Resonant Inelastic X-ray Scattering Interferometry

• Resonant inelastic x‐ray scattering interferometry is a phase-sensitive spectroscopic technique that exploits interference from equivalent atomic sites to probe quasi-molecular electronic excitations.
• It extends the double-slit experiment concept by measuring momentum-dependent interference patterns that differentiate bonding (even) and antibonding (odd) excitations.
• The method provides practical insights into both the spatial organization and ultrafast formation dynamics of excited states in complex quantum materials.

Searching arXiv for the specified papers to ground the article and verify citation details. Resonant inelastic x-ray scattering interferometry is an interferometric use of resonant inelastic x-ray scattering (RIXS) in which the measured intensity contains phase-sensitive information about electronic excited states. In the formulation demonstrated for insulating Ba₃CeIr₂O₉, the method constitutes an inelastic incarnation of Young’s double-slit experiment: the “slits” are the two equivalent $2p$ core levels on the two Ir sites of a structural dimer, and the resulting momentum-dependent interference pattern reveals the symmetry and character of quasi-molecular excited states (Revelli et al., 2019). A complementary interferometric perspective treats RIXS on the incident-energy axis, where overlapping core-hole resonances generate quantum-path interference patterns whose lineshapes encode whether an excitation is formed through constructive or destructive interference and, correspondingly, whether its creation dynamics are effectively “fast” or “slow” on sub-femtosecond time scales (Wray et al., 2016).

1. Double-slit concept in resonant inelastic x-ray scattering

Young’s optical experiment involves two narrow slits at positions $\pm R/2$ illuminated by a coherent wave, with the two emerging beams interfering to produce a sinusoidal intensity pattern $I(q)\propto \cos^2(q\cdot R/2)$. In the RIXS realization reported for Ba₃CeIr₂O₉, the analogous two paths arise because an incident photon can create a core hole on either of the two equivalent Ir sites in a structural dimer, and the inelastically scattered photon is emitted from whichever site hosted the intermediate core hole (Revelli et al., 2019).

The essential condition for interference is the absence of “which-path” information. If the final electronic excitation is a quasi-molecular orbital delocalized over both Ir sites, the two amplitudes interfere in momentum space, yielding a double-slit-type oscillation as a function of transferred momentum $q$ along the dimer axis. In this setting, the interference pattern is not a measure of the ground-state real-space structure in the elastic-scattering sense, but a measure of the symmetry and character of electronic excited states (Revelli et al., 2019).

This construction extends the conceptual reach of diffraction. Elastic scattering captures structural information from interference of elastically scattered waves, whereas the inelastic RIXS analogue exposes the internal structure of excited states. The reported interpretation is that RIXS interferometry can therefore reveal the real-space organization of excitations in solids containing dimers, trimers, ladders, or other superstructures (Revelli et al., 2019).

2. Scattering formalism and interference terms

Within the dipole and fast-collision approximations, the RIXS cross section for a ground state $|g\rangle$ and final states $|f\rangle$ is written as

$$I(q,\omega)=\sum_f |A_f(q)|^2 \delta(\hbar\omega-E_f),$$

with scattering amplitude

$A_f(q)\propto \langle f|T^\dagger(q)|g\rangle,$

and

$$T^\dagger(q)=\sum_R e^{i\,q\cdot R} D_R^\dagger(\epsilon_{\mathrm{out}}^*)D_R(\epsilon_{\mathrm{in}}).$$

Here, $R$ runs over all Ir sites contributing to the delocalized $\pm R/2$0, $\pm R/2$1 is the local dipole operator at site $\pm R/2$2 with photon polarization $\pm R/2$3, $\pm R/2$4 is the momentum transfer, and $\pm R/2$5 is the energy loss (Revelli et al., 2019).

For a two-site dimer with $\pm R/2$6 and $\pm R/2$7, and with identical local matrix elements $\pm R/2$8, the amplitude becomes

$\pm R/2$9

After squaring and summing over final states in a small energy window, the intensity assumes the form

$I(q)\propto \cos^2(q\cdot R/2)$0

or, more generally,

$I(q)\propto \cos^2(q\cdot R/2)$1

The definitions given are $I(q)\propto \cos^2(q\cdot R/2)$2, representing the $I(q)\propto \cos^2(q\cdot R/2)$3-independent background from two incoherent slits, and $I(q)\propto \cos^2(q\cdot R/2)$4, representing the amplitude of interference, with sign determined by the parity of $I(q)\propto \cos^2(q\cdot R/2)$5. The dimer separation is $I(q)\propto \cos^2(q\cdot R/2)$6, and for the reported geometry $I(q)\propto \cos^2(q\cdot R/2)$7 along the $I(q)\propto \cos^2(q\cdot R/2)$8-axis (Revelli et al., 2019).

