Emergent ultrafast phenomena in correlated oxides and heterostructures

The capability of inducing, detecting, and reversibly switching the magnetization of materials on ultrafast timescales (from pico- to femto-seconds) and at atomic distances is an extremely urgent task for material scientists [25]. The inescapable need and the ever-growing urgency are simply dictated by the daily increasing technological demand to overcome the limitations, in terms of speed and density, of today's magnetic memories and, more in general, digital electronics (logic gates, storage devices, ...). This quest goes well beyond searching for a new material, which can optimize the actual technology, and regards the concurrent development of new ideas, tools and overall technologies to control the magnetic response of a functional material. One very promising route exploits ultrafast light pulses to photo-stimulate the emergence of novel magnetic states on timescales much shorter than the internal thermalization [26]. In particular, by exploiting the interplay between the electronic and spin degrees of freedom, it is possible to induce transitions towards transient non-thermal states, whose magnetic properties are different from those exhibited in the equilibrium state. As an example, ultrafast optical techniques have been proposed as a tool to directly manipulate the exchange coupling J_exc in Mott insulators [27, 28] and have been used to induce femtosecond magnetic switching in manganites [29] by exploiting an initial quantum-coherent magnetic regime on timescales faster than the coupling with the incoherent environment. Moreover, the intrinsic capability of being reversible makes these all-optical routes usually preferable with respect to ordinary chemical or surface doping. Along this line of thinking, it is urgent to find new classes of SCMs that can be efficiently used as playgrounds for inducing transient non-thermal magnetic states by means of ultrafast optical techniques.

Mott insulators are always characterized by a very strong antiferromagnetic coupling driven by the virtual hopping (${t}_{{\rm{h}}}$) of the electrons on neighboring sites, with an energy cost equal to the on-site Coulomb repulsion U. The exchange energy in Mott insulators, which can assume very large values (up to $\approx 150$ meV), is given by J_exc = 2${t}_{{\rm{h}}}^{2}/U$ and regulates the dynamics of the localized spins on the correlated sites. Because J_exc originates from the electrostatic Coulomb repulsion and the Pauli principle, rather than from magnetic dipole forces, it can be directly modified by the change of the population of the Hubbard bands induced by photostimulation with ultrashort light pulses tuned across the Mott gap [27, 28]. Spin–orbit Mott insulators are emerging as a new interesting class of magnetic correlated materials. In these systems, the strong SOC gives rise to Mott insulating phases in which the local moments have also orbital character [30, 31], thus paving the way to the simultaneous control of both the magnetic state and the orbital occupation on the ultrafast timescale.

The family of $5d$ TMOs, such as iridium oxides, is considered as one of the most promising systems to investigate the ultrafast charge/orbital/magnetic dynamics in spin–orbit Mott insulators. In $5d$ TMOs one of the fundamental assumptions shared by almost all microscopical and phenomenological theories for strongly correlated systems—the presence of a single energy scale dominating over all others—dramatically fails. The increased average radius of the $5d$ orbitals, as compared to that of the $3d$ TMOs, simultaneously leads to the reduction of the on-site Coulomb repulsion (down to $U\sim 1$ eV) and to the increase of the hopping integrals (spin-orbit coupling strength up to ${t}_{{\rm{h}}}\sim 0.4$ eV) and, consequently, of the bandwidth W [32]. As a consequence of the closeness of the U and W energy scales, neither a local Mott-like picture nor a delocalized Wannier scenario is fully appropriate to capture even the most basic physics of these compounds. As a further element, the large atomic number of the $5d$ metal ions makes the SOC extremely relevant (spin-orbit coupling strength up to ${\lambda }_{{\rm{SOC}}}\sim 0.7$ eV) for the electronic and magnetic properties of the system [30, 33, 34]. The topology of the lattice, combined with the actual valence of the transition-metal ion, determines the orbital and spin effective interactions and, consequently, the possible orderings and critical temperatures of the system. As an example, the interplay of the lattice topology and SOC can lead to bond-directional magnetic interactions (Kitaev-like terms in the effective-spin Hamiltonian) [35] that drive a strong frustration of the magnetic ground state and the possible emergence of exotic spin-liquid phases at low temperatures [33, 36–38].

In the following sections, we will provide two examples of $5d$ TMOs (the sodium iridate, Na₂IrO₃, and the strontium iridate, Sr₂IrO₄) in which the interaction between the SOC and the topology of the lattice links the dynamics of local magnetic moments to the creation, propagation and recombination of photo-injected charge excitations. The out-of-equilibrium approach permits both to better address the leading interactions that determine the equilibrium properties and to produce transient non-thermal states with an emergent magnetic dynamics due to the coupling of the photo-injected electron–hole excitations with an "artificial" magnetic environment. These findings open new routes in the investigation and, possibly, in the optical control of novel excitations and of exotic phases (e.g. spin-liquid) in $5d$ TMOs.

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In Na₂IrO₃, the edge-sharing IrO₆ octahedra form skewed (with respect to the crystallographic axes) planes of Ir ions arranged in a honeycomb lattice (see figure 2). The octahedral crystal-field of the oxygen cage splits the Ir–$5d$ orbitals in two manifolds: e_g, much higher in energy and out of the scope of the present analysis, and t_2g, at lower energy and occupied by 5 electrons [39–41]. The actual distortions lift the degeneracy of the t_2g manifold and affect the hopping amplitudes within and between the Ir hexagons [32, 42, 43]. The system exhibits an insulating gap of ∼340 meV [31] and quite complex magnetic properties: strong magnetic correlations below ${{\rm{\Theta }}}_{{\rm{corr}}}\sim 100\,{\rm{K}}$ and a long-range ordered zig-zag phase below ${T}_{{\rm{N}}}\sim 15$ K [41, 44, 45]. The large frustration parameter ${{\rm{\Theta }}}_{{\rm{corr}}}/{T}_{{\rm{N}}}\sim 7$ suggests the tendency to form a low-temperature spin-liquid phase out of which the zig-zag order emerges.

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2.2.1. Band-structure calculations

2.2.2. The non-equilibrium experimental approach

In Sr₂IrO₄, the corner-sharing IrO₆ octahedra form IrO₂ planes perpendicular to the crystallographic c-axis, where the Ir ions are arranged in a square lattice (see figure 3). Because of a staggered in-layer rotation of the octahedra, the unit cell, which coincides with the magnetic unit cell, is formed by four IrO₂ layers [30, 51–53]. Similarly to Na₂IrO₃, 5 electrons occupy the t_2g manifold induced by the octahedral crystal-field. Because of the simpler topology of the square lattice, which does not induce neither additional magnetic frustration nor privileged hopping paths, the localized J_eff picture is solid and widely accepted [30]: the SOC splits the six t_2g levels and returns two half-filled ${J}_{{\rm{eff}}}=1/2$ levels and four completely filled ${J}_{{\rm{eff}}}=3/2$ levels, further split in two doublets by a residual tetragonal (distortion) crystal field [54]. According to this scheme, the system behaves as a robust Mott insulator once the effect of the modest on-site Coulomb repulsion U is properly taken into account [30]. The charge gap, estimated by optical conductivity [55] and resistivity [53] measurements is ∼100 meV. The Sr₂IrO₄ magnetic properties can be efficiently described by a Heisenberg Hamiltonian [53, 54] with exchange terms up to third neighbors in plane (${J}_{{\rm{h}}}=60$, ${J}_{\mathrm{hh}}=-20$, and ${J}_{\mathrm{hhh}}=15$ meV) [53], while the out-of-plane term is as low as ${J}_{\perp }\approx 1$ μeV. The Heisenberg model accounts for the canted in-plane anti-ferromagnetic ordering with a critical Néel temperature of about 240 K [56].

