather than to TO phonons as has been proposed, we advance here a radically new mechanism. First, we argue that this broad absorption peak represents a "super-exciton" involving the promotion of one electron from the $t_{1u}$ half-filled state to a higher-energy empty $t_{1g}$ state, dramatically lowered in energy by the large dipole-dipole interaction acting in conjunction with Jahn Teller effect within the enormously degenerate manifold of $\big(t_{1u}\big)^2\big(t_{1g}\big)^1$ states. Both long-lived and entropy-rich because they are triplets, the IR-induced excitons act as a sort of cooling mechanism that permits transient superconductive signals to persist up to much larger temperatures.

Tables

Table 1. Table 1: Molecular terms of undistorted C 3 − 60 superscript subscript absent 60 limit-from 3 {}_{60}^{3-} calculated in the absence of Jahn Teller coupling within the ( t 1 ​ u ) 3 ∪ ( t 1 ​ u ) 2 ​ ( t 1 ​ g ) 1 superscript subscript 𝑡 1 𝑢 3 superscript subscript 𝑡 1 𝑢 2 superscript subscript 𝑡 1 𝑔 1 (t_{1u})^{3}\cup(t_{1u})^{2}(t_{1g})^{1} manifold according to the interaction model III of Ref.  Nikolaev&Michel . The zero of energy is set at the lowest A u 4 superscript subscript 𝐴 𝑢 4 {}^{4}\!A_{u} state and the single-particle energy difference between t 1 ​ g subscript 𝑡 1 𝑔 t_{1g} and t 1 ​ u subscript 𝑡 1 𝑢 t_{1u} , Δ ​ ϵ = 1153 ​ meV Δ italic-ϵ 1153 meV \Delta{\epsilon}=1153\,\text{meV} , is the same as in Nikolaev&Michel . The electronic configurations are labeled according to the irreducible representations of the icosahedral group I h subscript 𝐼 ℎ I_{h} . Since the t 1 ​ u subscript 𝑡 1 𝑢 t_{1u} and t 1 ​ g subscript 𝑡 1 𝑔 t_{1g} orbitals transform as atomic p 𝑝 p orbitals, they are in one to one correspondence with angular momentum states in O ​ ( 3 ) 𝑂 3 O(3) symmetry: A → S → 𝐴 𝑆 A\to S , T → P → 𝑇 𝑃 T\to P , H → D → 𝐻 𝐷 H\to D and G + T → F → 𝐺 𝑇 𝐹 G+T\to F . Note the exceptionally wide spread of terms in the ( t 1 ​ u ) 2 ​ ( t 1 ​ g ) 1 superscript subscript 𝑡 1 𝑢 2 superscript subscript 𝑡 1 𝑔 1 (t_{1u})^{2}(t_{1g})^{1} configuration.

E(meV)	${\left(t_{1u}\right)}^3$	${\left(t_{1u}\right)}^2{\left(t_{1g}\right)}^1$
0	${}_{}{}^{4}A_u^{}$
285	${}_{}{}^{2}H_u^{}$
476	${}_{}{}^{2}T_{1u}^{}$
494		${}_{}{}^{4}H_g^{}$
525		${}_{}{}^{2}T_{1g}^{}$
618		${}_{}{}^{4}A_g^{}$
1109		${}_{}{}^{2}G_g^{}{+}^2{T}_{2g}$
1143		${}_{}{}^{4}T_{1g}^{}$
1280		${}_{}{}^{2}H_g^{}$
1496		${}_{}{}^{2}T_{1g}^{}$
1947		${}_{}{}^{2}H_g^{}$
2218		${}_{}{}^{2}A_g^{}$
2549		${}_{}{}^{2}T_{1g}^{}$

