$\textbf{D}=-\frac{15120\left(\alpha_{3}k_{B}^{7}V\zeta(7)\right)}{\pi^{2}c^{6}\hbar^{6}}$
$\textbf{E}=\frac{224\pi^{6}k_{B}^{8}V\left(4\alpha_{2}^{2}-\alpha_{4}\right)}{15c^{7}\hbar^{7}}$
$\textbf{K}=\frac{1451520k_{B}^{9}V\zeta(9)(9\alpha_{2}\alpha_{3}-\alpha_{5})}{\pi^{2}c^{8}\hbar^{8}}$
$\textbf{L}=-\frac{640\left(\pi^{8}k_{B}^{10}V\left(55\alpha_{2}^{3}-30\alpha_{2}\alpha_{4}-15\alpha_{3}^{2}+3\alpha_{6}\right)\right)}{11\left(c^{9}\hbar^{9}\right)}$
where $\zeta$ is the Riemann zeta function.

It is clear from this expansion that, contrarily to what is generally
believed, the coefficients of the specific heat series expansion are not
associated to a single coefficient of the dispersion relation series; E, for
instance, contains both $\alpha_{2}$ and $\alpha_{4}$.

The $\alpha_{i}$ coefficients have been determined in Section VII.1 by fits at
low wave-vectors. For the dispersion curve at saturated vapor pressure, the
coefficients c=238.3 m/s, $\alpha_{1}$=0, $\alpha_{2}$=1.55Å-2,
$\alpha_{3}$=-4.04Å-3, $\alpha_{4}$=2.30Å-4, $\alpha_{5}$=0, $\alpha_{6}$=0
provide a good fit of the dispersion curve for k$<$0.5 Å-1. The corresponding
coefficients of the specific heat series are therefore: A=0.0831, C=-0.0548,
D=0.0653, E=0.0603, K=-0.262, L=0.141, when temperatures are expressed in
kelvin and CV in J/(mol.K).

Strong oscillations in the amplitude of successive terms of the series
expansions are present for temperatures $\approx$1 K. They originate from an
attempt to describe the dispersion relation by a power series expansion around
k=0 while (as seen in Fig. 26) the phase velocity has a very linear portion,
quite extended, far from k=0. This quite unusual behavior has several
consequences. Clearly, for k$>$0.5Å-1, the higher order coefficients will vary
with the order of the polynomial, and once a polynomial providing a good
description of the dispersion relation is chosen, it must be consistently used
in the thermodynamic calculations. The previous discussion may appear as
academic, but it is at the root of the problems encountered in the analysis of
specific heat measurements Phillips  _et al._ (1970); Greywall (1978, 1980),
as is illustrated in Fig. 54. The partial contribution due to the phonons
estimated by these authors (green dashed line for Phillips et al., blue dash-
dotted line for Greywall), has the right order of magnitude, but it deviates
from the present neutron scattering result (black dashed line) above 0.5 K; in
particular, the temperature dependence is clearly different.

Figure 54: CV/T3 showing the phonon contribution to the specific heat and the
onset of the roton contribution. Dots: CV measurements Phillips  _et al._
(1970); Greywall (1978, 1980) with their fitted phonon contributions (see
legend). Present work: a) red dotted line: analytical result with K=L=0; b)
red dash-dotted line: analytical result with K$\neq$0 and L$\neq$0; c) black
solid line: numerical integration of full dispersion curve; d) dashed line:
numerical integration over the phonon region (0$<$k$<$kM). See Fig. 52 for
additional details.

The _analytical_ calculations of the specific heat using series expansions of
the dispersion curve display strong deviations with respect to the full
numeric result, beginning at temperatures as low as 0.4 K. Furthermore, the
series expansions used in the previous analysis of specific heat data Phillips
_et al._ (1970); Greywall (1978, 1980), being truncated, cannot be directly
used to extract the dispersion relation coefficients. As shown in the figure,
removing the terms K and L strongly affects the fits. Handled with care, the
analytical expansions are still very useful for the calculation of phonon
thermal properties in different contexts.

####  VIII.2.2 Roton region

The analytical calculation of the roton contribution to the thermodynamical
properties was already performed by Landau in his seminal papers on roton
excitations Landau (1941, 1947). In the vicinity of the roton minimum, the
excitation energies are described by Eq. 18. In the limiting case where
BR=CR=0, Landau obtains the expression

$C_{V}^{R}=C_{\textit{L}}e^{-\frac{\Delta_{R}}{k_{B}T}}\left(1+\frac{k_{B}T}{\Delta_{R}}+\frac{3}{4}\left(\frac{k_{B}T}{\Delta_{R}}\right)^{2}\right)$
(23)

with
$C_{\textit{L}}=\frac{\Delta_{R}^{2}k_{R}^{2}V\sqrt{\mu_{R}m_{4}}}{\sqrt{2}\pi^{3/2}\sqrt{k_{B}}T^{3/2}\hbar}$

The analysis of the measured specific heats with Landau’s analytical
expression is therefore sensitive to a combination of several roton parameters
(energy, wave-vector and effective mass).

Landau also assumes that $\left|k-k_{R}\right|<<k_{R}$. The analytical
calculation, however, can be done without this last assumption. We calculate
here the average energy of the rotons ER and their specific heat C${}_{V}^{R}$
with Eq. 20 using Eq. 18 with BR=CR=0, including in the integrand all terms of
the expansion of $(k-k_{R})^{2}$. The calculation is performed by integration
over wave-vectors. We find an additional term

$C_{V}^{addit}=C_{\textit{L}}e^{-\frac{\Delta_{R}}{k_{B}T}}\left(\frac{T}{2\tau_{R}}\right)\left(1+\frac{3k_{B}T}{\Delta_{R}}+\frac{15}{4}\left(\frac{k_{B}T}{\Delta_{R}}\right)^{2}\right)$
(24)

where we introduced the parameter $\tau_{R}$ which has the dimensions of
temperature: $\tau_{R}$ $=\frac{k_{R}^{2}\hbar^{2}}{2k_{B}\mu_{R}m_{4}}$. The
additional term is similar to Landau’s expression, with an additional
coefficient T/(2$\tau_{R}$). Since $\tau_{R}$ $\approx$ 158 K, this additional
term is of the same magnitude as the third term in Eq. 23 for temperatures
above 1 K, and they are both very small.