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2.3.1. Direct observation of the ultrafast dynamics of magnetic correlations

The two paradigmatic examples of the honeycomb-lattice Na₂IrO₃ and the square-lattice Sr₂IrO₃ demonstrate that the creation of non-thermal electronic populations can be exploited to photoinduce transient magnetic states, which are potentially tunable and reversible. The nature of these transient states crucially depends on the topology of the lattice, on the strength and directionality of the short-range magnetic interactions and on the lifetime and orbital character of the charge excitations. Here we summarize the key concepts that can be exploited for developing novel schemes for the ultrafast manipulation of magnetic states in $5d$ TMOs:

• The interplay of SOC, correlations and topology of the lattice, typical of $5d$ TMOs and relativistic correlated materials in general, allows to map the local demagnetization processes into specific optical transitions in the near-infrared/visible energy range. The other way around, the resonant photostimulation of high energy charge states with specific orbital components can be exploited to selectively trigger magnetic excitations.
• The competition between different magnetic configurations—e.g. zig-zag and stripy antiferromagnetic phases, typical of frustrated correlated magnets, can be exploited for achieving a controlled magnetic switching via the selective excitation of charge states coupled to magnetic correlations with different net magnetization in terms of spatial arrangement and polarization.

In the near future, we foresee that the use and development of ultrafast techniques will become more and more fundamental to understand how the magnetic dynamics of correlated 5d materials can be functionalized and manipulated by light. The combination of ultrafast photo-doping with the chemical surface [58] doping will provide new opportunities to discover additional emergent properties of scientific and technological interest.

In solids, the simplest example of coherent transport is given by the Bloch oscillations. When electrons, described by Bloch wavefunctions, are free from any scattering process, they respond to a steady electric field E in an oscillatory way, with period T_B = 2 $\pi {\hslash }$/aeE (a is the lattice constant, e the electronic charge) [61]. Even though in bulk materials this period is much larger than the dephasing time, thus making oscillations vary hard to observe, the recent advances in the atomic-scale synthesis of oxide heterostructures [19] have opened very intriguing scenarios. For example, the polar nature of many TMO heterostructures, such as LaAlO₃/SrTiO₃ and LaVO₃/SrTiO₃, gives rise to intrinsic electric fields [62] corresponding to a potential slope of the order of 0.08 eV Å⁻¹. In these systems, the corresponding period of the Bloch oscillations is ${{T}}_{{\rm{B}}}\sim 12$ fs, which is very close to both the expected dephasing time and the time necessary to eventually collect the charges across few atomic monolayers. This observation suggests the possibility of exploiting quantum-coherent pathways to enhance the conversion efficiency in suitable engineered few-monolayers systems, provided the excitations are collected faster than the time necessary to completely lose the initial quantum-coherence.

3.1.1. Decoherence and the density matrix

In order to properly introduce the concept of dephasing of a generic pure quantum state $| \psi \rangle$, it is necessary to make use of the density matrix, $\widehat{\rho }$(t) = $| \psi \rangle \langle \psi |$. For sake of simplicity, we limit the discussion to a two-level system, that accounts for the ground ($| {\psi }_{0}\rangle$) and excited ($| {\psi }_{e}\rangle$) states. Any form of quantum-coherent excitation prepares the system into a superposition of the ground and excited states: $| \psi \rangle$ = a₀(t)$| {\psi }_{0}\rangle$ + a_e(t)$| {\psi }_{e}\rangle$. Assuming that ${a}_{n}$(t) = ${c}_{{n}}$(t) exp(-${\rm{i}}{\omega }_{n}$t), where ${c}_{n}$(t) are slowly varying complex functions of time and ${\omega }_{n}$ = ${E}_{n}$/ℏ are the eigenfrequencies of the ground $({n}=0)$ and excited $({n}=e)$ states, the matrix representation of $\widehat{\rho }$ reads:

$$\begin{eqnarray}\widehat{\rho }({t})=\left[\begin{array}{cc}{\rho }_{00} & {\rho }_{0e}\\ {\rho }_{e0} & {\rho }_{{ee}}\end{array}\right]=\left[\begin{array}{cc}| {c}_{0}({t}){| }^{2} & {c}_{0}({\rm{t}}){c}_{e}^{* }({t}){{\rm{e}}}^{-{\rm{i}}{\omega }_{0e}{t}}\\ {c}_{e}({\rm{t}}){c}_{0}^{* }({t}){{\rm{e}}}^{{\rm{i}}{\omega }_{0e}{t}} & | {c}_{e}({t}){| }^{2}\end{array}\right],\end{eqnarray} \tag{ 1 }$$

where ${\omega }_{0e}$ = (E₀ – E_e)/ℏ. In this form, the diagonal terms describe the time-dependent populations of the ground and excited states, while the off-diagonal ones define the coherences of the system, which can be related to the macroscopic polarization $\overrightarrow{P}$(t) = $N/V{\overrightarrow{\mu }}_{0e}({\rho }_{0e}+{\rho }_{e0})$, via the atomic dipole moments ${\overrightarrow{\mu }}_{0e}$ = $\langle {\psi }_{0}| \overrightarrow{\widehat{\mu }}| {\psi }_{e}\rangle$ with density N/V. Importantly, the density matrix maintains exactly the same physical meaning in the case of statistical ensembles, where the matrix elements should be calculated as the averages over the population indexed by s: e.g. ${\overline{\rho }}_{0e}$ = 1/$N{\sum }_{s}{\rho }_{0e}^{(s)}$. Even though the microscopic treatment of the time-evolution of $\widehat{\rho }$ strongly depends on the specific system considered and goes beyond the scope of the present work, the decoherence processes can be accounted for in a phenomenological way by introducing a decoherence timescale, ${\tau }_{D}$, that depends on the interaction with the environment and produces the exponential decay of the off-diagonal matrix elements:

$$\begin{eqnarray}\widehat{\overline{\rho }}({t})=\left[\begin{array}{cc}{\overline{| {c}_{0}({t})| }}^{2} & \overline{{c}_{0}{c}_{e}^{* }}{{\rm{e}}}^{-{\rm{i}}{\omega }_{0e}{t}}{{\rm{e}}}^{-{t}/{\tau }_{D}}\\ \overline{{c}_{e}{c}_{0}^{* }}{{\rm{e}}}^{{\rm{i}}{\omega }_{0e}{t}}{{\rm{e}}}^{-{t}/{\tau }_{D}} & {\overline{| {c}_{e}({t})| }}^{2}\end{array}\right]\end{eqnarray} \tag{ 2 }$$

${\tau }_{D}$ thus characterizes the evolution of a macroscopically quantum-coherent state into an incoherent statistical ensemble (${\rho }_{0e},{\rho }_{e0}\to$0) and is determined by all the quasi-elastic primary relaxation processes that destroy the macroscopic coherence without altering the population of the levels. The precise knowledge of ${\tau }_{D}$ in solids represents the prerequisite for the design of experiments and theories aimed at measuring and describing the decoherence processes and the transport properties on the timescales shorter than those necessary to achieve the incoherent transport regime.