The phase $I(q)\propto \cos^2(q\cdot R/2)$9 is identified with excitation symmetry: $q$0 for “antibonding” (even) excitations and $q$1 for “bonding” (odd) excitations in the general cosine form. In the explicit $q$2 and $q$3 fit convention used for the experimental analysis, these appear as $q$4 or $q$5 in the fitting function. This suggests that the experimentally extracted phase is a direct proxy for site-exchange parity, rather than merely a phenomenological fit parameter (Revelli et al., 2019).

3. Experimental realization in Ba₃CeIr₂O₉

The prototype experiment was performed at the Ir $q$6-edge in Ba₃CeIr₂O₉, a material with structural Ir dimers and strong spin-orbit coupling. RIXS spectra were measured as a function of $q$7, the momentum transfer along the dimer axis, with energy resolution $q$8. Three intra-$q$9 peaks, labeled $|g\rangle$0, $|g\rangle$1, and $|g\rangle$2, were observed between $|g\rangle$3 and $|g\rangle$4 (Revelli et al., 2019).

For each fixed peak-energy window, the intensity $|g\rangle$5 was obtained by integrating the RIXS counts over that window and normalizing by its width $|g\rangle$6. The resulting intensity oscillated sinusoidally with period $|g\rangle$7, where $|g\rangle$8. To quantify this behavior, the reported fitting form was

$|g\rangle$9

with $|f\rangle$0 or $|f\rangle$1, and $|f\rangle$2, $|f\rangle$3, $|f\rangle$4, and $|f\rangle$5 included to account for geometry-dependent envelope and background (Revelli et al., 2019).

The observed periodicity is a direct consequence of the dimer spacing and therefore acts as a structural interferometric scale. The envelope and background terms indicate that the raw momentum dependence is not a pure structure factor but a modulated quantity affected by experimental geometry and noninterfering contributions. A plausible implication is that extraction of the intrinsic phase information depends less on the absolute intensity than on the position of nodes and maxima in $|f\rangle$6-space (Revelli et al., 2019).

4. Phase, parity, and bonding character of excited states

The most distinctive outcome of the dimer experiment is the identification of excitation symmetry from the phase of the interference pattern. A feature with $|f\rangle$7, corresponding to $|f\rangle$8, has a maximum at $|f\rangle$9 and is assigned even (antibonding) symmetry of the final-state wavefunction under site exchange. This behavior was observed for peak $$I(q,\omega)=\sum_f |A_f(q)|^2 \delta(\hbar\omega-E_f),$$0 (Revelli et al., 2019).

By contrast, a feature with $$I(q,\omega)=\sum_f |A_f(q)|^2 \delta(\hbar\omega-E_f),$$1, corresponding to $$I(q,\omega)=\sum_f |A_f(q)|^2 \delta(\hbar\omega-E_f),$$2, has a node at $$I(q,\omega)=\sum_f |A_f(q)|^2 \delta(\hbar\omega-E_f),$$3 and is assigned odd (bonding) symmetry. This behavior was observed for peaks $$I(q,\omega)=\sum_f |A_f(q)|^2 \delta(\hbar\omega-E_f),$$4 and $$I(q,\omega)=\sum_f |A_f(q)|^2 \delta(\hbar\omega-E_f),$$5. The stated conclusion is that the $$I(q,\omega)=\sum_f |A_f(q)|^2 \delta(\hbar\omega-E_f),$$6-phase directly encodes the parity of the quasi-molecular orbital: bonding excitations change sign under exchange of the two Ir sites and produce a $$I(q,\omega)=\sum_f |A_f(q)|^2 \delta(\hbar\omega-E_f),$$7-modulation, whereas antibonding excitations are symmetric and produce a $$I(q,\omega)=\sum_f |A_f(q)|^2 \delta(\hbar\omega-E_f),$$8-modulation (Revelli et al., 2019).