In order to improve the accuracy of Landau’s analytic description, it is
necessary to take into account the asymmetry (cubic term) and the deformation
of the minimum (quartic term). Unfortunately, the corresponding expressions
are extremely complex and cumbersome, they are not of practical interest. At
the analytical level, we are therefore left with equations providing a rough
approximation. As seen in Fig. 55, the difference between Landau’s
approximation and the roton contribution calculated numerically with the
dispersion curve grows substantially with the temperature. The figure also
illustrates the fact that phonons, defined as excitations below the maxon
wave-vector, have a non-negligible contribution to the thermodynamic
properties at higher temperatures than commonly believed: the phonon and roton
contributions become equal at 0.77 K for the heat capacity, for instance.

Figure 55: The total heat capacity (black solid line), and phonon (blue dash-
dotted line), roton (red dashed line) contributions to the specific heat
calculated numerically with the measured dispersion relation (present work).
Short-dash black line: Landau formula for the rotons contribution. Crosses:
specific heat measurements Phillips  _et al._ (1970).

###  VIII.3 Normal fluid density

The normal fluid density is given Landau (1941, 1947); Fetter and Walecka
(2003) by the expression

$\rho_{n}=\left(\frac{\hbar^{2}}{6\pi^{2}k_{B}T}\right)\int_{0}^{k_{end}}\frac{e^{\frac{\epsilon(k)}{k_{B}T}}}{e^{\frac{\epsilon(k)}{k_{B}T}}-1}k^{4}dk$
(25)

One can obviously integrate analytically the series expansions within the
limitations discussed previously. In order to take full advantage of the
measured dispersion curve $\epsilon(k)$, it is convenient to perform a
numerical integration, as done above for the specific heat. The normal fluid
densities calculated using the dispersion curve are given in Table 8 in the
temperature range up to 1.3 K. The result is compared in Fig. 56 to selected
experimental data Donnelly and Barenghi (1998). The latter cover the
temperature range from 1.2 K to the lambda point.

Figure 56: Normal fluid density calculated using the measured dispersion
relation. Experimental points by Maynard, Tam and Ahlers, and Singass and
Ahlers (data-base in Ref. Donnelly and Barenghi, 1998)

.

The percent difference between previous data Brooks and Donnelly (1978);
Donnelly and Barenghi (1998) and our calculated curve is shown in Fig. 57, on
the one hand for the experimental data-base, and on the other hand for the
tabulated values. The latter were calculated below 1 K using spline
approximations to different neutron data sets. We see that for the purposes of
calculating $\rho_{n}$, our result matches accurately the direct measurements
at about 1.25 K. Above this temperature, the dispersion curve itself becomes
temperature dependent, a problem that is beyond the scope of the present
article. It is therefore extremely satisfactory to see that these two sets of
data agree particularly well. It is also clear that neutron techniques extend
considerably towards lower temperatures the range where this parameter has
been accurately determined. The temperature range below 1 K is a particularly
fertile playground for objects all possible sizes immersed in helium and
sensitive to its excitations, from nano-oscillators to huge particle detectors
Guénault  _et al._ (2019, 2020, 2019, 2020); Guo and McKinsey (2013);
Morishita  _et al._ (1989); Schutz and Zurek (2016); Maris  _et al._ (2017);
Knapen  _et al._ (2017).

Figure 57: Deviation plot for the normal fluid density. We show the percent
difference between Maynard’s experimental data, or tabulated (essentially,
calculated) values Brooks and Donnelly (1978); Donnelly and Barenghi (1998),
and the values calculated using the dispersion relation, taken as the
reference (this work). The ‘simple Landau model’ (see text) deviates
substantially from our data.

Landau’s general formula, Eq. 25, yields when applied to the first term of the
series expansions for the phonon and the roton dispersion relation (the so-
called ‘simple Landau model’ Landau (1941, 1947)) :
$\rho_{n}^{(L)}=\rho_{n}^{Ph(L)}+\rho_{n}^{R(L)}$, with

$\rho_{n}^{Ph(L)}=\frac{2\pi^{2}{k_{B}}^{4}T^{4}}{45c^{5}\hbar^{3}}$
$\rho_{n}^{R(L)}=\frac{\hbar
k_{R}^{4}(\mu_{R}m_{4})^{1/2}}{3\sqrt{2}\pi^{3/2}(k_{B}T)^{1/2}}e^{-\frac{\Delta_{R}}{k_{B}T}}$

Since these formulas are frequently used, it is interesting to check their
validity range. A comparison between the expressions above and our complete
numerical calculation with the measured dispersion relation, is shown in Fig.
57. Deviations on the order of 10% are found below 1 K, even around 0.5 K
where this expression is often assumed to give a good approximation. The
deviation is even larger if one considers the phonon and roton contributions
separately, as shown in Fig. 58. The superfluid densities are defined as
$\rho_{s}=\rho-\rho_{n}$ (see Table 8).

