Unraveling Time- and Frequency-Resolved Nuclear Resonant Scattering Spectra Lukas Wol1and J org Evers1y 1Max-Planck-Institut f ur Kernphysik Saupfercheckweg 1 69117 Heidelberg Germany

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Unraveling Time- and Frequency-Resolved Nuclear Resonant Scattering Spectra

Lukas Wolﬀ1, ∗and J¨org Evers1, †

1Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany

(Dated: December 13, 2022)

Owing to their extremely narrow line-widths and exceptional coherence properties, M¨ossbauer

nuclei form a promising platform for quantum optics, spectroscopy and dynamics at energies of

hard x-rays. A key requirement for further progress is the development of more powerful measure-

ment and data analysis techniques. As one approach, recent experiments have employed time- and

frequency-resolved measurements, as compared to the established approaches of measuring time-

resolved or frequency-resolved spectra separately. In these experiments, the frequency-dependence

is implemented using a tunable single-line nuclear reference absorber. Here, we develop spectroscopy

and analysis techniques for such time- and frequency-resolved Nuclear Resonant Scattering spectra

in the frequency-frequency domain. Our approach is based on a Fourier-transform of the experimen-

tally accessible intensities along the time axis, which results in complex-valued frequency-frequency

correlation (FFC) spectra . We show that these FFC spectra not only exhibit a particularly simple

structure, disentangling the diﬀerent scattering contributions, but also allow one to directly access

nuclear target properties and the complex-valued nuclear resonant part of the target response. In

a second part, we explore the potential of an additional phase control of the x-rays resonantly scat-

tered oﬀ of the reference absorber for our scheme. Such control provides selective access to speciﬁc

scattering pathways, allowing for their separate analysis without the need to constrain the parame-

ter space to certain frequency or time limits. All results are illustrated with pertinent examples in

Nuclear Forward Scattering and in reﬂection oﬀ of thin-ﬁlm x-ray cavities containing thin layers of

M¨ossbauer nuclei.

I. INTRODUCTION

M¨ossbauer nuclei are established as a versatile tool

for highly-sensitive spectroscopic studies of condensed

matter systems [1–3]. Owing to their extremely narrow

line-widths and exceptional coherence properties [4–6],

they also form a promising platform for quantum optics

at hard x-ray energies [7–11]. In the future, state-of-

the-art and upcoming x-ray light sources with unprece-

dented quality and intensity may open up a regime of

nonlinear [12–14] and non-equilibrium phenomena [15]

which has only started to be explored experimentally

[16]. However, exploring this regime will require ad-

vanced measurement and data analysis techniques to ac-

cess a broader range of observables, and to compare the-

oretical predictions with experimental observations. Ex-

amples include spectroscopy beyond the linear response

regime [17], photon-correlation measurements [18], or

methods to study the time-resolved sample dynamics af-

ter external stimuli, potentially on a per-shot basis [19].

As a ﬁrst step towards this goal, recent experiments

employed a time- and frequency-resolved measurement

of the scattered x-ray intensity [20–23], as compared to

the established approaches of measuring time-resolved

or frequency-resolved spectra separately [2], the latter

of which can be obtained, for instance, by probing the

nuclear absorption using pure nuclear Bragg reﬂections

to produce highly monochromatic x-ray light on the

scale of the nuclear line-width [24]. Alternatively, the

∗lukas.wolﬀ@mpi-hd.mpg.de

†joerg.evers@mpi-hd.mpg.de

frequency-selectivity can be achieved using a heterodyne

approach by adding an additional single-line reference

absorber, in the following referred to as analyzer, on a

velocity drive up- or downstream of the target under

investigation [25, 26], see Fig. 1. This method is also

used to perform time- and frequency-resolved data ac-

quisition, which provides a number of signiﬁcant advan-

tages over other detection schemes using single-line refer-

ence absorbers, even though it does not require changes

to the experimental setup apart from the electronics.