3.1.2. Excitonic resonances in correlated materials

In order to observe coherent phenomena in solids, it is necessary to find well defined excitonic transitions, whose dephasing time can be accessed by ultrafast techniques. While the study of low-temperature excitons in semiconductors is a well established field [63], correlated materials and TMOs constitute an almost unexplored world, though very promising. Let us consider the antiferromagnetic Mott (or charge-transfer, CT) insulators, such as the copper oxides parent compounds. The lowest excitation is the CT of a localized Cu–3${d}_{{x}^{2}-{y}^{2}}$ hole to its neighboring O–2${p}_{x,y}$ orbitals, with an energy cost ${{\rm{\Delta }}}_{{\rm{CT}}}\,\simeq$ 1.5–2 eV. In the optical conductivity, this process is revealed by a typical CT edge, which defines the onset of optical absorption by particle–hole excitations. However, the local CT excitation is a many-body process and can be dressed by structural deformations or rearrangements of the spin-background (lattice- or spin-polarons), or even form a bound state (Hubbard exciton (HE)), thus forming a local exciton which can hop in the lattice and in the ordered antiferromagnetic background. This physics is entirely contained in the ubiquitous structures that are observed at the CT edge of TMOs (see figure 4) and whose nature is still subject of intense debate [64–68]. From the linewidth of these resonances, which ranges between 0.2 and 0.4 eV, it is possible to estimate the upper bound of the excitonic lifetimes (1.5–3 fs). Although very short, these values are close to the current limit of ultrafast technologies that pushed the temporal resolution well below 10 fs [69–71]. However, the total linewidths in TMOs are usually severely affected by the inhomogeneous broadening related to the presence of defects or intrinsic electronic/lattice inhomogeneities typical of correlated materials [72]. The real dephasing time, which corresponds to the inverse of the homogenous contribution to the total linewidth, should be evaluated either by starting from microscopic theories or by developing suitable multi-dimensional ultrafast techniques, as discussed in section 3.4. We stress that similar physics is present in many correlated TMOs, such as vanadium oxides (LaVO₃, V₂O₃) and Ir-based relativistic Mott insulators (Sr₂IrO₄, Na₂IrO₃). The possibility of investigating the decoherence of excitonic resonances at the edge of the Mott (CT) gap is thus very general to many classes of TMOs that can be exploited for applications.

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3.1.3. Time-domain models

3.1.4. Time-domain experiments

As discussed in the previous section, the typical decoherence timescale for electron–hole excitations at ambient temperature in TMOs can range between 4 fs and several tens of femtoseconds, depending on the specific characteristics (charge carrier density, hopping ${t}_{{\rm{h}}}$) of the system considered. The recent advances in ultrafast science made these timescales directly achievable, thus providing the platform for the study of coherent-transport phenomena in correlated materials.

From the technological standpoint, the possibility of tailoring the coherent dynamics on ultrafast timescales would have a dramatic impact on the working principle of few-atomic-layers oxide-based devices. To properly make the case, we start from a paradigmatic example, i.e. LaVO₃/SrTiO₃ (LVO/STO) heterostructures (see figure 5), that have been recently suggested for photovoltaic applications [62]. At 300 K, LVO is a orthorhombic paramagnetic Mott insulator (direct gap of $\simeq 1.1$ eV) with 2 electrons in the crystal-field split 3d–t_2g orbitals. The fundamental optical excitation is the transition from LHB to UHB, which corresponds to the transfer of an electron from one V atom to its neighboring V-site. At 140 K, LVO transforms into a monoclinic structure with an antiferromagnetic C-type order (c-axis ferromagnetic stacking of antiferromagnetic ab layers). Interestingly, the orthorhombic LVO is one of the few Mott insulators which exhibits large quantum fluctuations at room temperature [86], in the sense that the orbital occupation of the t_2g orbitals is strongly fluctuating with a frequency range which is determined by quantum effects instead of k_BT. However, these strong incoherent orbital fluctuations are almost completely suppressed in the monoclinic 140 K phase, when they turn into a phase-coherent condensate (orbital ordering) in which the orbital occupation assumes a well defined (phase stable) pattern in space and time. The exploitation of the LVO properties is based on the possibility of creating heterostructures in which the chemical/structural composition of each layer is controlled at the atomic scale [19]. Let us consider the multilayer structure constituted by 4 LVO unit cells interfaced to SrTiO₃, as suggested in [62]. DFT calculations (see figure 5) show the layer-by-layer density DOSs of the V–3d_xy states. Interestingly, the shift of the bands between neighboring layers indicates the formation of a potential gradient between the (LaO)⁺–(VO₂)⁻ planes, that is estimated as high as 0.08 eV Å⁻¹. Furthermore, the intrinsic presence of conducting states (of d_xy character) at the LVO/STO interfaces provides a natural way of collecting the electron–hole excitations generated within the LVO layers. The working principle of the LVO/STO heterostructure can be described by the following steps: (i) the absorption of an above-gap photon creates a many-body excitation $| {\psi }_{e}({\bf{r}})\rangle$ that consists in a local doublon (doubly occupied V–3d_xy orbital) and holon (unoccupied V–3d_xy orbital) pair; (ii) the charged doublons and holons, subject to the intrinsic E_in, hop across the few LVO atomic layers and are collected by the two opposite interface conductive layers, thus generating a photocurrent.

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The efficiency of such a device is crucially based on the pathways available to collect the initial electron–hole excitons. Given that the exciton recombination time (>100 fs, see [79]) is large as compared to the time necessary to collect the charges, the quasi elastic scattering processes, which dephase the initial wavefunction without changing the population, are expected to play a crucial role. While a large scattering rate would be detrimental for the collection of the charges, the absence of any scattering process would lead to Bloch oscillations without efficiently establishing any photocurrent. This naive expectation has been recently demonstrated by non-equilibrium dynamical mean-field theory (DMFT) simulation of the transport properties of a multilayer Mott insulator in which doublons and holes are created by photoexcitation [87].

The natural question emerging from these considerations is whether the timescale necessary to collect the charges is compatible with the emergence of transport phenomena along quantum walks, which can be faster and more efficient than classical ones [88, 89]. Considering the hopping terms estimated by local density approximation calculations combined with DMFT [86], we note that the most efficient channel for collecting charges along the c-axis (perpendicular to the (VO₂)⁻ planes) in the orthorombic phase is mainly related to the hopping between adjacent 3d_xz orbitals, with a hopping integral ${t}_{{xz},{xz}}\simeq 190$ meV. In the low-temperature monoclinic phase ($T\lt 140$ K), the most favorable hopping is given by the overlap between 3d_xz and 3d_yz orbitals with an hopping integral ${t}_{{xz},{yz}}\simeq 150$ meV. Assuming that the photo-exciton is created in one of the internal LVO layers, the time necessary to collect the charges (${\tau }_{{\rm{coll}}}$) is given at the most by two times the inverse hopping integral, i.e. ${\tau }_{{\rm{coll}}}$ = 6–8 fs. This value is very close to the ${\tau }_{D}$ estimated in section 3.1, thus suggesting that the quantum laws may greatly affect the transport properties of nanoscaled TMO devices, such as LVO/STO heterostructures.

Unraveling the impact of quantum coherent effects on the macroscopic electronic properties of heterostructures at ambient conditions is one of the challenges of solid state physics and materials science. The local many-body charge excitations spread throughout and across the metal–oxygen planes until they are collected by the interface conductive states. During this process, the quantum excitations lose their coherence via the coupling with the fluctuating environment constituted by the spin, charge and lattice background. In this frame, the final goal would be the development of strategies to tune the transport regime from coherent to incoherent and exploit quantum effects to enhance the transport or create quantum-protected states that could be exploited for quantum technology applications. Many of the fundamental ideas to treat this problem have been developed in the context of quantum transport in bio-systems at physiological temperatures [90]. In this section we will briefly review the concepts that have been developed in the recent years to describe the quantum effects in bio-systems and we will discuss how these ideas could be exploited to achieve the full quantum-coherent control of transport in heterostructures and nanoscaled materials.

In the simplest quantum-transport models, complex interacting systems are treated as quantum networks of two-level systems (sites), mutually coupled via hopping amplitude and interacting with both a thermal bath and a continuum of states (the collector). The incoherent hopping motion of the charge excitation corresponds to the case in which ${\tau }_{D}$ is shorter than the hopping time and only the population of the multi-levels, i.e., the diagonal terms of the density matrix, contributes to the transport. Quantum-coherent effects can emerge, for example, when ${\tau }_{D}$ is much larger than the typical hopping time and, as a consequence, the off-diagonal terms of the density matrix can give rise to novel effects mainly related to the interference of the wavefunctions describing each site. Considering the case of photo-stimulated excitons propagating in a quantum network coupled to the environment, many concepts have been introduced to describe the way local excitations are generated, transported and collected in photosynthetic light harvesting complexes [90]. The study of coherent effects in bio-systems at physiological temperatures and the role of the coupling with the environment was largely boosted by the discovery that quantum coherence is preserved in Fenna–Matthews–Olson (FMO) bacteriochlorophyll complexes [91] and in marine algae [92] on timescales of the order of hundreds femtoseconds. As a consequence, the first stage of the energy transfer within the light harvesting complexes can be dominated by wave-like phenomena, that possibly enhance the collection efficiency by allowing the simultaneous probing of large areas of phase space to find the most efficient path.