Figure 58: Normal fluid density deviation plot: percent difference between the ‘simple Landau model’ normal densities Landau (1941, 1947); Fetter and Walecka (2003), and our numerical calculation with the full dispersion curve. Phonon, roton, and total normal density deviations are defined as 100 ($\rho_{n}^{Ph(L)}$-$\rho_{n}^{Ph}$)/$\rho_{n}^{Ph}$, 100 ($\rho_{n}^{R(L)}$-$\rho_{n}^{R}$)/$\rho_{n}^{R}$, and 100 $((\rho_{n}^{Ph(L)}+\rho_{n}^{R(L)}$) - $\rho_{n}$)/$\rho_{n}$, respectively.  $T$ | $\rho$ | $\rho_{n}$ | $\rho_{n}^{Ph}$ | $\rho_{n}^{R}$
---|---|---|---|---
K | g/cm3 | g/cm3 | g/cm3 | g/cm3
0 | 0.14514 | – | – | –
0.05 | 0.14514 | 1.103E-10 | 1.103E-10 | 3.94E-74
0.10 | 0.14514 | 1.754E-09 | 1.754E-09 | 6.65E-37
0.15 | 0.14514 | 8.799E-09 | 8.799E-09 | 1.566E-24
0.20 | 0.14514 | 2.749E-08 | 2.749E-08 | 2.305E-18
0.25 | 0.14514 | 6.626E-08 | 6.626E-08 | 1.130E-14
0.30 | 0.14514 | 1.355E-07 | 1.355E-07 | 3.213E-12
0.35 | 0.14514 | 2.476E-07 | 2.474E-07 | 1.798E-10
0.40 | 0.14514 | 4.196E-07 | 4.159E-07 | 3.649E-09
0.45 | 0.14514 | 6.945E-07 | 6.567E-07 | 3.772E-08
0.50 | 0.14514 | 1.230E-06 | 9.872E-07 | 2.432E-07
0.55 | 0.14514 | 2.540E-06 | 1.427E-06 | 1.113E-06
0.60 | 0.14514 | 5.939E-06 | 1.997E-06 | 3.943E-06
0.65 | 0.14513 | 1.419E-05 | 2.722E-06 | 1.147E-05
0.70 | 0.14513 | 3.220E-05 | 3.631E-06 | 2.857E-05
0.75 | 0.14513 | 6.769E-05 | 4.757E-06 | 6.293E-05
0.80 | 0.14513 | 1.316E-04 | 6.141E-06 | 1.254E-04
0.85 | 0.14513 | 2.381E-04 | 7.835E-06 | 2.303E-04
0.90 | 0.14512 | 4.048E-04 | 9.901E-06 | 3.95E-04
0.95 | 0.14512 | 6.519E-04 | 1.242E-05 | 6.39E-04
1.00 | 0.14512 | 0.001002 | 1.549E-05 | 9.87E-04
1.05 | 0.14512 | 0.001480 | 1.922E-05 | 0.001461
1.10 | 0.14511 | 0.002112 | 2.375E-05 | 0.00209
1.15 | 0.14511 | 0.002925 | 2.923E-05 | 0.00289
1.20 | 0.14512 | 0.003945 | 3.585E-05 | 0.00391
1.25 | 0.14512 | 0.005201 | 4.378E-05 | 0.00516
1.30 | 0.14512 | 0.006720 | 5.322E-05 | 0.00667
Table 8: Total density $\rho$ (from Ref. Donnelly and Barenghi, 1998) and
normal fluid densities (total, phonon and roton contributions) as a function
of temperature, calculated numerically using the measured dispersion relation
(this work). See Supplemental Material at [URL will be inserted by publisher]
for more detailed tables.

###  VIII.4 Number density of phonons and rotons

The number density of phonons $N_{Ph}$ and rotons $N_{R}$ as a function of
temperature, of interest for instance in the ballistic regime, is given by the
expressions:

$N_{Ph}=(2\pi^{2})^{-1}\int_{0}^{k_{M}}(e^{\frac{\epsilon(k)}{k_{B}T}}-1)^{-1}k^{2}dk$

$N_{R}=(2\pi^{2})^{-1}\int_{k_{M}}^{k_{end}}(e^{\frac{\epsilon(k)}{k_{B}T}}-1)^{-1}k^{2}dk$

The result is shown Fig. 59 (see Supplemental Material at [URL will be
inserted by publisher] for data tables.), converted to the number of phonons
and rotons per helium atom, in order to emphasize the importance of the
excitation density as the temperature exceeds T$\approx$1.3 K. Near the lambda
point, the roton density, an atomic-size excitation, approaches one per atom,
as expected.

Figure 59: Number of phonon and roton excitations per atom. The numerical
calculation performed with the measured low temperature dispersion curve, is
accurate for T$<$1.3 K.

##  IX Conclusions

The main result of this work is the determination of the dispersion relation
$\epsilon(k)$ of superfluid 4He in the whole wave-vector range at saturated
vapor pressure, and in a large range for pressures up to 24 bar. Earlier
neutron data were scarce, of low resolution, often contradictory. Roton and
maxon parameters were affected by the instrumental resolution and the wave-
vector range over which the dispersion curve was averaged out, leading to a
substantial spread in the values found in the literature; as the experimental
techniques progressively improved, lower roton energies, wave-vectors and
effective masses were obtained. Motivated by the lack of high accuracy data,
Donnelly and coworkers Brooks and Donnelly (1978); Donnelly and Barenghi
(1998) designed a ‘recommended dispersion relation’, a spline curve making its
way across the large error bars of some of the data available at that time.

A _measured_ dispersion curve is now available. It is characterized by small
error bars, and it agrees well with measurements performed by other
techniques: ultrasound and compressibility at low wave-vectors, as well as
heat capacity, a technique providing an important global test over a large
wave-vector range.

Our neutron measurements have been performed using time of flight (TOF)
techniques with a 105 pixels detector matrix. Statistical errors are in
general negligible in this work. Considerable effort has been devoted to
understand the sources of systematic errors and the resulting corrections.
This also allowed us to identify problems that affected earlier measurements.

The energy scale of our spectrometer has been refined by a calibration at a
single point, the roton energy $\Delta_{R}$=0.7418 meV determined with an
uncertainty of 1 $\mu$eV by Stirling Stirling (1983); Glyde (1994); Stirling
(1991). This is the main source of uncertainty of our results for
$\epsilon$(k). It affects them in a global way, and they can therefore be
corrected (the energies should be corrected proportionally, as indicated by
Eq. 13, and the wave-vectors recalculated using Eq. 11) if a more accurate
value of $\Delta_{R}$ becomes available. The remaining systematic errors
originate from small defects in the IN5 spectrometer construction. Comparing
data at two different neutron energies has allowed us to show that the
corresponding effects are smaller than the global systematic error quoted
above. They can be seen as oscillations in the curves, in particular around
the roton minimum. No effort was done to suppress these by averaging or
smoothing, since they are a good indicator, for further applications of the
present data, of residual systematic errors.

