Witnessing Quantum Entanglement Using Resonant Inelastic X-ray Scattering
Brookhaven National Laboratory（布鲁克海文国家实验室）；Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；Harvard University（哈佛大学）；Advanced Photon Source, Argonne National Laboratory（阿贡国家实验室先进光子源）；University of Cologne（科隆大学）
Tianhao Ren, Yao Shen, Marton Lajer, Sophia F. R. TenHuisen, Jennifer Sears, Wei He, Mary H. Upton, Diego Casa, Petra Becker, Matteo Mitrano, Mark P. M. Dean, Robert M. Konik
Although entanglement is both a central ingredient in our understanding of quantum many-body systems and an essential resource for quantum technologies, we only have a limited ability to quantify entanglement in real quantum materials. Thus far, entanglement metrology in quantum materials has been limited to measurements involving Hermitian operators, such as the detection of spin entanglement using inelastic neutron scattering. Here, we devise a method to extract the quantum Fisher information (QFI) from non-Hermitian operators and formulate an entanglement witness for resonant inelastic x-ray scattering (RIXS). Our approach is then applied to the model iridate dimer system Ba$_3$CeIr$_2$O$_9$ and used to directly test for entanglement of the electronic orbitals between neighboring Ir sites. We find the entanglement can be detected if we account for the expected symmetries, parity, and electron number conservation, of the dimer system. We also consider the roles that the incident and outgoing x-ray polarizations and the incident photon energy play in entanglement detection. Our protocol provides a new handle for entanglement detection in quantum materials.
We demonstrate how quantum entanglement can be directly witnessed in the quasi-1D Heisenberg antiferromagnet KCuF$_3$. We apply three entanglement witnesses --- one-tangle, two-tangle, and quantum Fisher information --- to its inelastic neutron spectrum, and compare with spectra simulated by finite-temperature density matrix renormalization group (DMRG) and classical Monte Carlo methods. We find that each witness provides direct access to entanglement. Of these, quantum Fisher information is the most robust experimentally, and indicates the presence of at least bipartite entanglement up to at …
We develop a formalism to study indirect resonant inelastic x-ray scattering (RIXS) in systems with itinerant electrons, accounting for the attraction between valence electrons and the positively-charged core hole exactly, and apply this formalism to the hole-doped cuprate superconductors. We focus on the relationship between RIXS lineshapes and band structure, including broken symmetries. We show that RIXS is capable of distinguishing between competing order parameters, establishing it as a useful probe of the pseudogap phase.
We report a Cu K-edge resonant inelastic x-ray scattering (RIXS) study of momentum-dependent charge excitations in Nd$_{2-x}$Ce$_x$CuO$_4$ ($x$ = 0.075 and 0.18). The peak position and width of the excitations coincide excellently with those observed in Cu L$_3$-edge RIXS. It demonstrates that the same charge excitations are observed at the two edges.
Ru M$_3$-edge resonant inelastic x-ray scattering (RIXS) measurements of RuCl$_3$ with 27 meV resolution reveals a spin-orbit exciton without noticeable splitting. We extract values for the spin-orbit coupling constant ($λ=154\pm2$ meV) and trigonal distortion field energy ($\left|Δ\right|<65$ meV) which support the $j_{\rm eff}=1/2$ nature of RuCl$_3$. We demonstrate the feasibility of M-edge RIXS for $4d$ systems, which allows ultra high-resolution RIXS of $4d$ systems until instrumentation for L-edge RIXS improves.
The scattering cross section of the resonant inelastic light scattering is represented as a correlation function in the Keldysh-Schwinger functional integral formalism. The functional integral approach enables us to compute the cross section in the Feynman diagram perturbation theory where many-body effects can be fully incorporated. This approach is applied to the one G-phonon Raman scattering of graphene, and the result is shown to agree with the one previously obtained by the conventional Fermi golden rule formula. Also, this approach is generalized to the systems in non-equilibrium conditi…
Phonons, the quantum mechanical representation of lattice vibrations, and their coupling to the electronic degrees of freedom are important for understanding thermal and electric properties of materials. For the first time, phonons have been measured using resonant inelastic x-ray scattering (RIXS) across the Cu K-edge in cupric oxide (CuO). Analyzing these spectra using an ultra-short core-hole lifetime approximation and exact diagonalization techniques, we can explain the essential inelastic features. The relative spectral intensities are related to the electron-phonon coupling strengths.