Equations

$$\begin{array}{rl}E\left({}^2{T}_{1u}\right)-E\left({}^4{A}_u\right)& =10J_{\text{H}},\\ E\left({}^2{H}_u\right)-E\left({}^4{A}_u\right)& =6J_{\text{H}}.\end{array}$$

$$\begin{split}\delta\sigma_{1}(\omega)&\propto\int_{0}\!d{\epsilon}\,\mathcal{A}_{\text{exc}}({\epsilon})\,\Bigg{[}\,\theta({\epsilon}-\omega)\,b({\epsilon}-\omega)\,\chi"({\epsilon}-\omega)+\theta(\omega-{\epsilon})\,\Big{(}1+b(\omega-{\epsilon})\Big{)}\chi"(\omega-{\epsilon})\\ &\phantom{\propto\int_{0}\!d{\epsilon}\,\mathcal{A}_{\text{exc}}({\epsilon})\,\Bigg{[}}\quad-b({\epsilon}+\omega)\,\chi"({\epsilon}+\omega)\,\Bigg{]}\,,\end{split}$$

$${\dot{N}}_{\text{exc}}(t)=\int dϵ{\mathcal{A}}_{\text{exc}}(ϵ){\dot{n}}_{\text{exc}}(ϵ,t),$$

$$\dot{\mathcal{E}}(t)=\int d{\epsilon}\,\big{(}\omega-{\epsilon}\big{)}\,\mathcal{A}_{\text{exc}}({\epsilon})\,\dot{n}_{\text{exc}}({\epsilon},t)=\big{(}\omega-E_{\text{exc}}\big{)}\,\dot{N}_{\text{exc}}(t)\,,$$

$$T(t)\,\dot{\mathcal{S}}(t)=\dot{\mathcal{E}}(t)-\mu(t)\,\dot{\mathcal{N}}(t)=\int d{\epsilon}\,\big{(}\omega-{\epsilon}+\mu(t)\big{)}\,\mathcal{A}_{\text{exc}}({\epsilon})\,\dot{n}_{\text{exc}}({\epsilon},t)=\Big{(}\omega-E_{\text{exc}}+\mu(t)\Big{)}\,\dot{N}_{\text{exc}}(t)\,,$$

Full text

Cooling quasiparticles in A3C60 fullerides by excitonic mid-infrared absorption

Andrea Nava

International School for Advanced Studies (SISSA), Via Bonomea 265, I-34136 Trieste, Italy

  

Claudio Giannetti

Interdisciplinary Laboratories for Advanced Materials Physics (ILAMP), Università Cattolica del Sacro Cuore, Brescia I-25121, Italy

  

Antoine Georges

Centre de Physique Théorique, Ećole Polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau, France

Collége de France, 11 place Marcelin Berthelot, 75005 Paris, France

Department of Quantum Matter Physics, University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva 4, Switzerland

  

Erio Tosatti

International School for Advanced Studies (SISSA), Via Bonomea 265, I-34136 Trieste, Italy

CNR-IOM Democritos, Via Bonomea 265, I-34136 Trieste, Italy

International Centre for Theoretical Physics (ICTP), Strada Costiera 11, I-34151 Trieste, Italy

  

Michele Fabrizio

International School for Advanced Studies (SISSA), Via Bonomea 265, I-34136 Trieste, Italy

**Long after its discovery superconductivity in alkali fullerides A3C60 still challenges conventional wisdom. The freshest inroad in such ever-surprising physics is the behaviour under intense infrared (IR) excitation. Signatures attributable to a transient superconducting state extending up to temperatures ten times higher than the equilibrium $T_{c}\sim$ 20 K have been discovered in K3C60 after ultra-short pulsed IR irradiation – an effect which still appears as remarkable as mysterious. Motivated by the observation that the phenomenon is observed in a broad pumping frequency range that coincides with the mid-infrared electronic absorption peak still of unclear origin, rather than to TO phonons as has been proposed, we advance here a radically new mechanism. First, we argue that this broad absorption peak represents a ”super-exciton” involving the promotion of one electron from the $t_{1u}$ half-filled state to a higher-energy empty $t_{1g}$ state, dramatically lowered in energy by the large dipole-dipole interaction acting in conjunction with Jahn Teller effect within the enormously degenerate manifold of \big{(}t_{1u}\big{)}^{2}\big{(}t_{1g}\big{)}^{1} states. Both long-lived and entropy-rich because they are triplets, the IR-induced excitons act as a sort of cooling mechanism that permits transient superconductive signals to persist up to much larger temperatures. **