In quantum-coherent models, the transport efficiency is strongly affected by the interaction with different environments, such as the coupling with the collector and the thermal bath. In general, the problem is treated by suitable master equations [89] that describe the time evolution of both the diagonal (${\rho }_{{ii}}$) and off-diagonal (${\rho }_{{ij}}$, with $i\ne j$) terms of the density matrix, where i(j) labels the excited site. One of the simplest master equations which can be used to model the effects of different environments on excitonic transport is the Haken-Strobl master equation [93]:

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial {\rm{t}}}{\rho }_{{ij}}({\rm{t}})=-\displaystyle \frac{{\rm{i}}}{{\hslash }}{[{H}_{{\rm{eff}}}\rho -\rho {H}_{{\rm{eff}}}^{\dagger }]}_{{ij}}-(1-{\delta }_{{ij}})\displaystyle \frac{{\rho }_{{ij}}}{{\tau }_{D}},\end{eqnarray} \tag{ 3 }$$

where H_eff is an effective non-Hermitian Hamiltonian which describes the interaction with different dissipative environments, such as the collector where the excitation can be trapped. The effective non-Hermitian Hamiltonian can be written as H_eff = H₀–${\rm{i}}{\gamma }_{{\rm{ct}}}Q$. Here H₀ is the Hermitian Hamiltonian describing the closed system (which is the network of coupled two-level sistes) and also includes the static (time-independent) disorder induced by the environment; Q describes the dissipation in the continuum channel (collector or electromagnetic field) with strength ${\gamma }_{{\rm{ct}}}$. The imaginary part (${{\rm{\Gamma }}}_{r}$) of the rth complex eigenvalue of H_eff represents the decay width of the correspondent eigenstate. The quantum dephasing (the decay of the off-diagonal matrix elements) is accounted for by the second right term of equation (3) and represents the effect of the external bath which the network is coupled to. Physically, the dephasing is induced by time-dependent fluctuations of the site energies with amplitude proportional to 1/${\tau }_{D}$.

As the above simple model shows, quantum transport is affected by different parameters: the dephasing rate, the strength of static disorder and the strength of the coupling to the collector. Finding the optimal parameters is a difficult task since their interplay is highly non-trivial. Here we summarize the main concepts that emerged form the study of quantum transport in bio-molecules and that are expected to be of great relevance for the transport in heterostructures:

• Superradiance occurs when the energy levels of a quantum system are coherently coupled with a common decay channel. In short, it implies the existence of states with a cooperatively enhanced decay rate and it has been shown to have important effects on transport efficiency in open systems: for example, maximal efficiency in energy transport in the FMO system is achieved in the vicinity of the superradiance transition (ST). Superradiance, is usually reached only above a critical coupling strength with the continuum (in the overlapping resonance regime): ${\rm{\Gamma }}/{\rm{\Delta }}\geqslant 1$ where ${\rm{\Gamma }}=\langle {{\rm{\Gamma }}}_{r}\rangle$ is the average decay width and Δ is the mean level spacing of the closed system. Since the ST is due to the coherent constructive interference between the paths to the collector, one might expect that any consequence of ST would disappear in the presence of dephasing and relaxation provided by the thermal bath. On the contrary, strong opening-assisted quantum enhancement of transport for the FMO is seen also at room temperature where the quantum transfer is up to two times faster than that obtained by an incoherent calculation [94]. The existence of superradiant states, that are mainly localized in the vicinity of the collectors, is intrinsically accompanied by that of subradiant states which are long-lived and effectively decoupled from the environment. The existence of subradiant solutions opens the possibility of designing protocols to protect the quantum coherence even in systems coupled with a fluctuating environment [95].
• The segregation of specific solutions with a spatial pattern that can mostly match the collector, provides an emerging mechanism that enahnces the directionality of the transport process [90]. For example, it has been shown that FMO complexes may work as rectifiers for unidirectional energy flow from the peripheral light-harvesting antenna to the reaction center complex by taking advantage of quantum coherent effects that drive an efficient trapping by the pigments facing the reaction center complex [96].
• Energy mismatches in disordered materials lead to destructive interference of the wavefunction and, subsequently, to localization of the quantum wavefuntions [97], which is in general detrimental to the transport efficiency. Nevertheless, if the static disorder is sufficiently strong, quantum-coherent effects may significantly enhance transport in open systems even in the deep classical regime (i.e. where the decoherence rate is greater than the inter-site hopping amplitude) [94].
• The interaction with an external environment is not always detrimental to transport. Indeed, as suggested by natural photosynthetic bio-systems, quantum coherence may be enhanced or regenerated by the interaction with the environment [88, 98–100]. Furthermore, the dephasing induced by the environmental fluctuations can greatly enhance the transport efficiency for it counteracts the coherence-driven localization. The coupling to a common decay channel, such as the collector, has been also found to promote an open-assisted energy transfer in quantum networks and light-harvesting complexes [93, 94].

The extension of these concepts to solid state systems would pave the way to the observation and exploitation of quantum-coherent phenomena in materials for technological applications. An interesting example of coherent control has been provided by the theoretical investigation of the interaction of two consecutive coherent light pulses with a one-dimensional Mott insulator [101]. The formation of bound states of charge carriers, i.e., holon-doublon excitons, is included in the model by considering an extended Hubbard model in which, besides the on-site Coulomb repulsion U, nearest neighbor electron–electron interactions (V) are also included. The main result is that, when the decoherence time of optically active excitations is of the order of the pulse length, the many-body system can be mapped onto an effective two-level system, and its dynamics can be rationalized in terms of a simple Rabi model [101]. As a consequence, the final energy after the interaction with the second pulse can be either enhanced or quenched depending on the time delay between the first and second pulses, as shown in figure 6(b). This effect, shown in figure 6, is the counterpart of the quantum-coherent time evolution of a two-level system interacting with an electro-magnetic field, which is described by the optical Bloch equations [102]. This result suggests the possibility of coherently controlling the electronic state of a correlated material by suitable excitation with light pulses as short as ${\tau }_{D}$.

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TMOs provide the ideal platform to extend the concepts of superradiance, directionality, localization and noise-assisted transport that have been introduced for addressing the transport properties of natural light-harvesting systems [100]. The starting point is that the energy of the ubiquitous CT transition usually matches most of the visible spectrum (1–3 eV) and can be thus used to directly probe and control the fundamental electronic decoherence processes on the sub-10 fs timescale [69]. Furthermore, correlated materials are characterized by the unique property that the dynamics of the quasiparticles at the Fermi level and of the high-energy excitations involving the Hubbard band (the CT transition in copper oxides) are strongly intertwined [73]. As a consequence, the decoherence dynamics of holon-doublon excitons is expected to be strongly affected by the onset of ordered states (either local or long-ranged), such as magnetic or charge-ordered phases. Therefore, an interesting and completely unexplored mechanism that could help in tailoring the ultrafast electronic decoherence in solids is the possibility of tuning the fluctuations of the thermal bath in the vicinity of a phase transition. In this regime the relevant incoherent fluctuations (spin, phonons, charges) transform into a phase-coherent condensate, such as magnetic, charge- and orbitally ordered states or high-temperature superconducting condensates. Besides the example of LVO, in which the quantum orbital fluctuations in the high-temperature orthorombic phase condense into an orbitally ordered states at 140 K (see section 3.2), another paradigmatic example of the potentiality of TMOs is given by the Nd${}_{2-x}$Ce_xCuO₄ family of copper oxides. When tuning the electron doping concentration (x), it is possible to move away from the antiferromagnetic insulating phase and explore interesting states such as: (i) bad metallic phase with long-range antiferromagnetic correlations [103]; (ii) insulating or bad metallic states with strong short-range antiferromagnetic fluctuations [103]; (iii) short-range (coherence length $\simeq 2$ nm) and fluctuating charge order [104], in which the electronic charges spontaneously tend to break the translational symmetry and form phase-ordered patterns; (iv) unconventional superconductivity at high temperatures, in which the condensation of the charge carriers into a phase coherent condensate induces the opening of a $\simeq 40$ meV gap with d-wave symmetry.