I present a tractable theory for the Resonant Inelastic X-ray Scattering (RIXS) spectral function of magnons. The low-energy transition operator is written as a product of local spin operators times fundamental x-ray absorption spectra. This leads to simple selection rules for the magnetic cross section. The scattering cross section linear (quadratic) in spin operators is proportional to the magnetic circular (linear) dichroic absorption. RIXS is a novel tool to measure magnetic quasi particles (magnons) and the incoherent spectral weight, as well as multiple magnons up to very high energy los…
Orbital degrees of freedom play a major role in the physics of many strongly correlated transition metal compounds. However, they are still very difficult to access experimentally, in particular by neutron scattering. We propose here how to reveal orbital occupancies of the system ground state by magnetic resonant inelastic x-ray scattering (RIXS). This is possible because, unlike in neutron scattering, the intensity of the magnetic excitations in RIXS depends essentially on the symmetry of the orbitals where the spins are in.
We develop a framework to evaluate the time-dependent resonant inelastic X-ray scattering (RIXS) signal with the use of non-equilibrium dynamical mean field theory simulations. The approach is based on the solution of a time-dependent impurity model which explicitly incorporates the probe pulse. It avoids the need to compute four-point correlation functions, and can in principle be combined with different impurity solvers. This opens a path to study time-resolved RIXS processes in multi-orbital systems. The approach is exemplified with a study of the RIXS signal of a melting Mott antiferromagn…
Effective symmetry-based transition operators for resonant inelastic X-ray scattering (RIXS) are derived that show how the scattering between different states depends on the polarization of the incoming and outgoing X-rays. In spherical symmetry, the effective operators can be rewritten in terms of spin operators, although the expressions depend on the nature of the ground state. For lower symmetries, the combined action of the crystal field and the spin-orbit interaction breaks up the spin space and spin operators are no longer appropiate operators. By taking iridium compounds as an example, …
The recent discovery that resonant inelastic x-ray scattering can probe single-magnon (SM) dispersions in transition metal (TM) oxides when the x-ray energy is tuned to the TM $L$-edge, has put this technique on a par with inelastic neutron scattering. It is generally presumed that selection rules forbid SM-scattering at oxygen (O) $K$-edges. However, based on a symmetry analysis and exact diagonalization study, we show that SM-scattering at O K-edges becomes allowed when (i) spin-orbit coupling is present in the TM d-shell and (ii) inversion symmetry at the O-site is broken. For cuprates the …
We report strongly momentum-dependent local charge screening dynamics in CE-type charge, orbital, and spin ordered La$_{0.5}$Sr$_{1.5}$MnO$_4$, based on Mn K-edge resonant inelastic x-ray scattering data. Through a comparison with theoretical calculations, we show that the observed momentum dependence reflects highly localized, nearest neighbor screening of the transient local charge perturbation in this compound with an exciton-like screening cloud, rather than delocalized screening. The size of the screening cloud is estmated to be about 0.4-0.5 interatomic distances.