Superconducting alkali doped fullerenes A3C60 are molecular compounds where several actors play together to determine an intriguing physical behaviour. The high icosahedral symmetry of C60 implies, prior to intermolecular hybridisation, a large degeneracy of the molecular orbitals, thus a strong electronic response to JT molecular distortions lowering that symmetry. In particular, the $t_{1u}$ LUMO, which accommodates the three electrons donated by the alkali metals, is threefold degenerate and JT coupled to eight fivefold-degenerate molecular vibrations of $H_{g}$ symmetry, which mediate the pairingGunnarsson-review . The JT effect, favouring low spin, is partly hindered by (Coulomb) Hund’s rule exchange, which favours high spin. Therefore the overall singlet pairing strength $g$, though still sizeable, is way too small compared to the charging energy of each C${}^{3-}_{60}$ to justify by simple arguments why A3C60 are $s$-wave superconductors. The explanation of this puzzle proposed in Science2002 ; nostroRMP and vindicated by recent experiments emphasises the crucial role of a parent Mott insulating state where the JT coupling effectively inverts Hund’s rules, the molecular ground state therefore turning to spin $S=1/2$ rather than $S=3/2$ FabrizioPRB1997 . A $S=1/2$ antiferromagnetic insulating phase is indeed the ground state in over-expanded NH3K3C60Durand2003 ; Kitano and in Cs3C60Prassides2012 at ambient pressure. In the metallic state, attained under pressure in Cs3C60 and at ambient pressure in K3C60 and Rb3C60, the incipient Mott localisation slows down the coherent motion of quasiparticles while undressing them from charge correlations. As a result, the singlet pairing strength $g$ eventually overwhelms the quasiparticle Coulomb pseudopotential and, on approaching the Mott transition, the system is effectively driven towards the top of the universal Tc vs. $g$ curve Robaszkiewicz , where the critical temperature reaches the maximum possible value at a given non-retarded attraction $T_{c}^{\text{MAX}}\sim 0.055\,g$. Thus, according to the theory of Ref. nostroRMP , the peak $T_{c}\sim 38~{}\text{K}$ reached by Cs3C60 at $\sim 7~{}\text{kbar}$ Ganin ; Alloul2013 is actually the highest attainable at equilibrium in fullerides.

This equilibrium upper limit has been far surpassed in out-of-equilibrium conditions in a recent remarkable pump-probe experiment on K3C60Cavalleri2016 . After irradiation by an intense femtosecond infrared pulse between 80 and 200 meV, K3C60 showed a transient regime of some picoseconds where the optical properties looked like those of a superconductor, alas up to a temperature $T\gtrsim 200$ K, ten times higher than the equilibrium $T_{c}\sim 20$ K, see Fig. 1(b). This tantalising observation has already elicited various theoretical efforts GeorgesPRB2016 ; Demler2016 ; Georges2016-2 ; Millis2016 ; Mazza2017 , where it was mainly assumed, as in the original work Cavalleri2016 , that TO phonon IR absorption acts as the crucial ingredient increasing the pairing efficiency. Here we follow another route directly inspired by experimental features, which leads to a totally different perspective.

First of all, the transient ”superconducting” gap does increase Cavalleri2016 , yet not as much as the transient $T_{c}$, see Fig. 1(b). More importantly, we note in Ref. Cavalleri2016 that the transient reduction of optical conductivity (suggestive of a transiently enhanced superconducting state) is broadly distributed over the IR pumping frequency range from 80 to 200 meV, see Fig. 1(a). Although that includes the two highest $T_{1u}$ IR-active modes near 150 and 170 meVMihaly1993 , the enhancement does not especially peak there, extending instead to lower frequencies, see Fig. 1(a). There is instead an intriguing similarity between a long known DegiorgiPRB1994 ; Degiorgi-ADP1998 broad absorption peak that characterises the equilibrium IR response of K3C60 and Rb3C60. This peak is present and strong in the equilibrium optical data of Ref. Cavalleri2016 , centred around $\sim$ 50 meV and $\sim$ 100 meV broad, see Fig. 1(b). Given these characteristics, the underlying excitation is not a phonon, and can only be electronic; yet, nobody seems to know exactly what it is Gelfand1993 ; DeshpandePRB1994 ; Gunnarsson&EyertPRB1998 ; Chibotaru-Meroedrico .