The measured dispersion curve has allowed us to calculate the thermodynamic
properties of superfluid helium in the temperature range below 1.3 K, where
the non-interacting-excitations picture is valid. Our results provide the most
accurate values for the heat capacity (direct measurements are affected by
delicate issues of thermometry and temperature scales), the normal fluid
density, and other thermodynamical properties. They also provide independently
the contributions from excitations in different wave-vector ranges, and in
particular the phonon and the roton contributions. This information is of
interest, for instance, for low temperature transport phenomena, the damping
of nano-resonators in helium below 1 K Guénault  _et al._ (2019, 2020),
particle detection Schutz and Zurek (2016); Maris  _et al._ (2017); Knapen
_et al._ (2017), and several other effects determined by ballistic phonons and
roton quasiparticles.

In addition to these numerical calculations, we developed analytical
expressions for the thermodynamical properties. We found that specific heat
measurements were analyzed using inconsistent low temperature series
expansions; no disagreement is observed between the neutron results and the
specific heat data using our coherent expressions.

The results on pure superfluid 4He have analogues in other systems. Rotons and
maxons are studied in 4He in reduced dimensions, confined geometries, droplets
spectrometry Krotscheck and Navarro (2002); Bryan and Sokol (2018); Nicolis
and Penco (2018); Bossy  _et al._ (2019); Moroshkin and Kono (2019); Escartín
_et al._ (2019), but also in 3He Godfrin  _et al._ (2012); Pines (2016), in
cold atomic gases Santos  _et al._ (2003); Chomaz  _et al._ (2018); McCormack
_et al._ (2020), classical liquids Trachenko and Brazhkin (2015), etc.: they
are a general feature of many-body interacting systems.

Throughout this manuscript, we have considered ‘elementary excitations’ (poles
of the fully renormalized single-particle propagator) and ‘statistical quasi-
particles’ (elementary excitation energies defined as the functional
derivative of the total energy with respect to the distribution function), in
the sense described by Balian and de Dominicis Balian and de Dominicis (1971);
Carneiro and Pethick (1975); Carneiro (1979), as essentially identical
concepts. The description of the thermodynamic properties using a self-
consistent temperature dependent dispersion relation has been explored by
Donnelly and Roberts Donnelly and Roberts (1977), it describes
phenomenologically the thermodynamic properties. The other route, presently
explored by different types of many-body microscopic calculations, seems more
promising. The difference between the measured specific heat and that
calculated for non-interacting quasi-particles using the dispersion curve has
a simple behavior (see Fig. 50), and provides information about roton-roton
interactions Cohen (1960); Fåk  _et al._ (2012). Additional contributions have
been predicted Carneiro and Pethick (1975); Carneiro (1979) for the heat
capacity, it would be interesting to have a theoretical estimate of their
magnitude and make a quantitative comparison with the present data.

##  X Acknowledgements

We are grateful to S. Triqueneaux and X. Tonon for their help with the
experiments, to B. Guérard and P. van Esch for helpful discussions on the IN5
detectors properties, and to H. Maris for pointing out Ref. Rugar and Foster,
1984 to us. This work was supported by the European Microkelvin Platform. The
research leading to these results has received funding from the European
Union’s Horizon 2020 Research and Innovation Programme, under Grant Agreement
no 824109.