Resonant inelastic x-ray scattering (RIXS) at the oxygen $K$-edge has recently accessed multi-spinon excitations in the one-dimensional antiferromagnet (1D-AFM) \sco, where four-spinon excitations are resolved separately from the two-spinon continuum. This technique, therefore, provides new opportunities to study fractionalized quasiparticle excitations in doped 1D-AFMs. To this end, we carried out exact diagonalization studies of the doped $t$-$J$ model and provided predictions for oxygen $K$-edge RIXS experiments on doped 1D-AFMs. We show that the RIXS spectra are rich, containing distinct t…
Although orbital degrees of freedom are a factor of fundamental importance in strongly correlated transition metal compounds, orbital correlations and dynamics remain very difficult to access, in particular by neutron scattering. Via a direct calculation of scattering amplitudes we show that instead magnetic resonant inelastic x-ray scattering (RIXS) does reveal orbital correlations. In contrast to neutron scattering, the intensity of the magnetic excitations in RIXS depends very sensitively on both the symmetry of the orbitals that spins occupy, and on photon polarizations. We show in detail …
We report a high-resolution resonant inelastic soft x-ray scattering study of the quantum magnetic spin-chain materials Li2CuO2 and CuGeO3. By tuning the incoming photon energy to the oxygen K-edge, a strong excitation around 3.5 eV energy loss is clearly resolved for both materials. Comparing the experimental data to many-body calculations, we identify this excitation as a Zhang-Rice singlet exciton on neighboring CuO4-plaquettes. We demonstrate that the strong temperature dependence of the inelastic scattering related to this high-energy exciton enables to probe short-range spin correlations…
Resonant inelastic x-ray scattering (RIXS) spectra at the Fe K-edge were measured for a single crystal of the iron oxypnictide superconductor PrFeAsO0.7 (Tc=42 K). They disclose a weak, broad feature centered around 4.5 eV energy loss, which is slightly resonantly enhanced when the incident energy is tuned in the vicinity of the 4p white line. We tentatively ascribe it to the charge-transfer excitation between As 4p and Fe 3d.
We analyze resonant inelastic x-ray scattering (RIXS) at the Cu K edge in a typical orbital-ordered compound KCuF3 on the basis of a microscopic theory. Spectral shape and its dependence on polarization direction and momentum transfer of photons are explained consistently with experimental data within our microscopic calculation. According to our microscopic orbital-resolving analysis, high-energy spectral weights (above 5 eV) originate from charge-transfer excitations related to the Cu-dγorbitals, while the low-energy weights (below 2 eV) originate from the d-d orbital excitations among the f…
Using the three-orbital Hubbard model, we investigate the low-energy excitation spectra in the antiferromagnetic phase of Ca$_2$RuO$_4$. We calculate the dynamical susceptibilities in the low-energy region by the random phase approximation and find that the anisotropic dispersion of the transverse mode is in good agreement with the spectra recently reported by inelastic neutron scattering experiments. By the fast-collision approximation, we simulate the resonant inelastic x-ray scattering (RIXS) spectra of the Ru $L_3$ edge from the dynamical susceptibilities. We show that the dispersion of th…
We present a theoretical investigation of the origin of Raman-like and fluorescencelike (FL) features of resonant inelastic x-ray scattering (RIXS) spectra. Using a combination of local-density approximation + dynamical mean-field theory and a configuration interaction solver for Anderson impurity model, we calculate the $L$-edge RIXS and x-ray absorption spectra of high-valence transition-metal oxides LaCuO$_3$ and NaCuO$_2$. We analyze in detail the behavior of the FL feature and show how it is connected to the details of electronic and crystal structure. On the studied compounds we demonstr…
We calculate the resonant inelastic X-ray scattering (RIXS) response of the Kitaev honeycomb model, an exactly solvable quantum-spin-liquid model with fractionalized Majorana and flux excitations. We find that the fundamental RIXS channels, the spin-conserving (SC) and the non-spin-conserving (NSC) ones, do not interfere and give completely different responses. SC-RIXS picks up exclusively the Majorana sector with a pronounced momentum dispersion, whereas NSC-RIXS also creates immobile fluxes, thereby rendering the response only weakly momentum dependent, as in the spin structure factor measur…