Intriguingly, it now appears that the superconducting enhancement follows rather closely the shape of this IR absorption feature. Our first task is therefore to understand this excitation which might provide a precious clue to superconductivity enhancement in alternative to the resonance with infrared-active TO modes.

In A3C60 the conduction electrons occupy the narrow band originated by the threefold degenerate $t_{1u}$ LUMO of C60. The Coulomb interaction projected onto the $t_{1u}$ manifold includes a charge repulsion, the Hubbard $U\sim 1~{}\text{eV}$, plus a quadrupole-quadrupole electronic interaction providing an intra-molecular Hund’s rule exchange $J_{\text{H}}>0$. The latter splits the twenty possible $(t_{1u})^{3}$ configurations of C${}_{60}^{3-}$, assumed at first with nuclei rigidly frozen in their ideal icosahedral positions, as

[TABLE]

The highest-spin state, ${}^{4}\!A_{u}$, has therefore the lowest energy, see Table 1. Once the nuclei defreeze, and the molecular ion can distort, the resulting JT energy $E_{JT}$ strongly competes against exchange $J_{\text{H}}$ , since now the quadrupole operators of the $t_{1u}$ electrons couple with the quadrupole of the $H_{g}$ vibrational modes, but with opposite sign. nostroRMP In C${}_{60}^{3-}$, the JT effect actually prevails over Coulomb exchange, effectively inverting Hund’s rules. The real ground state thus becomes the low-spin ${}^{2}T_{1u}$ multipletManiniPRB94-1 ; ManiniPRB94-2 ; O'BrienPRB96 ; DunnPRB2005 ; SigristPRB2007 ; Shahab2016 .

Next, what about the $(t_{1u})^{2}(t_{1g})^{1}$ configuration? Within each C${}_{60}^{3-}$ molecule, the lowest dipole-allowed excitation corresponds to transferring one electron from the $t_{1u}$ LUMO to the $t_{1g}$ LUMO+1, which is also threefold degenerate and whose single-particle energy level lies $\Delta{\epsilon}\sim 1.2~{}\text{eV}$ above. High as this energy is, the $(t_{1u})^{2}\,(t_{1g})^{1}$ subspace comprises as many as 90 states, hence many times more susceptible to exchange splitting and JT effects than the lowest energy $(t_{1u})^{3}$ subspace. In addition, the Coulomb interaction projected onto the enlarged $t_{1u}$–$t_{1g}$ manifold also includes a dipole-dipole interaction, which is stronger than the quadrupole-quadrupole. Through a fully quantitative multipole expansion of the Coulomb interaction, Nikolaev and Michel found (omitting JT couplings)Nikolaev&Michel that the split $(t_{1u})^{2}\,(t_{1g})^{1}$ subspace spans a gigantic $2\,\text{eV}$ range, four times wider than the splitting $10J_{\text{H}}\simeq 476\,\text{meV}$ of the $(t_{1u})^{3}$, see Table 1.

The two lowest $(t_{1u})^{2}\,(t_{1g})^{1}$ states with symmetry ${}^{4}H_{g}$ and ${}^{2}T_{1g}$ lie at only $494\,\text{meV}$ and $525\,\text{meV}$, respectively, above the ${}^{4}\!A_{u}$ ground state Negri1992 , and that is before JT coupling. After allowing for JT, there is a further lowering, and the situation becomes richer Rai1996 ; ChibotaruPRB1996 . The quadrupole moment of the $t_{1g}$ LUMO+1 has opposite sign to the $t_{1u}$ LUMO, and its absolute value is 2.6 times larger, which makes JT couplings much more effective. In particular, and unlike the $(t_{1u})^{3}$ manifold, the JT effect in $(t_{1u})^{2}\,(t_{1g})^{1}$ is stronger in the high-spin $S=3/2$ subspace than in low-spin $S=1/2$. The reason is that in the $S=3/2$ subspace the $H_{g}$ vibrations couple together the lowest energy ${}^{4}H_{g}$ with the ${}^{4}A_{g}$ term, which is a mere $124\,\text{meV}$ above, see Table 1. In the $S=1/2$ subspace, by contrast, the lowest energy ${}^{2}T_{1g}$ is only coupled to states higher than $600\,\text{meV}$ above, which reduces the effect.