On the one hand, it allows one to apply several es-

tablished data analysis methods using a single experi-

mental data set only. For example, late-time integra-

tion methods established as a standard analysis approach

for x-ray cavities probed in reﬂection [25, 27, 28] or re-

lated stroboscopic methods [29–31] can be employed by

integrating the two-dimensional data set along part of

the time axis. Importantly, the time- and frequency-

resolved spectra allow one to improve the recovery of

the target spectra by optimizing the integration range

throughout the data analysis [32, 33]. Similarly, oﬀ-

resonant methods can be employed in spectral regions

with large detuning between analyzer and target, which

may even provide access to the complex-valued target re-

sponse [29, 30, 34, 35]. These methods typically exploit

the interference between particular scattering pathways,

which can be studied in the time [34]- or frequency do-

main [35]. However, these established methods share the

drawback that they only make use of select regions of the

recorded two-dimensional spectra. On the other hand,

the two-dimensional data set allows for a much more

stringent comparison between theory and experiment.

One reason for this are the rich interference structures,

which are lost in the usual one-dimensional data by the

arXiv:2210.09848v2 [quant-ph] 12 Dec 2022

Analyzer

Detector

Doppler Drive

Phase

Control

Intensity

X-ray Pulse

Forward Scattering Grazing Incidence

Target

−200 −100 0 100 200

Detuning ∆(units of γ)

100

150

200

Frequency ν(units of γ)

−200 −100 0 100 200

Detuning ∆(units of γ)

100

150

200

Time (ns)

FIG. 1. Schematic of a setup to record time- and frequency-resolved Nuclear Resonant Scattering spectra in forward and

grazing incidence geometry. The goal is to characterize the resonant target response. An additional single-line analyzer mounted

on a Doppler drive is used to introduce a variable frequency dependence. Resonant and non-resonant scattering in analyzer

and target lead to four interfering scattering pathways contributing to the detection signal. An example for the experimentally

accessible scattered intensity as function of analyzer frequency and arrival time of the scattered photons is shown in the lower

right panel. The corresponding frequency-frequency correlation (FFC) spectrum studied in this work is obtained via a Fourier

transform along the time axis. An example FFC spectrum for the data in the lower-right panel is shown in the lower-left

panel. It exhibits clear horizontal and diagonal linear structures which can be interpreted in terms of spectral correlations

within the combined analyzer-target-system. In the second part of this work, an additional phase control between resonant

and non-resonant response of the analyzer is considered to disentangle the diﬀerent scattering pathways.

integration over the other axis (see bottom-right panel

of Fig. 1 for an example). Interestingly, these structures

encode intensity and phase of the target response. In

Refs. 20 and 21, the complex-valued target response was

determined by ﬁtting theory models to the entire two-

dimensional spectrum, thereby using all recorded data

rather than only parts of it. However, this approach is

computationally demanding as compared to other meth-

ods, and requires model ﬁts in order to extract the desired

target properties.

This raises the question, if the two-dimensional spec-

tra can also be analyzed in diﬀerent ways, which ideally

provide access to the desired target properties in a more

transparent way, without requiring global ﬁts to the en-

tire spectra, but still allow one to exploit the time- and

energy correlations in the spectra, and to make use of

large parts or even the entire experimental data set in

the analysis.

Motivated by this, here, we develop spectroscopy and

analysis techniques which are based on the Fourier trans-

form of experimentally accessible time- and frequency-

resolved intensities along the time axis. The resulting

complex-valued frequency-frequency correlation (FFC)