We derive general expressions for resonant inelastic x-ray scattering (RIXS) operators for $t_{2g}$ orbital systems, which exhibit a rich array of unconventional magnetism arising from unquenched orbital moments. Within the fast collision approximation, which is valid especially for 4$d$ and 5$d$ transition metal compounds with short core-hole lifetimes, the RIXS operators are expressed in terms of total spin and orbital angular momenta of the constituent ions. We then map these operators onto pseudospins that represent spin-orbit entangled magnetic moments in systems with strong spin-orbit co…
Collective spin excitations in a quantum spin ladder probed by high-resolution Resonant Inelastic X-ray Scattering
利用高分辨Cu $L_3$边缘共振非弹性X射线散射（RIXS）研究了自旋梯型化合物Sr $_{14}$ Cu $_{24}$ O $_{41}$的磁激发.我们的研究结果表明，RIXS耦合到集体自旋激发从一个量子自旋液体基态。与非弹性中子散射（INS）相比，RIXS截面在整个布里渊区（BZ）上仅适度变化，这表明在小动量转移时也具有高灵敏度。两个三重子的能隙为$100\pm 30$ meV。我们的研究结果得到了有限尺寸团簇的有效哈伯德模型计算的支持。
We report a Cu K-edge resonant inelastic x-ray scattering (RIXS) study of orbital excitations in KCuF3 . By performing the polarization analysis of the scattered photons, we disclose that the excitation between the eg orbitals and the excitations from t2g to eg exhibit distinct polarization dependence. The polarization dependence of the respective excitations is interpreted based on a phenomenological consideration of the symmetry of the RIXS process that yields a necessary condition for observing the excitations. In addition, we show that the orbital excitations are dispersionless within our …
We present a study of the resonant inelastic scattering response of \ybin\ excited at the tender Yb $M_5$ X-ray edge. In the high-temperature, paramagnetic phase, we observe a multiplet structure which can be understood at an ionic level. Upon cooling through the valence transition at $T_v\sim$ 40$K$, we observe a strong renormalization of the low-energy spectra, indicating a sensitivity to the formation of an intermediate valence phase at low temperatures. Similar spectrum renormalization has been observed in the optical conductivity, which suggests that the low-energy electronic structure po…
Ruddlesden-Popper and reduced Ruddlesden-Popper nickelates are intriguing candidates for mimicking the properties of high-temperature superconducting cuprates. The degree of similarity between these nickelates and cuprates has been the subject of considerable debate. Resonant inelastic x-ray scattering (RIXS) has played an important role in exploring their electronic and magnetic excitations, but these efforts have been stymied by inconsistencies between different samples and the lack of publicly available data for detailed comparison. To address this issue, we present open RIXS data on La4Ni3…
We establish the spectral line shape of orbital excitations created by resonant inelastic X-ray scattering for the model Mott insulator CoO. Improved experimental energy resolution reveals that the line shapes are strikingly different from expectations in a first principles-based atomic multiplet model. Extended theoretical simulations are performed to identify the underlying physical origins, which include a pronounced thermal tail reminiscent of anti-Stokes scattering on the energy gain side of excitations, and an essential contribution from interatomic many-body dynamics on the energy loss …
Collective excitations contain rich information about photoinduced transient states in correlated systems. In a Mott insulator, charge degrees of freedom are frozen, but can be activated by photodoping. The energy-momentum distribution of the charge excitation spectrum reflects the propagation of charge degrees of freedom, and provides information about the interplay among various intertwined instabilities on the time scale set by the pump. To reveal charge excitations out of equilibrium, we simulate time-resolved x-ray absorption and resonant inelastic x-ray scattering using a Hubbard model. …
The low energy excitations in Na2IrO3 have been investigated using resonant inelastic x-ray scattering (RIXS). A magnetic excitation branch can be resolved, whose dispersion reaches a maximum energy of about 35 meV at the Γ-point. The momentum dependence of the excitation energy is much larger along the Γ-X direction compared to that along the Γ-Y direction. The observed dispersion relation is consistent with a recent theoretical prediction based on Heisenberg-Kitaev model. At high temperatures, we find large contributions from lattice vibrational modes to our RIXS spectra, suggesting that a s…
We determine how the elementary excitations of iridium-oxide materials, which are dominated by a strong relativistic spin-orbit coupling, appear in Resonant Inelastic X-ray Scattering (RIXS). Whereas the RIXS spectral weight at the L2 x-ray edge vanishes, we find it to be strong at the L3-edge. Applying this to Sr2IrO4, we observe that RIXS, besides being sensitive to local doublet to quartet transitions, meticulously maps out the strongly dispersive delocalized excitations of the low-lying spin-orbit doublets.