We further note that the new $t\otimes H$ $(t_{1u})^{2}\,(t_{1g})^{1}$ JT problem within configurations ${}^{4}H_{g}$ and ${}^{4}A_{g}$ is equivalent to that of $C_{60}^{2-}$ in the $S=0$ subspace of the $(t_{1u})^{2}$ manifold, which involves the configurations ${}^{1}H_{g}$ and ${}^{1}A_{g}$ and where the JT energy gain is known to be maximum ManiniPRB94-1 ; ManiniPRB94-2 ; O'BrienPRB96 ; SigristPRB2007 . On the other hand, the Coulomb exchange splitting $E(^{4}A_{g})-E(^{4}H_{g})=124\,\text{meV}$ of $(t_{1u})^{2}\,(t_{1g})^{1}$ $S=3/2$ subspace is smaller than $E(^{1}A_{g})-E(^{1}H_{g})=6J_{\text{H}}=286\,\text{meV}$ of the $S=0$ $(t_{1u})^{2}$ case, implying a larger JT energy gain.

Accurate estimates of the molecular terms within the enlarged $(t_{1u})^{3}\cup(t_{1u})^{2}(t_{1g})^{1}$ manifold and in presence of JT coupling to the $H_{g}$ vibrations would require a precise knowledge of all Hamiltonian parameters that are involved. That’s a tall order, because, while the frequencies of the $H_{g}$ modes are known from experiments, and different calculations of JT energy $E_{JT}$ more or less agree, the individual values of the coupling constants with the $t_{1u}$ electrons are hard to establish IwaharaPRB2010 without resorting to photoemission experiment Gunnarsson1995 . Also questionable is whether the simple linear coupling to vibrations, as usually assumed, is sufficient, as has been pointed out BunnPRB2008 ; DunnPRB2013 . Besides that, there are so far no direct estimate of the $H_{g}$ vibration coupling constants with the $t_{1g}$ electrons, obviously not extractable from photoemission. We mentioned that the $t_{1g}$ quadrupole moment is larger in absolute value than the $t_{1u}$ one, which would suggest stronger vibration coupling constants, as indeed observed in electronic structure calculations of isolated molecular anions Green1996 . Moreover, $t_{1g}$ electrons couple preferentially to higher frequency $H_{g}$ vibrations with tangential character, while $t_{1u}$ electrons to lower frequency radial vibrations, which might also imply a larger $t_{1g}$ JT energy BunnPRB2008 . One should finally note that, given the large size of the $(t_{1u})^{2}(t_{1g})^{1}$ subspace, even small variations of the many Coulomb exchange parametersNikolaev&Michel and vibrational coupling constants may lead to appreciably different results.

For these reasons we opt for a less ambitious approach and, following Ref. nostroRMP , we treat the JT problem within the anti-adiabatic approximation, were all effects depend only on the value of the total JT energy gain $E_{\text{JT}}$, whose value for $t_{1u}$ electrons is far less uncertain than the value of each vibrational coupling constant IwaharaPRB2010 , see the Supplementary Notes for details. We use the model III interaction parameters of Nikolaev and MichelNikolaev&Michel , with a 14% reduction to account for screening effects of nearby moleculesGunnarssonPRL92 , and we further assume, in accordance with the density functional results of Green1996 , that the $t_{1g}$ LUMO+1 JT energy is $1.25^{2}$ larger than the $t_{1u}$ LUMO one. In the left panel of Fig. 2 we show the low lying molecular terms as function of $E_{\text{JT}}$ IwaharaPRB2010 .