spectra exhibit particularly simple signatures comprising

horizontal and diagonal structures, which can be associ-

ated to diﬀerent contributing scattering processes. These

signatures (see the bottom-left panel of Fig. 1 for an ex-

ample) facilitate a selective analysis of the diﬀerent scat-

tering contributions. We in particular focus on two anal-

ysis approaches. First, we discuss linear ﬁts to the di-

agonal structures in the frequency-frequency correlation

spectra, which allow one to extract nuclear resonance en-

ergies, as well as spectral line features such as collective

energy shifts and superradiant line broadenings. Second,

we show that horizontal or vertical sections through the

diagonal structure provide access to amplitude and phase

of the nuclear resonant part of the target response, cross-

correlated with the analyzer response. This retrieval of

the response without contributions from the oﬀ-resonant

scattering in the target is of particular interest for x-ray

cavity targets containing M¨ossbauer nuclei, since their

spectra typically are strongly aﬀected by the interference

of electronic and nuclear response, in dependence on the

x-ray incidence angle. We further show that an addi-

tional control of the relative phase between the electronic

and nuclear response of the analyzer allows one to dis-

entangle diﬀerent scattering pathways, thereby facilitat-

ing their selective analysis without imposing additional

constraints such as a large analyzer-target detuning. All

approaches are illustrated using examples of practical rel-

evance.

The manuscript is structured as follows: The next Sec-

tion brieﬂy describes a generic experimental setup used

to record time- and frequency-resolved spectra including

phase- and frequency-control of the nuclear reference ab-

sorber. Further, we derive expressions for the frequency-

frequency correlation spectra in linear response theory,

and discuss them in particular limiting cases. Section III

presents the two analysis approaches for the diagonal

structures, including numerical examples in forward scat-

tering and cavity reﬂection. Section IV introduces the

phase control of the analyzer as an additional control pa-

rameter, and discusses its implications for the analysis of

the diagonal structure. Finally, Sec. V summarizes the

results.

II. LINEAR-RESPONSE FORMALISM AND

SPECTRAL CORRELATIONS IN NUCLEAR

RESONANT SCATTERING

For our analysis, we consider the setup shown schemat-

ically in Fig. 1. A temporally short and spectrally broad

x-ray pulse delivered by an accelerator-based x-ray source

is used to probe a target containing M¨ossbauer nuclei.

Both, forward scattering geometries and reﬂection from

x-ray cavities will be considered. Additional frequency

information is gained by introducing a single-line refer-

ence absorber, which can be tuned in frequency by ∆

via a Doppler drive. Throughout this paper, frequen-

cies are given in units of the target single-nucleus line-

width γ. Each target features a near-instantaneous elec-

tronic response ∝δ(t), and a delayed nuclear response,

which we denote as Si(t), with i∈ {a, t}for analyzer

and target [4, 5, 36–39]. The two-stage setup thus gives

rise to four diﬀerent scattering channels [34, 40], as il-

lustrated in Fig. 1. In the experiment, the time- and

frequency-resolved intensity of the scattered light is mea-

sured, which gives rise to two-dimensional spectra as il-

lustrated in the bottom right panel of Fig. 1 [20, 21]. A

Fourier transform along the time axis then leads to the

frequency-frequency correlation spectra considered here.

We note that Ref. 35 employed a similar Fourier trans-

form in order to select a particular frequency region for

a subsequent analysis in the time domain. The bottom

left part of Fig. 1 shows the real value as an example,

clearly exhibiting the horizontal and diagonal structures.

In Sec. IV, we will further consider the possibility of con-

trolling the relative phase φbetween the electronic and

nuclear response of the analyzer.

A. Time- and frequency resolved spectra in the

linear response formalism

A theoretical description of time- and frequency-

resolved Nuclear Resonant Scattering spectra can be

given employing the linear response formalism (see e.g.

[4, 5]) which is justiﬁed by the narrow line-width char-

acteristic of M¨ossbauer transitions that typically leads

to low excitation in state-of-the-art experiments, even

at high-brilliance third-generation synchrotron sources.

Neglecting polarization eﬀects [41], each target iin the

beam path can be described by a scalar transfer func-

tion ˆ

Ti(ω) in the frequency domain or (impulse) response

function Ti(t) in the time domain. Here and in the fol-

lowing, the “hat” denotes quantities in the frequency do-

main. Then, the outgoing ﬁeld is given by

Eout(ω) = ˆ

Ti(ω)ˆ

Ein(ω),(1)

Eout(t) = (Ti∗Ein)(t) (2)

in the frequency- or time domain, respectively. Next to

the convolution ∗in Eq. (2), we also deﬁne the cross-

correlation ?,

(f∗g)(x) = Z∞

−∞

dyf(x−y)g(y),(3)