## References

  * Nozières and Pines (2018) P. Nozières and D. Pines,  _The Theory of Quantum Liquids_ , Advanced Book Classics (CRC Press, 2018).
  * Fetter and Walecka (2003) A. L. Fetter and J. D. Walecka,  _Quantum Theory of Many-Particle Systems_ (Dover Publications, Mineola, NY, 2003).
  * Thouless (2014) D. J. Thouless,  _The Quantum Mechanics of Many-Body Systems_ , 2nd ed. (Dover Publications, Mineola, NY, 2014).
  * Landau (1941) L. Landau, J. Phys. (Moscow) 5, 71 (1941).
  * Landau (1947) L. Landau, J. Phys. (Moscow) 11, 91 (1947).
  * Bogoliubov (1947) N. N. Bogoliubov, J. Phys. U.S.S.R 11, 23 (1947).
  * Jackson  _et al._ (1981) A. D. Jackson, B. K. Jennings, R. A. Smith, and A. Lande, Phys. Rev. B 24, 105 (1981).
  * Nozières (2004) P. Nozières, J. Low Temp. Phys. 137, 45 (2004).
  * Volovik (2003) G. E. Volovik,  _The Universe in a Helium Droplet_ , Vol. 117 (Oxford University Press on Demand, 2003).
  * Wen (2004) X.-G. Wen,  _Quantum Field Theory of Many-Body Systems: from the Origin of Sound to an Origin of Light and Electrons_ (Oxford University Press on Demand, 2004).
  * Glyde (1994) H. R. Glyde,  _Excitations in Liquid and Solid Helium_ , Oxford Series on Neutron Scattering in Condensed Matter (Clarendon, Oxford, 1994).
  * Glyde (2017) H. R. Glyde, Rep. Prog. Phys. 81, 014501 (2017).
  * Aldrich and Pines (1976) C. H. Aldrich and D. Pines, J. Low Temp. Phys. 25, 677 (1976).
  * Lee and Lee (1975) D. K. Lee and F. J. Lee, Phys. Rev. B 11, 4318 (1975).
  * Feenberg (1969) E. Feenberg,  _Theory of Quantum Fluids_ (Academic, New York, 1969).
  * Krotscheck (1986) E. Krotscheck, Phys. Rev. B 33, 3158 (1986).
  * Boronat and Casulleras (1994) J. Boronat and J. Casulleras, Phys. Rev. B 49, 8920 (1994).
  * Ceperley (1995) D. M. Ceperley, Rev. Mod. Phys. 67, 279 (1995).
  * Boronat and Casulleras (1997) J. Boronat and J. Casulleras, Europhys. Lett. 38, 291 (1997).
  * Pistolesi (1998a) F. Pistolesi, Phys. Rev. Lett. 81, 397 (1998a).
  * Fabrocini  _et al._ (2002) A. Fabrocini, S. Fantoni, and E. Krotscheck,  _Introduction to Modern Methods of Quantum Many–Body Theory and their Applications_ , Advances in Quantum Many–Body Theory, Vol. 7 (World Scientific, Singapore, 2002).
  * Krotscheck and Navarro (2002) E. Krotscheck and J. Navarro,  _Microscopic Approaches to Quantum Liquids in Confined Geometries_ , Advances in Quantum Many–Body Theory, Vol. 4 (World Scientific, Singapore, 2002).
  * Vitali  _et al._ (2010) E. Vitali, M. Rossi, L. Reatto, and D. E. Galli, Phys. Rev. B 82, 174510/1 (2010).
  * Ferré and Boronat (2016) G. Ferré and J. Boronat, Phys. Rev. B 93, 104510 (2016).
  * Griffin (1993) A. Griffin,  _Excitations in a Bose-condensed Liquid_ , Vol. 4 (Cambridge University Press, 1993).
  * Guénault  _et al._ (2019) A. M. Guénault, A. Guthrie, R. P. Haley, S. Kafanov, Y. A. Pashkin, G. R. Pickett, M. Poole, R. Schanen, V. Tsepelin, D. E. Zmeev, E. Collin, O. Maillet, and R. Gazizulin, Phys. Rev. B 100, 020506(R) (2019).
  * Guénault  _et al._ (2020) A. M. Guénault, A. Guthrie, R. P. Haley, S. Kafanov, Y. A. Pashkin, G. R. Pickett, V. Tsepelin, D. E. Zmeev, E. Collin, R. Gazizulin, and O. Maillet, Phys. Rev. B 101, 060503(R) (2020).
  * Guo and McKinsey (2013) W. Guo and D. N. McKinsey, Phys. Rev. D 87, 115001 (2013).
  * Schutz and Zurek (2016) K. Schutz and K. M. Zurek, Phys. Rev. Lett. 117, 121302 (2016).
  * Maris  _et al._ (2017) H. J. Maris, G. M. Seidel, and D. Stein, Phys. Rev. Lett. 119, 181303 (2017).
  * Knapen  _et al._ (2017) S. Knapen, T. Lin, and K. M. Zurek, Phys. Rev. D 95, 056019 (2017).
  * Beauvois  _et al._ (2016) K. Beauvois, C. E. Campbell, J. Dawidowski, B. Fåk, H. Godfrin, E. Krotscheck, H.-J. Lauter, T. Lichtenegger, J. Ollivier, and A. Sultan, Phys. Rev. B 94, 024504 (2016).
  * Beauvois  _et al._ (2018) K. Beauvois, J. Dawidowski, B. Fåk, H. Godfrin, E. Krotscheck, J. Ollivier, and A. Sultan, Phys. Rev. B 97, 184520 (2018).
  * Campbell  _et al._ (2015) C. E. Campbell, E. Krotscheck, and T. Lichtenegger, Phys. Rev. B 91, 184510 (2015).
  * Beauvois  _et al._ (2019) K. Beauvois, H. Godfrin, E. Krotscheck, and R. Zillich, J. Low Temp. Phys. 197, 113 (2019).
  * Junker and Elbaum (1977) W. R. Junker and C. Elbaum, Phys. Rev. B 15, 162 (1977).
  * Rugar and Foster (1984) D. Rugar and J. S. Foster, Phys. Rev. B 30, 2595 (1984).
  * Maris and Massey (1970) H. J. Maris and W. E. Massey, Phys. Rev. Lett. 25, 220 (1970).
  * Maris (1977) H. J. Maris, Rev. Mod. Phys. 49, 341 (1977).
  * Sridhar (1987) R. Sridhar, Physics Reports 146, 259 (1987).
  * Maris (1973) H. J. Maris, Phys. Rev. A 8, 1980 (1973).
  * Kemoklidze and Pitaevskii (1970) M. P. Kemoklidze and L. P. Pitaevskii, Zh. Exsp. Teor. Fiz. 59, 2187 (1970), [Sov. Phys. JETP 32, 1183 (1971)].
  * Feenberg (1971) E. Feenberg, Phys. Rev. Lett. 26, 301 (1971).
  * Aldrich  _et al._ (1976) C. Aldrich, C. Pethick, and D. Pines, J. Low Temp. Phys 25, 691 (1976).
  * Abraham  _et al._ (1970) B. M. Abraham, Y. Eckstein, J. B. Ketterson, M. Kuchnir, and P. R. Roach, Phys. Rev. A 1, 250 (1970).
  * Donnelly and Barenghi (1998) R. J. Donnelly and C. F. Barenghi, J. Phys. Chem. Ref. Data 27, 1217 (1998), https://doi.org/10.1063/1.556028 .
  * Tanaka  _et al._ (2000) E. Tanaka, K. Hatakeyama, S. Noma, and T. Satoh, Cryogenics 40, 365 (2000).
  * Greywall (1978) D. S. Greywall, Phys. Rev. B 18, 2127 (1978).
  * Phillips  _et al._ (1970) N. E. Phillips, C. G. Waterfield, and J. K. Hoffer, Phys. Rev. Lett. 25, 1260 (1970).
  * Greywall (1980) D. S. Greywall, Phys. Rev. B 21, 1329(E) (1980).
  * Brooks and Donnelly (1978) J. S. Brooks and R. J. Donnelly, J. Phys. Chem. 6, 51 (1978).
  * Cowley and Woods (1971) R. A. Cowley and A. D. B. Woods, Can. J. Phys. 49, 177 (1971), http://dx.doi.org/10.1139/p71-021 .
  * Woods  _et al._ (1977) A. D. B. Woods, P. A. Hilton, R. Scherm, and W. Stirling, J. Phys. C 10, L45 (1977).
  * Svensson  _et al._ (1975) E. Svensson, P. Martel, and A. Woods, Phys. Lett. 55, 151 (1975).
  * Svensson  _et al._ (1976) E. Svensson, P. Martel, V. Sears, and A. Woods, Can. J. Phys. 54, 2178 (1976).