We develop a formalism for calculating resonant inelastic x-ray scattering (RIXS) spectra in systems of itinerant electrons with arbitrary band structures, accounting for the effect of the positively-charged core hole exactly. We apply this formalism to the cuprate superconductors and obtain quantitative agreement with experimental data over a wide range of dopings. We reproduce the dispersing peaks and non-trivial polarization dependence found in RIXS experiments on several materials. Thus we demonstrate that features previously attributed to collective magnetic modes can be explained by band…
More than 50 years ago the electron-hole attraction was proposed to drive narrow gap semiconductors or semimetals to a new phase, the excitonic insulator. The experimental proof of its existence in bulk materials remains elusive. In strongly correlated insulators, the proximity of the excitonic insulator phase is reflected by the presence of dispersive electron-hole excitations with a small gap above a singlet ground state. Recently, such an excitation spectrum was proposed to be realized in perovskite oxide LaCoO$_3$. In this Letter we use Co $L_3$-edge resonant inelastic X-ray scattering to …
We analyze theoretically the resonant inelastic x-ray scattering (RIXS) at the Ni K edge in the stripe-ordered state of La_{2-x}Sr_xNiO_4 at x=1/3. In the calculation of RIXS spectra, the stripe-ordered ground state is described within the Hartree-Fock approximation by using a realistic tight-binding model for Ni3dγand O2p_{x, y} orbitals, and the electron correlations in the electronic excitation processes are taken into account within the random-phase approximation. The calculated RIXS spectrum shows a tail toward the low-energy region when the momentum transfer of photons equals the stripe …
Momentum Dependent Charge Excitations of Two-Leg Ladder: Resonant Inelastic X-ray Scattering of (La,Sr,Ca)14Cu24O41
利用共振非弹性X射线散射研究了（La，Sr，Ca）14Cu 24 O 41的动量相关电荷激发.在正方形晶格的情况下，动量依赖性的莫特间隙激发的阶梯表现出很小的变化后，空穴掺杂，表明空穴对的形成。基于哈伯德模型的理论计算定性地解释了这一特征。此外，实验数据表明，带内激发作为连续强度低于莫特间隙，它出现在所有的动量转移同时。带内激发的强度与阶梯的空穴浓度成正比，这与光学电导率测量结果一致。
The community currently lacks a complete understanding of how resonant inelastic x-ray scattering (RIXS) experiments probe the electron-phonon ($e$-ph) interaction in solids. For example, most theoretical models of this process have focused on dispersionless Einstein phonons. Using a recently developed momentum average (MA) variational approximation for computing RIXS spectra of band insulators, we examine the influence of both electron and phonon dispersion in the intermediate state of the scattering process. We find that the inclusion of either, and their mutual interplay, introduces signifi…
We develop a method to study the effect of the superconducting transition on resonant inelastic X-ray scattering (RIXS) signal in superconductors with an order parameter with an arbitrary symmetry within a quasiparticle approach. As an example, we compare the direct RIXS signal below and above the superconducting transition for p-wave type order parameters. For a p-wave order parameter with a nodal line, we show that, counterintuitively, the effect of the gap is most noticeable for momentum transfers in the nodal direction. This phenomenon may be naturally explained as a type of nesting effect…
We present a study of resonant inelastic X-ray scattering (RIXS) spectra collected at the rare-earth $L$ edges of divalent hexaborides YbB$_6$ and EuB$_6$. In both systems, RIXS-active features are observed at two distinct resonances separated by $\sim10$ eV in incident energy, with angle-dependence suggestive of distinct photon scattering processes. RIXS spectra collected at the divalent absorption peak strongly resemble the unoccupied 5$d$ density of states calculated using density functional theory, an occurrence we ascribe to transitions between weakly-dispersing 4$f$ and strongly dispersi…