Wrapping up the concepts discussed in this section, we argue that TMOs provide the unique possibility of exploiting the physics of correlated materials to tune the decoherence processes in atomically thin heterostructures and devices. The exploitation of noise-assisted quantum-coherent phenomena such as super-transfer [105] and superradiant transitions [93] to increase the performances and efficiency of TMO-based devices would dramatically impact the current technology. However, some fundamental steps are still required to transform these opportunities into a real technological breakthrough:

• Tuning the decoherence. Coherent transport phenomena emerge in a limited region of the phase-space of the parameters ${\tau }_{D}$ and ${\gamma }_{{\rm{ct}}}Q$. In order to control and tune quantum-coherent processes, it is necessary to develop protocols to span the entire phase space and continuously move from the incoherent to the coherent regime. The most promising strategy is to tune ${\tau }_{D}$ by exploiting the modification of the electronic coupling with the bosonic bath in the vicinity of a phase transition. In this perspective, TMOs provide a very interesting playground since they are always on the verge of multiple phase transitions as a consequence of the local correlations of the metal d electrons. By finely tuning the temperature and doping, it is possible to turn strongly incoherent fluctuations of the charge, spin, lattice and orbital degrees of freedom into long-range ordered (and phase-coherent) states, such as magnetic phases, charge/orbital ordering and superconductivity, at temperatures ranging from 20 to 300 K.
• Designing effective collectors. One of the crucial parameters for the emergence of superradiant solutions in quantum networks is the coupling to the collectors, ${\gamma }_{{\rm{ct}}}Q$. The state-of-the art technologies for the synthesis of oxide heterostructures [19] enabled the layer-by-layer control of the chemical/physical properties of nanoscaled devices. For example, the spontaneous emergence of conductive layers at the interface of insulating oxides, such as LaAlO₃/SrTi₃ and LaVO₃/SrTi₃, provide a simple scheme for the design of 2D collectors that reside few atomic layers apart [106] and are characterized by high carrier mobility. Furthermore, the density of the interface conductive layers can be changed by controlling their mutual distance and their common temperature [106, 107], thus providing additional degrees of freedom for tuning the coupling with the collectors.
• Tuning the hopping. The hopping amplitude, ${t}_{{\rm{h}}}$, has a twofold role in controlling the quantum-transport properties of few atomic-layers devices. On one hand, ${t}_{{\rm{h}}}$ determines the fundamental timescale (${\hslash }/{t}_{{\rm{h}}}$) of the charge-hopping process between two adjacent layers; on the other hand, it controls the mean level spacing (Δ) of the energy levels of the closed quantum network with respect to the energy of the non-interacting sites. Interestingly, ${t}_{{\rm{h}}}$ strongly depends on the lattice structure of the material and on the orbital occupation of the TM d-bands [86], which can be controlled either statically, e.g. by tuning the temperature or the chemical composition, or dynamically, e.g. by resonant photoexcitation of interband transition or lattice modes [108].
• Disorder. The local disorder is expected to strongly impact the transport efficiency in the coherent regime, up to the point of driving the complete charge localization [97]. The disorder can be easily controlled by exploiting the intrinsic presence of defects in TMOs or by creating suitably designed disordered structures that could be controlled at the atomic scale.
• Developing predictive models. A key step is the development of predictive models for the description of coherent photodissociation and charge-collection processes in correlated oxides. The spreading of photo-excited carriers in Mott insulating heterostructures has been recently studied by means of DMFT solutions of the inhomogeneous Hubbard model [87]. While DMFT fully treats the local correlations of the TM d-orbitals and the quantum nature of the many-body wavefunctions, only short-range spin fluctuations can be included as source of dissipative coupling with the environment. The coupling with phonons (restricted with specific dispersionless optical modes) could be included by means of the Hubbard–Holstein model. Considering the relevance of the coupling to the thermal bath and given the necessity of simulating in a simple way the spectral modifications of the noise, e.g. the transformation from an incoherent fluctuating bath to a phase-locked coherent state in the vicinity of charge and orbitally ordering phase transitions, the development of effective models based on appropriate master equations that can describe both coherent and incoherent hopping [89] is expected to boost the research in this field. The modeling of quantum-coherent processes in oxide heterostructures will guide the optimization of the experimental conditions necessary to observe and study coherent phenomena and will lead to the exploitation of these effects to improve the efficiency of transport in nanoengineered devices.

The development of strategies to exploit and control quantum coherent effects at the nanoscale crucially depends on the knowledge of the fundamental processes that regulate the dephasing of the electronic wavefuctions. Unfortunately, any attempt to investigate the electronic dephasing in solids at high-temperatures has been hindered so far by the difficulties in directly measuring the fundamental dephasing time of photoexcitations. For example, the linewidths measured by conventional optical spectroscopy are strongly affected by inhomogeneous broadening, which reflects the statistical distribution of impurities. This effect is particularly severe in TMOs, in which the strong correlations lead to intrinsic electronic and structural inhomogeneities. On the other hand, time-domain optical spectroscopies with ultrafast resolution emerged as a well-established tool to investigate the dynamics of correlated oxides [1]. Unfortunately, state-of-the-art ultrafast spectroscopies simply probe the population of a specific transition and remain blind to the fundamental decoherence processes. As a consequence, the relevant mechanisms that lead to the ultrafast loss of quantum coherence of charge excitons in metal oxides and the interplay between their dephasing and the onset of long-range ordered phases are still a major unexplored field.

In order to access the fundamental decoherence (or dephasing) processes described in the previous sections, it is mandatory to develop a technique with the following features:

• a time resolution as short as the typical decoherence timescale (few femtoseconds)
• capability of disentangling the real polarization dephasing from the population decay and from inhomogeneous effects.

One of the most promising approaches that could fulfill these requirements is the multi-dimensional spectroscopy [63, 109], which was developed over the past two decades in physical chemistry and atomic physics and is the optical analog of NMR used to study the decoherence of atomic wavefunctions and the energy transfer between vibrating molecules.

In particular, 2D spectroscopy directly accesses ${\rho }_{0e}$, being based on the coherent interaction of three light pulses mediated by the third-order susceptibility, ${\chi }^{(3)}$, of the material [63, 109]. In the simplest picture (see figure 7(a)), the first pulse creates a macroscopic polarization, leaving the system in the coherent superposition of $| {\psi }_{0}\rangle$ and $| {\psi }_{e}\rangle$. The polarization oscillates in time with frequency ${\omega }_{0e}$ and decays on the dephasing timescale ${\tau }_{D}$. The second pulse, at time t₁, converts the macroscopic polarization into a population in the excited state, which does not oscillate but is reminiscent of the initial phase of the polarization. The third pulse, at time t₂ converts back to a coherent polarized state that radiates the measured signal field at time t₃. The 2D spectrum (see figure 7(b)) is generated, for each fixed t₂, by scanning t₁ and t₃ and taking the Fourier transform. 2D spectroscopy is considered the ultimate nonlinear technique [110], since it contains all the information that can be retrieved by 0D (conventional optics) and 1D spectroscopies, such as single- and multi-color pump–probe and multi-pulse techniques without the probe-frequency resolution [111–113]. On the other hand, 2D spectroscopy also contains fundamental information that is absent in 0D and 1D, such as:

• the possibility of determining whether two resonances are coupled or uncoupled [109]
• the capability of discriminating between homogeneous and inhomogeneous linewidths [109]
• the possibility of reconstructing the exciton migration and the charge-separation process [114].