(f ? g)(x) = Z∞

−∞

dyf∗(y−x)g(y),(4)

for two complex-valued functions f, g of frequency or time

variables x, y. The convolution can be interpreted as ap-

plying a ﬁlter fto g, or a propagation of an input gat

point yto the ﬁnal output f∗gat point xby virtue of

the response function f. The cross-correlation f ? g on

the other hand can be thought of as scanning the func-

tions fand gfor similarities by introducing relative shifts

xbetween them. Both quantities and their interpreta-

tions will turn out to be instrumental for understand-

ing the diagonal and horizontal structures appearing in

Fourier-transformed time- and frequency-resolved spec-

tra, the real part of which is shown in the lower left plot

of Fig. 1 as an example.

The responses of nuclear targets in forward scatter-

ing and grazing incidence geometry comprise two funda-

mentally diﬀerent scattering processes: Prompt scatter-

ing nonresonant with the nuclear transition and coherent

resonant scattering delayed by the slow decay of the nu-

clear transition. On the time scale of the nuclear decay,

the prompt radiation can be described by a Dirac-δ(t)-

like pulse and thus the outgoing ﬁeld in Nuclear Resonant

Scattering is of the form

Eout(t) = α[δ(t) + Si(t)] ∗Ein(t).(5)

Here, the prefactor αaccounts for attenuation and dis-

persion imposed by the surrounding nonresonant mate-

rial and Si(t) denotes the nuclear resonant part of the

target’s response. A two-target setup formed by a refer-

ence absorber foil Ta(analyzer) and an unknown tar-

get Ttin forward scattering geometry as depicted in

Fig. 1 can be described by the total response function

T(t)=(Tt∗Ta)(t). The response of the reference ab-

sorber with tunable transition frequency ωa+ ∆ and rel-

ative phase φbetween the prompt and scattered part can

be written as

Ta(t, ∆, φ) = αaδ(t) + e−i∆teiφSa(t).(6)

Note that, typically, we will consider thin reference ab-

sorbers whose spectral features are more narrow than

those of the target absorber. However, the subsequent

analysis does not employ approximations of the reference

absorber’s response function based on this thin-analyzer

limit, and our numerical results below will exhibit eﬀects

beyond this limit. In the following, for notational brevity,

we will absorb the detuning and phase dependence into

the nuclear scattering response as

Sa(t, ∆, φ) = e−i∆teiφSa(t),(7)

and suppress the dependence on φthroughout this Sec-

tion as phase control will become of relevance only later

in Sec.IV.

With these considerations, the experimentally accessi-

ble time- and frequency-dependent intensity at the de-

tector can be expressed in terms of response functions

I(t, ∆) = |(T∗Ein)(t)|2

= (T∗Ein)∗(t) (T∗Ein)(t),(8)

where the ∆-dependence arises via Tain the response

function T. Such two-dimensional time- and frequency-

resolved spectra allow for a much more stringent compar-

ison of experimental data to theory predictions than the

corresponding one-dimensional time-spectra or energy-

spectra alone, and have been measured in recent experi-

ments [20, 21, 42]

B. Frequency-frequency correlation spectrum

In order to discuss spectral correlations, we deﬁne the

frequency-frequency correlation (FFC) spectrum as the

Fourier-transform of the experimentally accessible inten-

sity Eq. (8) along the time axis,

I(ν, ∆) = Z∞

−∞

dt eiνt I(t, ∆) (9)

=E2

0Z∞

−∞

dt eiνt T∗(t)T(t)