We report a resonant inelastic x-ray scattering study of charge excitations in the quasi-one-dimensional Mott insulator SrCuO2. We observe a continuum of low-energy excitations, in which a highly dispersive feature with a large sinusoidal dispersion (~1.1 eV) resides. We have also measured the optical conductivity, and studied the dynamic response of the extended Hubbard model with realistic parameters, using a dynamical density-matrix renormalization group method. In contrast to earlier work, we do not find a long-lived exciton, but rather these results suggest that the excitation spectrum co…
We investigate electronic excitations in La2-x(Br,Sr)xCuO4 using resonant inelastic x-ray scattering (RIXS) at the oxygen K edge. RIXS spectra of the hole-doped cuprates show clear momentum dependence below 1 eV. The spectral weight exhibits positive dispersion and shifts to higher energy with increasing hole concentration. Theoretical calculation of the dynamical charge structure factor on oxygen orbitals in a three-band Hubbard model is consistent with the experimental observation of the momentum and doping dependence, and therefore the dispersive mode is ascribed to intraband charge excitat…
We theoretically investigate spin dynamics and $L_3$-edge resonant inelastic X-ray scattering (RIXS) of Chromium with commensurate spin-density wave (SDW) order, based on a multi-band Hubbard model composed of 3$d$ and 4$s$ orbitals. Obtaining the ground state with the SDW mean-field approximation, we calculate the dynamical transverse and longitudinal spin susceptibility by using random-phase approximation. We find that a collective spin-wave excitation seen in inelastic neutron scattering hardly damps up to $\sim$0.6 eV. Above the energy, the excitation overlaps individual particle-hole exci…
Resonant inelastic x-ray scattering (RIXS) has become a powerful tool in the study of the electronic structure of condensed matter. Although the linewidths of many RIXS features are narrow, the experimental broadening can often hamper the identification of spectral features. Here, we show that the Maximum Entropy technique can successfully be applied in the deconvolution of RIXS spectra, improving the interpretation of the loss features without a severe increase in the noise ratio.
Vanadium dioxide is of broad interest as a spin-1/2 electron system that realizes a metal-insulator transition near room temperature, due to a combination of strongly correlated and itinerant electron physics. Here, resonant inelastic X-ray scattering is used to measure the excitation spectrum of charge, spin, and lattice degrees of freedom at the vanadium L-edge under different polarization and temperature conditions. These spectra reveal the evolution of energetics across the metal-insulator transition, including the low temperature appearance of a strong candidate for the singlet-triplet ex…
Generic low-energy spectral features related to $d$-$d$ excitations in nearly cubic vanadium perovskites are predicted for the expected $L$-edge resonant inelastic X-ray scattering (RIXS) measurements. Model Hamiltonian describing local electronic properties, including crystal-field effects, of vanadium 3d orbitals in the basic $t_{2g}^2$ configuration is formulated with the help of complementary {\it ab initio} quantum-chemical cluster calculations. In the presence of 2p-core hole, the local Hamiltonian includes strong 2p-3d electron interactions. As a prerequisite for evaluating RIXS transit…
High-resolution resonant inelastic X-ray scattering (RIXS) at the oxygen K-edge has been used to study the orbital excitations of Ca2RuO4 and Sr2RuO4. In combination with linear dichroism X-ray absorption spectroscopy, the ruthenium 4d-orbital occupation and excitations were probed through their hybridization with the oxygen p-orbitals. These results are described within a minimal model, taking into account crystal field splitting and a spin-orbit coupling λ_{so}=200~meV. The effects of spin-orbit interaction on the electronic structure and implications for the Mott and superconducting ground …