This technique already showed that quantum phase-coherence is preserved in biological systems immersed in room-temperature baths [90–92] and unveiled many-body excitonic effects in semiconductors at low temperature (10 K) [115]. The possibility of extending 2D spectroscopy in the x-rays and in the THz frequency domain is expected to open new routes for the coherent spectroscopy of spins, valence electrons, and and core electronic excitations.

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So far, the use of 2D spectroscopy for the study of solid state systems has been mainly hindered by the limited temporal resolution of most setups ($\approx 100$ fs) and by the low repetition rate (1 kHz) that forces the experiments to be performed at very high fluences and on semitransparent samples. However, the recent dramatic advances in fiber laser technology [71, 116] made available high-repetition rate sources (up to 2 MHz) with enough energy per pulse for producing and amplifying supercontinuum white-light. The use of suitably designed non-collinear optical parametric amplifiers allows to produce ultrashort broadband pulses with tunable frequency and repetition rate and whose frequency content can support pulses as short as 6 fs [71, 116]. To show the advantages of 2D spectroscopy with respect to broadband pump–probe techniques [1], we simulated the absorptive [109] 2D spectra (see figure 7(b)) expected for the typical CT excitations in correlated oxides, taking into account the expected temporal resolution of the experiment (6 fs). Assuming a total linewidth of 400 meV (see figure 4 for the actual optical conductivity of specific materials), the inhomogeneously (bottom panel of figure 7(b)) and homogeneously (top panel of figure 7(b)) broadened cases are clearly disentangled in the 2D spectra. In contrast, any 1D (or 0D) optical spectroscopy would only access the integral with respect to ${\omega }_{1}$ of the 2D spectrum and would not be able to discriminate between the two cases.

In light of the concepts presented in this section, we foresee that the achievement of a temporal resolution of $\approx \,6$ fs, which is of the order of the decoherence timescale in correlated oxides, in high-repetition rate 2D experiments would unlock the study and, possibly, the control of the quantum-coherent dynamics in oxide heterostructures. The possibility of exploiting quantum-coherent processes at ambient temperatures would stimulate the design of oxide heterostructures and nano-materials that preserve the quantum-coherence of the photo-stimulation and collect the photo-generated charges along quantum walks. The success in coherently converting photons into charge excitations in oxide-base devices would provide the building block to develop robust quantum systems that can be manipulated at ambient conditions, for future use in large-scale quantum-information processes and communication technologies [59]. These ideas and techniques could be naturally applied to a broad variety of functional oxides (e.g. manganites, vanadates, nickelates, iridates and iron oxides) that exhibit optically driven insulator-to-metal transitions, multiferroicity, magnetic superconductivity and to low dimensional solids (e.g. graphene and metal dichalcogenides).

In general, the balance of the energy exchanges is rationalized through a local conservation of energy equation:

$$\begin{eqnarray}&&C\displaystyle \frac{\partial T}{\partial t}({\bf{x}},t)=-{\rm{\nabla }}\cdot {\bf{q}}({\bf{x}},t)+S({\bf{x}},t),\end{eqnarray} \tag{ 4 }$$

where C is the volumetric heat capacity and $S({\bf{x}},t)$ is the volumetric heat source [119] externally provided. The classical Fourier's law is obtained by assuming a direct proportionality, at time t, between the heat flux and the temperature gradient, i.e., ${\bf{q}}({\bf{x}},t)=-{\kappa }_{T}\ {\rm{\nabla }}T({\bf{x}},t)$, ${\kappa }_{T}$ being the thermal conductivity. While Fourier's law is a good approximation for describing slow phenomena, it implicitly assumes that the onset of the heat flux and of the temperature gradient are simultaneous, which implies an infinite speed of heat propagation [119]. When considering fast timescales and small length-scales, two different scenarios can be analyzed:

• (i)  
  After the instantaneous establishment of the temperature gradient, a non-zero time, ${\tau }_{q}$, is necessary to generate the heat flux. This delay is related to the microscopic interactions that lead to a very fast local thermalization, while the heat flux is delayed by specific dynamical constraints. A typical example is given by strongly anisotropic materials, such as iridium or copper oxides, in which the in-plane thermalization is driven very quickly by the coupling with spin fluctuations and phonons, while heat transport perpendicular to the TMO planes is strongly quenched by the small interlayer coupling. In this case, the constitutive equation relating the heat flux and the thermal gradient is modified into the Cattaneo–Vernotte model [119, 120]: ${\bf{q}}({\bf{x}},t+{\tau }_{q})=-{\kappa }_{T}\ {\rm{\nabla }}T({\bf{x}},t)$.
• (ii)  
  In the opposite limit, a heat flux is instantaneously established causing a temperature gradient at the delayed time ${\tau }_{T}$. This condition, represented by the relation ${\bf{q}}({\bf{x}},t)=-{\kappa }_{T}\ {\rm{\nabla }}T({\bf{x}},t+{\tau }_{T})$, is attained, for example, when a fast ballistic propagation of the charge carriers anticipates the local thermalization of the system.

By combining the two previous limits into a single relation, it is possible to introduce the dual-phase-lag  model [119], that describes the general constitutive relation between the heat flux and the temperature gradient:

$$\begin{eqnarray}&&{\bf{q}}({\bf{x}},t+{\tau }_{q})=-{\kappa }_{T}\ {\rm{\nabla }}T({\bf{x}},t+{\tau }_{T}).\end{eqnarray} \tag{ 5 }$$

By substituting the first order term of the Taylor series in the time variable of equation (5) into the energy conservation law at time t (equation (4)) and assuming $S({\bf{x}},t)$ = 0, it is possible to obtain the generalized partial differential equation for the temperature field:

$$\begin{eqnarray}&&\left(\displaystyle \frac{{\tau }_{q}}{\alpha }\right)\displaystyle \frac{{\partial }^{2}T({\bf{x}},t)}{\partial {t}^{2}}-{{\rm{\nabla }}}^{2}T({\bf{x}},t)+\displaystyle \frac{1}{\alpha }\displaystyle \frac{\partial T({\bf{x}},t)}{\partial t}\\ &&\quad -{\tau }_{T}\displaystyle \frac{\partial {{\rm{\nabla }}}^{2}T({\bf{x}},t)}{\partial t}=0,\end{eqnarray} \tag{ 6 }$$

where $\alpha ={\kappa }_{T}/C$ is the thermal diffusivity. Interestingly, equation (6) recalls a wave equation in which $\sqrt{\alpha /{\tau }_{q}}$ is the speed of propagation and $\sqrt{\alpha {\tau }_{q}}$ represents the diffusion length. The damping of the temperature wave is provided by a diffusive term and a third-derivative one. Equation (6) reduces to the conventional diffusion equation in the limit ${\tau }_{q}$, ${\tau }_{T}\to$0, the latter implying an infinite speed of heat propagation. From equation (6) it is possible to extract the complex dispersion relation of the system, linking the wavevector k = $| {\bf{k}}|$ to the frequencies (ω) of the propagating temperature waves:

$$\begin{eqnarray}&&{k}^{2}(1+{\rm{i}}\omega {\tau }_{T})=\left(\displaystyle \frac{{\tau }_{q}}{\alpha }\right){\omega }^{2}\left(1-\displaystyle \frac{{\rm{i}}}{\omega {\tau }_{q}}\right).\end{eqnarray} \tag{ 7 }$$

Oscillating solutions of equation (6) can be sought under the assumption that k and ω entering in equation (7) are complex quantities. Having in mind the experimental configuration of ultrafast experiments, in which the laser impinges on one of the sides of the sample and acts as an initial volumetric heat source $S({\bf{x}},{\rm{t}}$ = 0), we are interested in studying how the given initial temperature profile evolves in time. As a consequence, we assume a real wavevector, while the frequency is given by the complex quantity $\tilde{\omega }={\omega }_{1}+{\rm{i}}{\omega }_{2}$, where $| {\omega }_{1}|$ = 2π/t_osc is the inverse of the oscillation period and ${\omega }_{2}$ = 1/t_damp is the inverse of the damping time. The onset of a wave-like regime is governed by the Q-factor Q = $| {\omega }_{1}|$/${\omega }_{2}$, which discriminates the overdamped ($Q\ll 1$) from the underdamped regimes ($Q\gg 1$). For simplicity, we will consider the one-dimensional problem in which the spatial coordinate, z, is perpendicular to the sample surface.