We can now consider the full multiplet spectrum for a realistic estimate of $E_{\text{JT}}=50-70\,\text{meV}$ IwaharaPRB2010 . The ${}^{2}T_{1u}$ ground state and the lowest ${}^{2}H_{u}$ excitation, whose role was recently discussed Shahab2016 , both belong to the $(t_{1u})^{3}$ manifold. The very next state however is the ${}^{4}\!A_{g}$ term, of $(t_{1u})^{2}\,(t_{1g})^{1}$ origin, dramatically pushed down close to the ground state by JT and dipole-dipole interaction, despite the 1 eV energy of the $t_{1g}$ LUMO+1. We also performed a different calculation, treating the JT coupling within the single mode approximation ManiniPRB98 and using a variational approach SigristPRB2007 that consists of a statically distorted wavefunction projected onto a state with well defined icosahedral symmetry (details are in the Supplementary Notes). In the right panel of Fig. 2 we show the energies thus obtained of the ${}^{2}T_{1u}$ and ${}^{4}\!A_{g}$ states as function of the distortion vector norm. As anticipated, the ${}^{4}\!A_{g}$ energy minimum is reached for a larger distortion than that of ${}^{2}T_{1u}$, which entails substantial Franck-Condon effects - further strengthened by the shape difference, bimodal for ${}^{2}T_{1u}$ ManiniPRB94-1 ; ManiniPRB94-2 and unimodal for ${}^{4}\!A_{g}$.

We propose that the IR peak observed in A3C60 corresponds precisely to the low lying ${}^{4}\!A_{g}$ state, the ${}^{2}T_{1u}\rightarrow^{4}\!A_{g}$ transition essentially turning into a genuine triplet exciton in the bulk material. The parity allowed but spin forbidden optical creation of this exciton can actually acquire oscillator strength and appear in the IR optical spectrum of a narrow-band nearly (antiferro)magnetic metal, through the simultaneous absorption/emission of a low energy spin-triplet particle-hole excitation, that is a paramagnon. For that it is important to recall that A3C60 are indeed narrow quasiparticle-band metals, close to a transition into an antiferromagnetic Mott insulator state, so much so that the transition is realised when the cation A merely changes from Rb to Cs. The absorption process is schematically shown in Fig. 3. The photon induces a virtual spin-conserving transition $t_{1u}\to t_{1g}$. This intermediate state then transforms into the triplet exciton by absorbing/emitting a paramagnon via intermolecular exchange. One should note that this absorption mechanism is of the very same nature to that introduced by Rice and Choi Rice&Choi , which is necessary to explain why uncharged $T_{1u}$ vibrations acquire oscillator strength and thus are observed in optics. The contribution of the ${}^{2}T_{1u}\rightarrow(^{4}\!A_{g}$ $\pm$ paramagnon) peak to the optical conductivity reads

[TABLE]

where $\mathcal{A}_{\text{exc}}({\epsilon})$ is the exciton absorption spectrum, $b({\epsilon})$ the Bose distribution function, and $\chi"({\epsilon})$ the imaginary part of the dynamical local spin susceptibility. Equation (2) suggests that the large width of the absorption peak, which experimentally corresponds to a timescale of about $7~{}\text{fs}$, is the result of a convolution between the paramagnon bandwidth and a Franck-Condon broadening, rather than a radiative lifetime of the exciton. In fact, the expectedly strong Franck-Condon effect must cause a large broadening in $\mathcal{A}_{\text{exc}}({\epsilon})$ corresponding to the non-radiative relaxation of the triplet exciton to a dark state whose lifetime might be much longer, possibly picoseconds or more, before eventual (phosphorescent) recombination. In agreement with this exciton-paramagnon interpretation, the IR absorption peak grows in importance and intensity from K3C60 to Rb3C60DegiorgiPRB1994 , the latter closer to Mott insulation (realised in Cs3C60), thus with stronger and narrower paramagnons.