  * Svensson  _et al._ (1978) E. C. Svensson, R. Scherm, and A. D. Woods, J. Phys. (Paris) Colloques 39-C6, 211 (1978).
  * Stirling (1983) W. G. Stirling, in  _$75^{\rm th}$ Jubilee Conference on Helium-4_, edited by J. G. M. Armitage (World Scientific, Singapore, 1983).
  * Stirling (1985) W. G. Stirling, in  _Proceedings of the Second International Conference on Phonon Physics_ , edited by J. Kollàr, N. Kroo, M. Meynhard, and T. Siklos (World Scientific, Singapore, 1985) p. 829.
  * Stirling (1991) W. G. Stirling, in  _Excitations in Two-Dimensional and Three-Dimensional Quantum Fluids_ , NATO Advanced Study Institute, Series B: Physics, Vol. 257, edited by A. F. G. Wyatt and H. J. Lauter (Plenum, New York, 1991) pp. 25–46.
  * Talbot  _et al._ (1988) E. F. Talbot, H. R. Glyde, W. G. Stirling, and E. C. Svensson, Phys. Rev. B 38, 11229 (1988).
  * Dietrich  _et al._ (1972) O. W. Dietrich, E. H. Graf, C. H. Huang, and L. Passell, Phys. Rev. A 5, 1377 (1972).
  * Stirling, W. G.  _et al._ (1978) Stirling, W. G., Copley, J. R. D., and Hilton, P. A., in  _IAEA Symposium, Neutron Inelastic Scattering, Vienna_ , Vol. III (1978) pp. 45–52.
  * Andersen (1991) K. H. Andersen,  _A neutron scattering study of liquid 4He_, Ph.D. thesis, Keele University (1991).
  * Fåk and Andersen (1991) B. Fåk and K. H. Andersen, Phys. Lett. A 160, 468 (1991).
  * Andersen  _et al._ (1992) K. H. Andersen, W. G. Stirling, R. Scherm, A. Stunault, B. Fåk, H. Godfrin, and A. Dianoux, Physica B: Condensed Matter 180, 851 (1992).
  * Andersen  _et al._ (1994) K. H. Andersen, W. G. Stirling, R. Scherm, A. Stunault, B. Fåk, H. Godfrin, and A. J. Dianoux, J. Phys. Condens. Matter 6, 821 (1994).
  * Andersen and Stirling (1994) K. H. Andersen and W. G. Stirling, J. Phys. Cond. Matter 6, 5805 (1994).
  * Gibbs (1996) M. R. Gibbs,  _The collective excitations of superfluid 4He: the dependence on pressure and the effect of restricted geometry_, Ph.D. thesis, Keele University (1996).
  * Gibbs  _et al._ (1999) M. R. Gibbs, K. H. Andersen, W. G. Stirling, and H. Schober, J. Phys.: Condens. Matter 11, 603 (1999).
  * Glyde  _et al._ (1998) H. R. Glyde, M. R. Gibbs, W. G. Stirling, and M. A. Adams, Europhysics Letters 43, 422 (1998).
  * Pearce  _et al._ (2001a) J. Pearce, R. Azuah, W. Stirling, R. Dimeo, P. Sokol, and M. Adams, J. Low Temp. Phys. 124, 573 (2001a).
  * Andersen  _et al._ (1996) K. H. Andersen, J. Bossy, J. C. Cook, O. G. Randl, and J.-L. Ragazzoni, Phys. Rev. Lett. 77, 4043 (1996).
  * Lovesey (1984) S. W. Lovesey,  _Theory of Neutron Scattering from Condensed Matter_ (Clarendon Press, Oxford, 1984).
  * Schober (2014) H. Schober, Journal of Neutron Research 17, 109 (2014).
  * Ollivier  _et al._ (2010) J. Ollivier, H. Mutka, and L. Didier, Neutron News 21, 22 (2010), https://doi.org/10.1080/10448631003757573 .
  * Ollivier and Mutka (2011) J. Ollivier and H. Mutka, J. Phys. Soc. Jpn 80, SB003 (2011), https://doi.org/10.1143/JPSJS.80SB.SB003 .
  * (77) LAMP (Large Array Manipulation Program) http://www.ill.eu/data_treat/lamp.
  * Davison (1966) W. D. Davison, Proc. Phys. Soc. London 87, 133 (1966).
  * Boronat (2002) J. Boronat, in  _Microscopic Approaches to Quantum Liquids in Confined Geometries_ , edited by E. Krotscheck and J. Navarro (World Scientific, Singapore, 2002) pp. 21–90.
  * Feynman (1954) R. P. Feynman, Phys. Rev. 94, 262 (1954).
  * Boninsegni and Ceperley (1996) M. Boninsegni and D. M. Ceperley, J. Low Temp. Phys. 104, 339 (1996).
  * Boronat and Casulleras (1998) J. Boronat and J. Casulleras, J. Low Temp. Phys. 110, 443 (1998).
  * Fåk and Bossy (1998a) B. Fåk and J. Bossy, J. Low Temp. Phys. 112, 1 (1998a).
  * Fåk and Bossy (1998b) B. Fåk and J. Bossy, J. Low Temp. Phys. 113, 531 (1998b).
  * Pistolesi (1998b) F. Pistolesi, J. Low Temp. Phys. 113, 597 (1998b).
  * Pearce  _et al._ (2001b) J. V. Pearce, R. T. Azuah, B. Fåk, A. R. Sakhel, H. R. Glyde, and W. G. Stirling, J. Phys.: Condens. Matter 13, 4421 (2001b).
  * Donnelly  _et al._ (1981) R. J. Donnelly, J. A. Donnelly, and R. N. Hills, J. Low Temp. Phys. 44, 471 (1981).
  * Donnelly and Roberts (1977) R. J. Donnelly and P. H. Roberts, J. Low Temp. Phys 27, 687 (1977).
  * Wiebes (1969) J. Wiebes, Ph.D. thesis, Kammerlingh Onnes Laboratory (1969).
  * Morishita  _et al._ (1989) M. Morishita, T. Kuroda, A. Sawada, and T. Satoh, J. Low Temp. Phys. 76, 387 (1989).
  * Bryan and Sokol (2018) M. S. Bryan and P. E. Sokol, Phys. Rev. B 97, 184511 (2018).
  * Nicolis and Penco (2018) A. Nicolis and R. Penco, Phys. Rev. B 97, 134516 (2018).
  * Bossy  _et al._ (2019) J. Bossy, J. Ollivier, and H. R. Glyde, Phys. Rev. B 99, 165425 (2019).
  * Moroshkin and Kono (2019) P. Moroshkin and K. Kono, Phys. Rev. B 99, 104512 (2019).
  * Escartín  _et al._ (2019) J. M. Escartín, F. Ancilotto, M. Barranco, and M. Pi, Phys. Rev. B 99, 140505(R) (2019).
  * Godfrin  _et al._ (2012) H. Godfrin, M. Meschke, H.-J. Lauter, A. Sultan, H. M. Böhm, E. Krotscheck, and M. Panholzer, Nature 483, 576 (2012).
  * Pines (2016) D. Pines, Rep. Prog. Phys. 79, 092501 (2016).
  * Santos  _et al._ (2003) L. Santos, G. V. Shlyapnikov, and M. Lewenstein, Phys. Rev. Lett. 90, 250403 (2003).
  * Chomaz  _et al._ (2018) L. Chomaz, R. M. van Bijnen, D. Petter, G. Faraoni, S. Baier, J. H. Becher, M. J. Mark, F. Waechtler, L. Santos, and F. Ferlaino, Nature Physics 14, 442 (2018).
  * McCormack  _et al._ (2020) G. McCormack, R. Nath, and W. Li, Phys. Rev. A 102, 023319 (2020).
  * Trachenko and Brazhkin (2015) K. Trachenko and V. Brazhkin, Rep. Prog. Phys. 79, 016502 (2015).
  * Balian and de Dominicis (1971) R. Balian and C. de Dominicis, Annals of Physics 62, 229 (1971).
  * Carneiro and Pethick (1975) G. M. Carneiro and C. J. Pethick, Phys. Rev. B 11, 1106 (1975).
  * Carneiro (1979) G. M. Carneiro, Phys. Rev. B 19, 215 (1979).
  * Cohen (1960) M. Cohen, Phys. Rev. 118, 27 (1960).
  * Fåk  _et al._ (2012) B. Fåk, T. Keller, M. E. Zhitomirsky, and A. L. Chernyshev, Phys. Rev. Lett. 109, 155305 (2012).