By substituting the complex frequency $\tilde{\omega }$ into the dispersion relation given by equation (7), it is possible to obtain the analytic expressions of ${\omega }_{1}$ and ${\omega }_{2}$ as a function of k. Without entering into the details of the calculations, that will be presented elsewhere [121], it is possible to show that the nature of the admitted solutions strongly depends on the ratio ${\tau }_{T}$/${\tau }_{q}$. While non-oscillatory solutions, characterized by ${\omega }_{1}=$ 0, are found for ${\tau }_{T}\geqslant {\tau }_{q}$, i.e. when the heat flux precedes the establishment of a thermal gradient, a wave-like behavior of the temperature propagation may emerge in the case ${\tau }_{T}\lt {\tau }_{q}$, i.e., when the temperature gradient is established before the onset of the heat flux. We can define a lower (k_low) and upper (k_up) bound for the wavevectors that can sustain oscillatory solutions:

$$\begin{eqnarray}&&{k}_{{\rm{low}}({\rm{up}})}=\sqrt{\displaystyle \frac{2}{\alpha {\tau }_{T}}\left(\displaystyle \frac{{\tau }_{q}}{{\tau }_{T}}\right)\left(1-\displaystyle \frac{1}{2}\displaystyle \frac{{\tau }_{T}}{{\tau }_{q}}-(+)\sqrt{1-\displaystyle \frac{{\tau }_{T}}{{\tau }_{q}}}\right)}.\end{eqnarray} \tag{ 8 }$$

In the range ${k}_{{\rm{low}}}\lt k\lt {k}_{{\rm{up}}}$, the allowed complex frequencies are:

$$\begin{eqnarray}{\omega }_{1} & = & \mp \sqrt{-\left[\displaystyle \frac{{\alpha }^{2}}{4}{\left(\displaystyle \frac{{\tau }_{T}}{{\tau }_{q}}\right)}^{2}{k}^{4}+\displaystyle \frac{\alpha }{{\tau }_{q}}\left[\displaystyle \frac{1}{2}\left(\displaystyle \frac{{\tau }_{T}}{{\tau }_{q}}\right)-1\right]{k}^{2}+\displaystyle \frac{1}{4{\tau }_{q}^{2}}\right]};\\ {\omega }_{2} & = & \displaystyle \frac{1}{2{\tau }_{q}}+\displaystyle \frac{\alpha }{2}\left[\displaystyle \frac{{\tau }_{T}}{{\tau }_{q}}\right]{k}^{2}\end{eqnarray} \tag{ 9 }$$

while, for $k\leqslant {k}_{{\rm{low}}}$ and $k\geqslant {k}_{{\rm{up}}}$ no oscillatory solutions are admitted, i.e. ${\omega }_{1}=0$.

The oscillating behavior of the temperature field, is better appreciated by inspecting the Q-factor:

$$\begin{eqnarray}Q(k)=\left\{\begin{array}{ll}\sqrt{\displaystyle \frac{4\alpha {\tau }_{q}{k}^{2}}{{(1+\alpha {\tau }_{T}{k}^{2})}^{2}}-1\ } & \ \ \ \ {\rm{for}}\ {k}_{{\rm{low}}}\lt k\lt {k}_{{\rm{up}}}\\ 0 & \ \ \ \ {\rm{for}}\ k\leqslant {k}_{{\rm{low}}}\ \vee k\geqslant {k}_{{\rm{up}}}.\end{array}\right.\end{eqnarray} \tag{ 10 }$$

The most favorable condition to observe underdamped oscillations corresponds to the maximum Q-factor:

$$\begin{eqnarray}&&{Q}_{{\rm{\max }}}=\sqrt{\displaystyle \frac{{\tau }_{q}}{{\tau }_{T}}-1},\end{eqnarray} \tag{ 11 }$$

that is obtained at the wavevector ${k}_{{Q}_{{\rm{\max }}}}=1/\sqrt{\alpha {\tau }_{T}}$. Equation (11) suggests that the condition ${\tau }_{q}\gg {\tau }_{T}$ is the prerequisite to observe wave-like effects of the temperature propagation in real materials.

As shown in section 4.2, emergent wave-like regimes of the temperature field may potentially be found in systems in which the condition ${\tau }_{q}\gg {\tau }_{T}$ holds, i.e., a thermal gradient is very quickly established, while the heat flux follows at a later time ${\tau }_{q}$.

Correlated TMOs provide a very interesting playground to investigate this physics. Their layered structure is characterized by strongly anisotropic conduction properties that can lead to a very fast in-plane thermalization and to a very slow out-of-plane heat propagation, thus favoring the emergence of the wave-like regime, as schematically shown in figure 8(a).

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Here, we will focus on copper oxides and, in particular, on the Bi₂Sr₂CaCu₂O₈ (Bi2212) family that exhibits the most anisotropic transport properties. In general, the conduction properties of copper oxides are dominated by the charge carriers that occupy the Cu–3d orbitals within the Cu–O planes (ab-planes, see figure 8(b)). The strong tendency to delocalization, driven by the very large in-plane hopping integral (${t}_{\parallel }\sim 0.3$ eV) is contrasted by the strong effective interactions. At moderate doping concentration (∼15%) the electronic interactions give rise to a strong coupling with the bosonic fluctuations of both magnetic and phononic origin [112]. The fastest scattering process is likely to be related to the coupling with the short-ranged antiferromagnetic fluctuations that persist when the system is doped [122]. It has been recently shown [74] that the excess kinetic energy provided by the pump pulse is completely released to the magnetic background within a timescale of ∼50 fs. It is thus reasonable to assume ${\tau }_{T}$ = 50 fs as the time necessary for the local thermalization between charge carriers and spin fluctuations and for the onset of the temperature gradient after an anisotropic pump excitation. The electron–phonon coupling eventually drives the complete thermalization between the electronic, magnetic and phononic degrees of freedom on a timescale of ∼1 ps. We underline that, even in bulk materials, the pump excitation is intrinsically anisotropic, being the energy absorption confined within a penetration length of ∼20–200 nm depending on the material. As we will discuss later, 'artificial' gradients can be induced by nanostructuring the material under investigation.

Notably, the transport properties of copper oxides are strongly anisotropic, being the interlayer hopping limited to ${t}_{\perp }\approx 15$ meV [127]. As a consequence, the strong in-plane scattering makes the c-axis conductivity almost completely incoherent. In Bi2212, the c-axis optical conductivity can be as small as 0.01 ${{\rm{\Omega }}}^{-1}$ cm⁻¹, in contrast to an in-plane conductivity of the order of 5000 ${{\rm{\Omega }}}^{-1}$ cm⁻¹ [128]. In a similar way, the electronic thermal conductivity is reduced from ∼1 Wm⁻¹K⁻¹ in the ab-plane to ∼3 × 10⁻⁴ Wm⁻¹K⁻¹ when considering the interlayer thermal transport. The natural consequence of this strong anisotropy is that any possible heat exchange perpendicular to the Cu–O planes is strongly delayed as compared to the in-plane thermalization that is effective on the sub-100 fs timescale.

This observation naturally suggests a possible building block for designing materials in which the wave-like propagation of the temperature field can be observed and exploited. Given the difficulty of estimating ${\tau }_{q}$ from both first-principles and experiments, we will consider the value of ∼1 ps as a lower bound. This value, which represents a conservative estimate, corresponds to the reasonable assumption ${\tau }_{q}$/${\tau }_{T}$ = t_∥/t_⊥.