This parity assignment is central to the method. It shows that the interferogram does not merely indicate delocalization over two sites; it separates classes of delocalized excitations according to their transformation under site exchange. In that sense, RIXS interferometry functions as a momentum-space symmetry analyzer for excited states. A plausible implication is that when several excitations overlap spectrally but differ in parity, momentum-resolved phase analysis can disentangle them even if their energies are close (Revelli et al., 2019).

5. Generalization beyond dimers

The formalism is explicitly generalized to a motif of $$I(q,\omega)=\sum_f |A_f(q)|^2 \delta(\hbar\omega-E_f),$$9 equivalent sites at positions $A_f(q)\propto \langle f|T^\dagger(q)|g\rangle,$0, for which the RIXS operator is

$A_f(q)\propto \langle f|T^\dagger(q)|g\rangle,$1

The corresponding intensity for a given set of final states is

$A_f(q)\propto \langle f|T^\dagger(q)|g\rangle,$2

where $A_f(q)\propto \langle f|T^\dagger(q)|g\rangle,$3. If all $A_f(q)\propto \langle f|T^\dagger(q)|g\rangle,$4 are equal in magnitude, the structure factor becomes

$A_f(q)\propto \langle f|T^\dagger(q)|g\rangle,$5

For a linear trimer with $A_f(q)\propto \langle f|T^\dagger(q)|g\rangle,$6 and $A_f(q)\propto \langle f|T^\dagger(q)|g\rangle,$7, one obtains

$A_f(q)\propto \langle f|T^\dagger(q)|g\rangle,$8

which yields characteristic three-slit fringes. For a ladder or bilayer, the stated two-dimensional interference form is $A_f(q)\propto \langle f|T^\dagger(q)|g\rangle,$9, revealing the superstructure periodicity in-plane (Revelli et al., 2019).

The stated application domain includes dimers, trimers, heptamers, ladders, bilayers, and Kitaev spin-liquids. By reading off periods, phases, and envelopes of the oscillations, the method can identify intersite quasi-molecular orbitals and resolve bonding-antibonding splittings; distinguish excitations of different symmetry, including dipole versus quadrupole allowed and spin-flip versus non-spin-flip; map low-dimensional correlations in ladders, bilayers, or Kitaev spin-liquids by observing interference patterns related to nearest-neighbor pairs; and probe Majorana fermion excitations or other exotic fractionalized modes via their characteristic structure factors (Revelli et al., 2019).

These extensions indicate that the interferometric content of RIXS is not restricted to a two-path geometry. Instead, the measurable quantity is the structure factor of the excitation matrix elements over a motif. This suggests that RIXS interferometry can be viewed as an excitation-resolved analogue of diffraction from electronically active clusters embedded in a crystal (Revelli et al., 2019).

6. Incident-energy interferometry and quantum-path phases

A second interferometric axis in RIXS is the incident photon energy $$T^\dagger(q)=\sum_R e^{i\,q\cdot R} D_R^\dagger(\epsilon_{\mathrm{out}}^*)D_R(\epsilon_{\mathrm{in}}).$$0. In the Kramers-Heisenberg picture, the RIXS intensity for a particular final state $$T^\dagger(q)=\sum_R e^{i\,q\cdot R} D_R^\dagger(\epsilon_{\mathrm{out}}^*)D_R(\epsilon_{\mathrm{in}}).$$1 is written as

$$T^\dagger(q)=\sum_R e^{i\,q\cdot R} D_R^\dagger(\epsilon_{\mathrm{out}}^*)D_R(\epsilon_{\mathrm{in}}).$$2

with $$T^\dagger(q)=\sum_R e^{i\,q\cdot R} D_R^\dagger(\epsilon_{\mathrm{out}}^*)D_R(\epsilon_{\mathrm{in}}).$$3 running over short-lived core-hole intermediate states of resonance energy $$T^\dagger(q)=\sum_R e^{i\,q\cdot R} D_R^\dagger(\epsilon_{\mathrm{out}}^*)D_R(\epsilon_{\mathrm{in}}).$$4 and inverse lifetime $$T^\dagger(q)=\sum_R e^{i\,q\cdot R} D_R^\dagger(\epsilon_{\mathrm{out}}^*)D_R(\epsilon_{\mathrm{in}}).$$5. When two or more core-hole resonances overlap energetically, the coherent sum over their complex amplitudes produces interference patterns visible as modulations of the RIXS intensity versus incident energy $$T^\dagger(q)=\sum_R e^{i\,q\cdot R} D_R^\dagger(\epsilon_{\mathrm{out}}^*)D_R(\epsilon_{\mathrm{in}}).$$6: constructive interference “spreads” the resonant profile, and destructive interference “pinches” it (Wray et al., 2016).