The next and central question in the present context is if and why this exciton peak should actually play a role in the apparent enhancement of Tc found by Ref. Cavalleri2016 where IR-pumping is roughly in the same frequency range. We start by noting that the experimental transient superconducting-like absorption spectra suggest, see Fig. 1(b), that the IR pump can act to sweep away the thermally excited quasiparticle states that, at equilibrium, are responsible for the gap filling-up and closing with the transition to the normal state. Things superficially seem as if the pump effectively cooled down quasiparticles. Following this hypothesis, we can qualitatively describe how the quasiparticle distribution should evolve first during the IR laser pulse, about 300 fs long. Within that short time lapse, the system is effectively isolated from the environment, with which it was in thermal equilibrium before the IR shot.

The IR pulse supplies the initial normal metal with energy, which is sunk in the exciton-paramagnon excitation as well as by the vibrations that are emitted during the molecular relaxation after the vertical Franck-Condon transition. If we assume that the quasiparticle collision rate is high enough, as expected by the poor Fermi-liquid character above Tc nostroRMP , then the quasiparticle subsystem will exit the laser shot time in an effective microcanonical ensemble identified by an energy $\mathcal{E}$ and quasiparticle number $\mathcal{N}$. At a later time the quasiparticles will eventually come to equilibrium with the excitons, the lattice and the molecular vibrations (the decay times of the eight $H_{g}$ modes into $t_{1u}$ particle-hole excitations range between 0.03 and 4 ps). Yet, in the long transient before that happens, we can legitimately define an entropy $\mathcal{S}(\mathcal{E},\mathcal{N})$ of the quasiparticle liquid and its effective temperature $T_{\text{eff}}^{-1}=\partial\mathcal{S}/\partial\mathcal{E}$. Moreover, if the quasiparticle collision integral is strong enough to establish local equilibrium during the whole pulse duration, we are additionally allowed to define an entropy S\big{(}\mathcal{E}(t),\mathcal{N}(t)\big{)} that depends on the quasiparticle energy, $\mathcal{E}(t)$, and number, $\mathcal{N}(t)$, at time $t$ after the pulse front arrives. The absorption process of Fig. 3 implies that the creation rate of excitons is

[TABLE]

where $\dot{n}_{\text{exc}}({\epsilon},t)$ is equal to the term in square brackets of Eq. (2) multiplied by a parameter that we fit from equilibrium optical data (see Supplementary Notes), with the Bose distribution function and magnetic susceptibility corresponding to the instantaneous local equilibrium conditions. Since at $\omega\leq E_{\text{exc}}$, for each extra exciton a quasiparticle is annihilated then $\dot{N}_{\text{exc}}(t)=-\dot{\mathcal{N}}(t)$. Moreover energy conservation implies that

[TABLE]

where $\omega$ is the laser frequency and the last equivalence holds if $\mathcal{A}_{\text{exc}}({\epsilon})\sim\delta({\epsilon}-E_{\text{exc}})$, which we shall assume hereafter for simplicity. Because of our assumption of local equilibrium, it follows that the quasiparticle entropy satisfies

[TABLE]

where $T(t)$ and \mu(t)=-T(t)^{-1}\big{(}\partial\mathcal{S}/\partial\mathcal{N})_{\mathcal{E}} are, respectively, the instantaneous temperature and chemical potential. The entropy is expected to be maximum when the number of $t_{1u}$ quasiparticles is equal to its initial value of three per molecule, so that $\mu(t)<0$ for any $\mathcal{N}(t)<\mathcal{N}(0)$. Through Eq. (5) we thus reach the conclusion that the quasiparticle entropy can indeed decrease, and so the effective temperature, especially for frequencies $\omega\leq E_{\text{exc}}$ when exciton creation requires absorption of thermal quasiparticle-quasihole triplet pairs. To simplify the calculation of $T_{\text{eff}}=T(t\simeq 300~{}\text{fs})$ at the end of the laser pulse, besides assuming $\mathcal{A}_{\text{exc}}({\epsilon})\sim\delta({\epsilon}-E_{\text{exc}})$, we also neglect the contribution from the change in quasiparticle density, i.e. we take $\mu(t)=0$ in (5), which implies that the entropy may decrease only below resonance. Furthermore we assume for $\chi^{\prime\prime}({\epsilon})$ the expression of non-interacting quasiparticles at half-filling and temperature $T(t)$ with a reduced bandwidth of 100 meV, and model the evolution of their distribution function by a Boltzmann type of equation (see Supplementary Notes). In Fig.4 we show $T_{\text{eff}}$ thus obtained for equilibrium sample temperatures $T=25,50,100,150,200,300$ K. The result of this modelling, crude but we believe inevitable, is that the effective temperature $T_{\text{eff}}$ can indeed be substantially lower than the equilibrium value – thermal triplet quasiparticle-quasihole pairs being absorbed so that IR pumping can reach the exciton energy.