# Time-resolved quantum beats in the fluorescence of helium resonantly excited
by XUV radiation

A. C. LaForge1,∗, A. Benediktovitch2,∗, V. Sukharnikov2,∗, Š. Krušič3, M.
Žitnik3, M. Debatin4, R. W. Falcone5, J. D. Asmussen6, M. Mudrich6, R.
Michiels7, F. Stienkemeier7, L. Badano8, C. Callegari8, M. Di Fraia8, M.
Ferianis8,9, L. Giannessi8,9, O. Plekan8, K. C. Prince8, C. Spezzani8, N.
Rohringer2,10, and N. Berrah1 1 Department of Physics, University of
Connecticut, Storrs, Connecticut, 06269, USA  2 Deutsches Elektronen-
Synchrotron (DESY), 22607 Hamburg, Germany  3 Jožef Stefan Institute, Jamova
cesta 39, SI-1001 Ljubljana, Slovenia  4 Experimentalphysik I, Universität
Kassel, D-34132 Kassel, Germany  5 Department of Physics, UC Berkeley,
Berkeley, California 94720, USA  6 Department of Physics and Astronomy, Aarhus
University, 8000 Aarhus C, Denmark  7 Physikalisches Institut, Universität
Freiburg, 79104 Freiburg, Germany  8 Elettra-Sincrotrone Trieste, 34149
Basovizza, Trieste, Italy  9 Nazionale di Fisica Nucleare - Laboratori
Nazionali di Frascati, Via E. Fermi 40, 00044 Frascati, Roma, Italy  10
Department of Physics, Universität Hamburg, Hamburg 22761, Germany  ∗ These
authors contributed equally to this work.
aaron.laforge<EMAIL_ADDRESS>
###### Abstract

We report on the observation of time-resolved quantum beats in the helium
fluorescence from the transition $1s3p\,\rightarrow\,1s2s$, where the initial
state is excited by XUV free electron laser radiation. The quantum beats
originate from the Zeeman splitting of the magnetic substates due to an
external magnetic field. We perform a systematic study of this effect and
discuss the possibilities of studying this phenomenon in the XUV and X-ray
regime.

##  1 Introduction

A system in a coherent superposition of states is a fundamental quantum-
mechanical phenomenon and is at the core of numerous chemical and biological
transient phenomena [1]. The understanding of these phenomena is of the utmost
importance and motivates the development of new experimental techniques and
theoretical modeling. The essential component of the corresponding
experimental techniques is the creation of the superposition of excited states
which is then probed with one or several pulses after a certain delay. This
approach is realized for numerous techniques such as multidimensional
spectroscopy [2], transient absorption spectroscopy [3], and phase-modulated
coherent spectroscopy [4], just to mention a few examples. The typical
requirement is for femto- to attosecond pulse durations, dictated by the
timescale of the process of interest. When the timescale of the process is
long enough, instead of a second probe pulse, time-resolved fluorescence
spectroscopy is a possible alternative technique. If the initial state is in a
coherent superposition of excited states, the intensity of emitted
fluorescence would show temporal oscillations, referred to as quantum beats.