Considering the electronic problem related to the heat propagation along the c-axis in Bi2212, we obtain α ∼10⁻⁸–6.5 × 10⁻⁷ m² s⁻¹ in the 300–20 K temperature range, as reported in table 2, while ${\tau }_{q}$/${\tau }_{T}\,\gt$ 20. From equation (8) and assuming ${\tau }_{q}$ = 1 ps, it is possible to calculate the upper (${k}_{{\rm{up}}}\,\sim \,$400–50 nm⁻¹ at T = 300–20 K) and lower (${k}_{{\rm{low}}}\,\sim \,$5–0.63 nm⁻¹ at T = 300–20 K) bounds of the wavevectors compatible with temperature oscillations. While 2π/k_up corresponds to a fraction of the interatomic distance and cannot be experimentally achieved, ${\lambda }_{T}$ = 2π/k_low = 1–10 nm provides the maximum wavelength that can sustain an oscillatory behavior. These values are compatible with the geometric constraints that can be applied by nanostructuring copper oxides and other TMOs. In figure 8(c) we report the dispersion relations of the oscillating solution at T = 20 K, as calculated by equation (9). At the wavevector $k\sim 5$ nm⁻¹, corresponding to ${\lambda }_{T}\sim 1.2\,\mathrm{nm}$, the oscillation frequency is ${\omega }_{1}\sim 4\,\mathrm{rad}$ ps⁻¹ and the Q-factor is compatible with the observation of oscillations with period of $\approx 1.6$ ps.

The physics discussed in the current section suggests the possibility of engineering realistic nanostructured systems in which, on the picosecond timescale, the temperature propagation is described by a wave equation instead of the typical diffusion law.

Obviously, this regime can be achieved and observed only on fast timescales [118], i.e. before the damping washes out coherent effects and restores the "classical" behavior. The possibility of controlling layer-by-layer the growth of TMOs [19] opens intriguing perspectives: by creating heterostructures with typical dimensions of few nanometers, it is indeed possible to exploit the novel wave-like regime of the temperature field. In these systems, the local temperature increase could be manipulated at the nanoscale by exploiting interference effects, thus opening an entirely new field in nano-thermodynamics. Furthermore, the possibility of creating superlattices with periodicity of the order of few atomic cells could lead to the design of artificial bandstructures for the temperature waves, thus opening to the development of waveguides, cavities and frequency filters for the temperature field.

Far from pretending of being exhaustive, the above discussion aims at demonstrating that, for realistic materials, it is possible to investigate an emergent wave-like regime for the temperature propagation on ultrafast timescales. The numbers reported in table 2 should be considered as an example of the physics that could be investigated at the nanoscale. The concepts here proposed can be easily extended to many different cases. Although in the previous discussion we focused on the electronic temperature of the specific copper oxide Bi₂Sr₂CaCu₂O₈, similar effects are expected also when considering the lattice temperature. For example, in Bi2212 the largest value of α for the phonon case (6.3 × 10⁻⁶ m² s⁻¹ at 20 K) leads to the possibility of observing wave-like regimes at smaller wavevectors, corresponding to length scales of the order of tens of nanometers. For sake of completeness we report the relevant data for the phonon case in table

Table 1.  Electronic specific heat (C), thermal conductivity (k_T) and thermal diffusivity (α) of copper and iridium oxides. C and k_T of nearly optimally doped (T_c = 85 K) Bi2212 crystals are taken from [123, 124, 125]. These values are representative of the specific heats and thermal conductivities of other families of copper oxides, such as YBa₂Cu₃O₇ and La${}_{2-x}$Sr_xCuO₄. C and k_T of Sr₂IrO₄ crystals are taken from [126]. These values are representative of the specific heats and thermal conductivities of other families of iridates, such as Na₂IrO₃.

								Diffusion
	T	C	kT	α	${\tau }_{T}$	${\tau }_{q}$	Velocity	length
	K	J m−3 K−1	Wm−1K−1	m2 s−1	ps	ps	nm ps−1	nm
Bi2Sr2CaCu2O8
(Bi2212)
Electrons	300	3.3 × 104	3.0 × 10−4	1.0 × 10−8	0.05	>1	<0.10	>0.10
	50	4.0 ×103	2.5 × 10−4	6.5 × 10−8	0.05	>1	<0.25	>0.25
	20	4.0 × 102	2.5 × 10−4	6.5 × 10−7	0.05	>1	<0.80	>0.80
Sr2IrO4
(SIO)
Electrons/	300	1.5 × 103	1.5×10−7	1 × 10−10	0.1	≈1000	≈3 × 10−4	≈0.3
magnetic	50	2.5 × 102	2.5×10−8	1 × 10−10	0.1	≈1000	≈3 × 10−4	≈0.3
fluctuations	20	1.0 × 102	1.0×10−8	1 × 10−10	0.1	≈1000	≈3 × 10−4	≈0.3

Table 2.  Lattice specific heat (C), thermal conductivity (k_T) and thermal diffusivity (α) of copper oxides. C and k_T of nearly optimally doped (T_c = 85 K) Bi2212 crystals are taken from [123, 124, 125]. These values are representative of the specific heats and thermal conductivities of other families of copper oxides, such as YBa₂Cu₃O₇ and La${}_{2-x}$Sr_xCuO₄.

								Diffusion
	T	C	kT	α	${\tau }_{T}$	${\tau }_{q}$	Velocity	length
	K	J m−3 K−1	Wm−1K−1	m2 s−1	ps	ps	nm ps−1	nm
Bi2Sr2CaCu2O8
(Bi2212)
Phonons	300	2.4 × 106	0.9	3.8 × 10−7	1	>10	<0.2	>2
	50	5.0 × 105	0.5	1.0 × 10−6	1	>10	<0.3	>3
	20	8.0 × 104	0.5	6.3 × 10−6	1	>10	<0.8	>8

Similar concepts can also be extended to the spin temperature in systems characterized by short-range magnetic correlations. As an example, let us consider iridium oxides, which are characterized by a strong anisotropy of the magnetic dynamics. As discussed in section 2, the ratio between the in-plane and out-of-plane magnetic exchange can be as high as J_∥/${J}_{\perp }\,\sim \,$6 × 10⁴. Under non-equilibrium conditions, the strong anisotropy leads to a very fast in-plane thermalization of the magnetic fuctuations (${\tau }_{T}\,\sim \,$100 fs) while the heat flux perpendicular to the Ir planes is strongly delayed by the small interlayer magnetic coupling. As a conservative estimation, we assume ${\tau }_{q}$ = 10${}^{4}{\tau }_{T}$ and we use the electronic heat capacity and thermal conductivity of Sr₂IrO₄ (SIO), reported in table 1, as the upper bound of the magnetic contribution. The large ${\tau }_{q}$/${\tau }_{T}$ turns out to be favorable to observe undulatory effects of the effective temperature of magnetic fluctuations in iridium oxides. At wavevectors compatible with the dimensions of the SIO unit cell along the c-axis ($\simeq 25$ Å), the oscillation frequency is about ${\omega }_{1}\,\simeq$ 5 $\,\times \,{10}^{-4}$ rad ps⁻¹, with $Q\,\gt$ 1. Many strategies could be devised in order to tune the wavelength and the period of the coherent oscillation of the magnetic temperature. For example, larger values of the α parameter could be obtained by increasing the thermal conductivity via chemical doping of the Mott insulating phase. Furthermore, the use of strongly frustrated magnetic systems in proximity of a 3D magnetic transition could provide an additional knob to tune the magnetic heat capacity. A paradigmatic system is Na₂IrO₃ that exhibits a zig-zag phase transition at T_N = 15 K. In the 70–15 K temperature range, the heat capacity is dominated by the contribution of the short-range 2D magnetic fluctuations, giving rise to ${C}_{{\rm{mag}}}$ ≃ 4 × 10⁴ J m⁻³ K⁻¹. The dramatic changes of the magnetic heat capacity in the vicinity of T_N could be exploited to manipulate the wave-like nature of the magnetic temperature propagation in heterostructures or nanoscaled devices.