Conclusions – The apparently tenfold critical temperature enhancement discovered by IR pumping in K3C60 Cavalleri2016 is explained by a novel mechanism. First, noting that the effect broadly overlaps in frequency with the unexplained equilibrium mid-infrared absorption peak observed in all A3C60 fullerides, that peak is argued, on the basis of single-molecule calculations, to correspond to the creation of a triplet exciton, Frank-Condon broadened and downshifted from its high LUMO–LUMO+1 energy by large intra-molecular interactions. Spin conservation requires this process to be accompanied by absorption/emission of a paramagnon.

Second, we propose that the transient $T_{c}$ enhancement occurs because, in the process of promoting quasiparticles into these long-lived triplet excitons, the laser pulse effectively cools down the quasiparticles system. This also explains why the experiment at 300 K Cavalleri2016 still shows a transient increase of reflectivity, even though the optical data cannot be fit by a model for a superconducting state.

Differently from other laser cooling techniques KETTERLE1996 , this mechanism relies on the triplet excitons generated by the laser pulse, which effectively act as charge and spin reservoir soaking up entropy from quasiparticles GeorgesPRA2009 .

While compatible with existing data, various aspects and implications of the present theory can be tested against further experiments. For one, the exciton and its spin-triplet nature could be tackled by magnetic fields and other spectroscopic tools, including e.g., detection of phosphorescence in pumped Cs3C60.

The possible existence and detection at ambient pressure of the same broad IR absorption peak near 50 meV in the Mott insulating A15-Cs3C60 at ambient pressure would provide support to our proposal of a light-induced intra-molecular exciton ${}^{4}\!A_{g}$ without charge transfer among nearby molecules as opposed to the alternative ${}^{4}\!A_{u}$, a term which besides spin is also parity forbidden and thus much weaker as it requires additional inter-molecular excitations. It may be noted, on the other hand, that the lack of inversion symmetry in merohedrally disordered fcc fullerides might partly allow the parity forbidden dipole transitions DeshpandePRB1994 ; Chibotaru-Meroedrico , mixing in case the spin-quartet ${}^{4}\!A_{u}$ state with the ${}^{4}\!A_{g}$. Our theory of pumping-induced cooling is sufficiently general and would apply to that case too.

Also important would be a re-examination of NMR data, where signatures of a $75\,\text{meV}$ spin-gap in Rb3C60 AlloulPRB2002 have been so far attributed to thermal population of the ${}^{4}A_{u}$ state, for the possible presence of another, possibly even lower energy ${}^{4}A_{g}$ spin-quartet state.

Finally, the role of the triplet exciton in the IR-pumping enhancement of Tc could be addressed in a variety of ways and of materials. The strongest candidate remains pressurised Cs3C60, which metallizes and superconducts above 5 kbar, and where the full range of parameters becomes available as a function of pressure. The ideal maximum equilibrium $T_{c}$=38 K of fullerides being achieved near 7 kbarGanin , it would be exciting to explore whether the transient $T_{c}$ might conceivably even be raised closer to room temperature.

Acknowledgments

We are very grateful to A. Cavalleri, L. F. Chibotaru, M. Capone, and A. Cantaluppi for comments and discussions, and to S.S. Naghavi for his help. We also acknowledge discussions with A. Isidori, M. Kim and G. Mazza. This work was supported by the European Union, under ERC FIRSTORM, contract N. 692670, ERC MODPHYSFRICT, contract N. 320796, and ERC QMAC, contract N. 319286.