=E2

2π(ˆ

T ? ˆ

T)(ν),(10)

where the initial pulse was written as Ein(t) = E0δ(t)

as it is typically orders of magnitude shorter than the

nuclear evolution time scales. Note that I(t, ∆) vanishes

for times t < 0 since the excitation occurs at t= 0. How-

ever, the symmetric integration will allow us to derive

simple analytical expressions for the case with detection

time gating in Sec. III D. Interestingly, the FFC spec-

trum Eq. (9) can be expressed as the (spectral) auto-

correlation of the total response function ˆ

T. The result

Eq. (10) can thus be regarded as a frequency-domain in-

stance of the Wiener-Khinchin theorem which relates the

Fourier transform of the power spectral density |T(t)|2

to the autocorrelation ( ˆ

T ? ˆ

T)(ν) (see, e.g., [43]). As

we will see in the following, the Fourier transformation

in Eq. (9) translates temporal interference eﬀects into

spectral correlations from which spectral features and

the phase information of the nuclear target can be ex-

tracted. However, it is important to note that the FFC

spectrum itself is not an intensity, as it is complex-valued

for general Fourier frequencies ν. The reason is that it

is derived via the Fourier transform from the experimen-

tally accessible real-valued intensity. Regarding previous

detection schemes using single-line reference absorbers

as mentioned in the introduction, we note that the FFC

spectrum reduces to the real-valued time-integrated spec-

trum for ν= 0 (cf. [25]). By only integrating over certain

parts of the time-axis, late-time and stroboscopic spec-

tra (see, e.g. [30]) can be recovered from the same ν= 0

contribution. In this sense, the FFC spectrum can be re-

garded as a generalization of these methods to arbitrary

Fourier frequencies νusing the same data set. The case

where the signal at early times is discarded, is discussed

in more detail in Sec. III D and appendix A. In the fol-

lowing, we focus on the ideal case in which all times are

available for data analysis in order to derive analytical

expressions for the most relevant features of these spec-

tra.

A numerical example for the FFC spectrum in Eq. (9)

is given in Fig. 1. The bottom-right panel shows the ex-

perimentally accessible time- and frequency-resolved in-

tensity I(t, ∆) in Eq. (8). The bottom-left panel shows

the real part of the FFC spectrum in Eq. (9), which is

dominated by a set of horizontal and diagonal spectral

features in the ∆−ν-plane. Note, that throughout this

paper, we only plot and analyze the positive νbranch

of the FFC spectrum since the inversion of the Fourier

frequency ν→ −νleads to complex conjugation of the

FFC signal. The main part of our FFC analysis, how-

ever, will be carried out on its real part or absolute value

and thus the negative νbranch contains only redundant

information.

For an interpretation of these diagonal features, we

rewrite the spectral auto-correlation function Eq. (10)

using the Fourier-domain response functions

T(ω) = ˆ

Tt(ω)ˆ

Ta(ω),(11)

Ta(ω, ∆, φ) = αah1 + ˆ

Sa(ω, ∆, φ)i(12)

FIG. 2. Decomposition of FFC spectra into single-target and two-target contributions. (a) shows the real part of the FFC

spectrum I(ν, ∆) after removal of the oﬀ-resonant background. It can be separated into two parts shown in (b) and (c). The

ﬁrst part in (b) is the sum of the individual responses of target and analyzer, corrected for the non-resonant absorption. The

horizontal structures are due to quantum beats in the target. As guide to the eye, the relevant mutual detunings between the

target transitions are indicated by dashed red lines. (c) is the diﬀerence between the spectra in (a) and (b). This spectrum

is dominated by diagonal structures, which originate from the interference of the resonant scattering oﬀ of the diﬀerent target

resonances with the resonant analyzer response. These results are obtained for a 2µm thick enriched α−57 Fe target with

hyperﬁne ﬁeld B= 33.3 T and a stainless-steel single-line analyzer as described in the main text. The level structure and the

relevant transitions in the target are shown in (d). The orange, red and green transitions describe the scattering of left-circularly,

linearly and right-circularly polarized light, respectively.

I(ν, ∆) = E2

0|αa|2

2πh(ˆ

Tt?ˆ

Tt)(ν)+(ˆ

Tt?ˆ

Ttˆ

Sa)(ν) (13)

+( ˆ

Ttˆ

Sa?ˆ

Tt)(ν)+(ˆ

Ttˆ

Sa?ˆ

Ttˆ

Sa)(ν)i.