Two limiting mechanisms are distinguished. In photon-operator RIXS, also described as “Raman-like,” $$T^\dagger(q)=\sum_R e^{i\,q\cdot R} D_R^\dagger(\epsilon_{\mathrm{out}}^*)D_R(\epsilon_{\mathrm{in}}).$$7, so the two-step resonance may be viewed as a single effective operator on the ground state, and no net destructive cancellation occurs when summing over intermediate states. This is described as a “fast” excitation, with constructive or neutral interference of the $$T^\dagger(q)=\sum_R e^{i\,q\cdot R} D_R^\dagger(\epsilon_{\mathrm{out}}^*)D_R(\epsilon_{\mathrm{in}}).$$8-channel phases. In shake-up RIXS, also described as “fluorescence-like” or “indirect,” $$T^\dagger(q)=\sum_R e^{i\,q\cdot R} D_R^\dagger(\epsilon_{\mathrm{out}}^*)D_R(\epsilon_{\mathrm{in}}).$$9, so only the incoherent square-modulus term survives when the channels are incoherently summed, and in the two-path limit this forces fully destructive interference, corresponding to a relative phase of $R$0 between the two resonances. This is described as a “slow” excitation because the core-hole potential or spin-orbit coupling must evolve before the final excitation can form (Wray et al., 2016).

To characterize the interference sign experimentally, the proposed metric is

$R$1

where $R$2 is the resonant part of the x-ray absorption or inelastic fluorescence yield at incident energy $R$3, and $R$4 is the RIXS intensity at energy loss $R$5 and incident energy $R$6. The recommended procedure is to evaluate $R$7 at the incident energy where $R$8 is largest, namely the leading edge of absorption, to maximize sensitivity and minimize self-absorption errors. By construction, net-constructive channels give $R$9, whereas net-destructive channels give $\pm R/2$00 (Wray et al., 2016).

Starting from the same Kramers-Heisenberg expression, the numerator of each path is written as $\pm R/2$01, with the denominator $\pm R/2$02. The real amplitudes $\pm R/2$03 encode dipole-matrix-element strengths, and the phases $\pm R/2$04 encode the quantum-path phases imparted by each intermediate resonance channel. On the leading edge of the lowest resonance, expansion of higher-lying channels into a single real term $\pm R/2$05 leads to a logarithmic derivative in which the sign of $\pm R/2$06 is determined by the sign of $\pm R/2$07, namely constructive versus destructive net phase (Wray et al., 2016).

The same framework introduces a time-domain interpretation. The core-hole inverse lifetime $\pm R/2$08 corresponds to a decay time $\pm R/2$09, described as a few hundred attoseconds at the transition-metal $\pm R/2$10-edge, with $\pm R/2$11–$\pm R/2$12–$\pm R/2$13. Shake-up processes require a finite ramp in time to develop, so if their formation rate $\pm R/2$14, the intensity scales as $\pm R/2$15 and interference is strongly destructive, whereas if $\pm R/2$16, the process becomes effectively instantaneous and interference neutralizes (Wray et al., 2016).

7. Experimental signatures, applications, and interpretive limits

The incident-energy interferometry was examined experimentally at multiple absorption edges. At the $\pm R/2$17-edge ($\pm R/2$18) of SrCuO$\pm R/2$19 and NiO at ALS MERLIN, the reported conditions were energy resolution $\pm R/2$20, $\pm R/2$21-incident polarization in the $\pm R/2$22 plane, $\pm R/2$23 scattering, and grazing incidence, while sweeping $\pm R/2$24 across the $\pm R/2$25 resonances and recording $\pm R/2$26. At the $\pm R/2$27-edge ($\pm R/2$28) of NiO, atomic-multiplet plus SIAM simulations were used for the $\pm R/2$29 and $\pm R/2$30 resonances. At the $\pm R/2$31-edge ($\pm R/2$32) of SrCuO$\pm R/2$33 at APS Sector 30, the reported conditions were $\pm R/2$34-incident polarization with $\pm R/2$35CuO plane, momentum transfer along the Cu–O chain, and resolution $\pm R/2$36, while sweeping $\pm R/2$37 from the well-screened pre-edge to the poorly screened peak and beyond (Wray et al., 2016).

Several characteristic patterns were identified. At the Cu $\pm R/2$38-edge of SrCuO$\pm R/2$39, the $\pm R/2$40 orbiton showed a single-peak resonance around $\pm R/2$41 and was inferred to have a $\pm R/2$42 relative phase between the $\pm R/2$43 and $\pm R/2$44 channels, corresponding to destructive interference, whereas the $\pm R/2$45 orbiton showed two well-separated resonance maxima at $\pm R/2$46 and $\pm R/2$47 energies with zero relative phase, corresponding to constructive interference. Fitting with the two-channel form of Kramers-Heisenberg returned $\pm R/2$48 for $\pm R/2$49 and $\pm R/2$50 for $\pm R/2$51. Time-domain simulation showed that the $\pm R/2$52 spin-flip orbiton is strongly suppressed for $\pm R/2$53 and only appears at longer times, whereas the $\pm R/2$54 orbiton appears almost immediately (Wray et al., 2016).

For NiO at the $\pm R/2$55-edge, four principal loss features were listed: $\pm R/2$56 spin-flips, $\pm R/2$57 $\pm R/2$58 “photon-operator” orbital excitation, $\pm R/2$59 shake-up $\pm R/2$60, and $\pm R/2$61 other states. The extracted $\pm R/2$62 from the simulated RIXS map reproduced which features carry destructive versus constructive interference. The text states that experimentally one can measure $\pm R/2$63 in the brightest windows and confirm that the $\pm R/2$64 peak has $\pm R/2$65 while the $\pm R/2$66 spin-flip mode has $\pm R/2$67 (Wray et al., 2016).

At the Cu $\pm R/2$68-edge of SrCuO$\pm R/2$69, three major features were identified at both zone center and boundary: the pre-gap feature at $\pm R/2$70, the Mott-gap continuum at $\pm R/2$71–$\pm R/2$72, and the charge-transfer band above $\pm R/2$73. Fits of the Mott-gap feature to a two-peak destructive versus constructive model clearly favored destructive interference. At the leading edge $\pm R/2$74, the reported values were $\pm R/2$75, $\pm R/2$76, and $\pm R/2$77. The interpretation given is that Mott-gap excitations form on the time scale of Cu–Cu hopping $\pm R/2$78, corresponding to $\pm R/2$79, which is comparable to or slower than $\pm R/2$80, hence strongly destructive; charge-transfer modes form faster, with $\pm R/2$81, and are less suppressed; and the near-zero $\pm R/2$82 of the pre-gap mode implies ultrafast formation, consistent with an exciton-like state that remains bound and has small dispersion (Wray et al., 2016).

One interpretive issue addressed explicitly concerns the anomalous interference of the pre-gap feature near the Mott-gap leading edge in one-dimensional SrCuO$\pm R/2$83. The feature appears only for momentum transfers near the zone boundary and had previously been hypothesized to be a $\pm R/2$84Mott gap $\pm R/2$85 two spinons$\pm R/2$86 shake-up. However, its $\pm R/2$87 indicates effectively neutral or weakly constructive interference, pointing instead to a very rapid formation channel, “more like a bound exciton than a continuum shake-up.” Its reduced dispersion, stated as $\pm R/2$88 versus $\pm R/2$89 for the Mott gap, and its shallow $\pm R/2$90 are therefore taken to suggest a distinct excitation symmetry (Wray et al., 2016).

Taken together, the momentum-space interferograms of delocalized final states and the incident-energy interference of overlapping intermediate resonances define two complementary interferometric modalities within RIXS. The first reads out spatial phase relations of the excited-state wavefunction, and the second reads out quantum-path phases and associated sub-femtosecond creation dynamics. In that combined sense, resonant inelastic x-ray scattering interferometry constitutes a phase-sensitive spectroscopic framework for extracting the symmetry, real-space structure, and dynamical formation character of excitations in correlated quantum materials (Revelli et al., 2019).