Eq.(12) shows the clear separation of nonresonant (elec-

tronic) scattering, which is approximately constant on

the scale of the nuclear resonances, and the frequency-

dependent nuclear resonant scattering in form of the

nuclear resonant response Sa(ω, ∆, φ) in the frequency-

domain. This separation is crucial for the evaluation

of the FFC spectrum in the form of Eq.(13), the sin-

gle parts of which can be interpreted as follows: the

ﬁrst term describes spectral correlations between diﬀer-

ent transitions in the target, as will be further discussed

below. The other three terms contain the contribution

Ttˆ

Sa=αt[ˆ

Sa+ˆ

Saˆ

St] combining target and analyzer. Its

ﬁrst addend ∼ˆ

Saarises from the aforementioned non-

resonant (electronic) scattering in the target followed by

resonant (nuclear) scattering processes in the analyzer

foil. It forms the basis of the heterodyne-type detec-

tion schemes which determine the target response using

interference between radiation emitted from the target

and analyzer, respectively (cf. [32, 34]). It will turn out

to be the main origin of the diagonal structure found in

FFC spectra. The second addend ∼ˆ

Saˆ

St=ˆ

Sa,t, known

as radiative coupling [4, 5, 40, 44], describes processes

with resonant scattering in both analyzer and target. In

the following, we will exploit that these two scattering

contributions are naturally separated owing to their dif-

ferent scattering amplitudes as a function of detuning

between analyzer and target and will focus on the ﬁrst

contribution, as it dominates the FFC spectra in the large

analyzer-target detuning limit.

C. Large analyzer-target detuning limit

The resonantly scattered part of the analyzer response

Sa(ω) is nonzero in the vicinity of the resonance fre-

quency ω=ωa+ ∆ only. For the same reason, the full

target response becomes spectrally ﬂat far-oﬀ nuclear res-

onance, i.e., ˆ

Tt(ω)≈αt. As a result, we can approximate

Tt(ω)ˆ

Sa(ω)≈αtˆ

Sa(ω) (14)

in the limit of large detunings ∆ + ωa−ωtbetween the

analyzer and target transitions with frequencies ωt. In

this approximation, the spectral auto-correlation func-

tion can be written as

Ioﬀ (ν, ∆) ≈E2

0|αa|2

2πh(ˆ

Tt?ˆ

Tt)(ν) + |αt|2(ˆ

Sa?ˆ

Sa)(ν)

+αt(ˆ

Tt?ˆ

Sa)(ν) + α∗

t(ˆ

Sa?ˆ

Tt)(ν)i.(15)

The ﬁrst two terms correspond to the full target response

F[|Tt(t)|2] and the resonantly scattered part of the ana-

lyzer response F[|Sa(t)|2], where Fdenotes the Fourier

transform. These contributions can be determined from

separate measurements of the time-dependent scattered

intensity of the target and the analyzer alone, i.e., in the

absence of the respective other target. Their complex-

valued contributions to Eq. (15) then follow from a

Fourier transformation analogous to Eq. (9).

This separation of the single-stage contributions ˆ

Tt?ˆ

and ˆ

Sa?ˆ

Safrom the FFC spectrum is illustrated in

Fig. 2. The full FFC spectrum in panel (a) decom-

poses into the single-stage response parts shown in (b)

and the residual two-target contributions in (c). Clearly,

the single-stage responses correspond to the horizontal

lines in (a) since they do not depend on ∆, whereas the

two-target parts give rise to the diagonal lines. Note

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UnravelingTime-andFrequency-ResolvedNuclearResonantScatteringSpectraLukasWol1,andJorgEvers1,y1Max-Planck-InstitutfurKernphysik,Saupfercheckweg1,69117Heidelberg,Germany(Dated:December13,2022)Owingtotheirextremelynarrowline-widthsandexceptionalcoherenceproperties,Mossbauernucleiformapromisingplat...

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Unraveling Time- and Frequency-Resolved Nuclear Resonant Scattering Spectra Lukas Wol1and J org Evers1y 1Max-Planck-Institut f ur Kernphysik Saupfercheckweg 1 69117 Heidelberg Germany

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