For fluorescence from an ensemble of identical non-interacting atoms, quantum
beats emerge from close lying electronic states [5, 6], e.g. from a multiplet
due to Zeeman splitting of an electronic level by an external magnetic field.
To illustrate the concept, consider a $V$-type level scheme [7], which is
prepared in a coherent superposition of excited states with a small energy
gap, $\hbar\Delta$, between them. The frequency of the beating will be
proportional to the energy gap between the excited states [6]. Due to the
coherence of the emitted fluorescence, the combination of respective waves
with slightly different frequencies results in the periodic modulation of the
intensity of fluorescence in time. An example of the beating phenomenon is
given in Fig. 1 (a). For the case of no external magnetic field, the
fluorescence lifetime is a simple exponential decay with a time constant,
$\tau$, which is given by the red line. The introduction of an external
magnetic field $B_{0}$ causes a splitting proportional to the magnetic field
strength $\hbar\Delta\sim\mu_{B}B_{0}$, producing a temporal interference in
the decay, given by the green dashed line. Due to their inverse relationship,
a doubling of the magnetic field reduces the period of the interference by
half, given by the dash-dotted blue line.

Since fluorescence quantum beating is a general phenomenon [7], it can be
observed for various kinds of transitions and photon energies. To initiate the
beating, the primary requirement is that the pump pulse needs to be shorter
than the lifetime of the excited levels and the beating period [5]. The first
proof-of-principle experimental demonstrations of optical fluorescence quantum
beats were performed with a shuttered spectral lamp [8, 9]. Subsequent
improvements in experimental techniques (e.g. impulsive electronic excitation
and pulsed lasers) transformed the fluorescence quantum beating technique into
a tool for high-precision optical spectroscopy [5, 6, 10]. In the optical
regime, quantum beats-related phenomena were observed for the resonant
fluorescence of atoms [11, 12] and molecules [13, 14, 15], from an ensemble of
two-level atoms [16, 17], in quantum well systems [18], and even in the
temporal profiles of superfluorescence [19, 20, 21]. In the XUV regime,
quantum beats were used as a tool to evaluate the polarization of synchrotron
radiation [22]. In the $\gamma$-ray regime, a phenomenon similar to quantum
beats exists from coherently scattered radiation from nuclei Mössbauer
transitions [23, 24, 25].

In this work, we demonstrate the quantum beating phenomenon from helium atoms
excited by seeded XUV free electron laser (FEL) pulses. The FEL was tuned to
the resonant transition from the $1s^{2}$ ground state to the $1s3p$ excited
state. The observed fluorescent emission occurs between the $1s3p$ and $1s2s$
excited states. Due to the coupling to an external magnetic field, the excited
state is split into its magnetic substates and an interference is observed in
the subsequent fluorescence intensity. Compared to previous work [22], we have
used orders of magnitude higher magnetic field strengths resulting in orders
of magnitude faster beating, which can only be observed with the aid of a
femtosecond streak-camera. The quantum beating effect is systematically
investigated by varying the experimental parameters. We additionally discuss
the prospects for quantum beating in the XUV and X-ray regime.

##  2 Experimental setup

The experiment was performed at the Low Density Matter endstation [26] of the
seeded FEL FERMI, in Trieste, Italy. The FEL photon energy was set to
$h\nu\,=\,23.00$ eV, corresponding to the $1s3p$ excited state in helium, and
tuned via the seed laser and the undulator gaps, with a pulse length of
approximately 100 fs full width half maximum (FWHM) [27, 28]. The FEL pulse
energy at the sample varied from approximately 0.6 $\mu$J to 66 $\mu$J with a
spot size of 35 $\mu$m FWHM, and was calculated from the value measured
upstream on a shot-by-shot basis by gas ionization and the nominal
reflectivity of the optical elements in the beam transport system [29]. A
cold, supersonic gas jet of He atoms was produced by expansion of high
pressure He gas through a pulsed, cryogenically cooled Even-Lavie nozzle. By
varying the expansion conditions (backing pressure and nozzle temperature),
the target density was varied in the range of 1012-1015 atoms/cm3. The atomic
beam was perpendicularly crossed by the FEL at the center of a magnetic bottle
spectrometer (MBS), which measures kinetic energies of electrons from their
time-of-flights, and applies an intense magnetic field [30]. The MBS was not
directly used in this experiment, but its movable permanent magnet was used to
induce the splitting of the $1s3p$ state. The FEL light was linearly polarized
perpendicular to the magnetic field of the MBS. To measure the fluorescence
lifetime, a streak camera (Hamamatsu Fesca-200) was mounted downstream of the
interaction region on the rear portion of the endstation. Focusing optics
coaxial with the FEL beam were used to collect the maximum amount of
fluorescence.

##  3 Results and Discussion

Figure 1: (a) Representative fluorescence lifetime with an exponential decay,
$\tau$, for no external magnetic field (solid red line), a magnetic field
$B_{0}$ (dashed green line), and a magnetic field $2B_{0}$ (dash-dotted blue
line). (b) Experimentally-measured fluorescence intensity for He for the
$1s3p\,\rightarrow\,1s2s$ transition. The nozzle temperature $T=30$ K and the
pulse energy $W=66\,\mu J$ were fixed.

Fig. 1 (b) shows an experimentally measured temporal profile of the
fluorecence intensity. To ensure the fluorescence originated from the
$1s3p\,\rightarrow\,1s2s$ transition ($504\,$nm), we used a 10 nm bandpass
filter centered at 505 nm and performed background subtraction measurements.
As a cross check, we installed a polarizing filter. When changing the
polarization from perpendicular to parallel with respect to the magnetic
field, the fluorescence intensity vanished, indicating that the emitted
fluorescence had the same polarization as the FEL excitation pulse. The
maximal range of measurement of the streak camera is 500 ps while the lifetime
of the $1s3p\,\rightarrow\,1s2s$ transition is $1/\Gamma\,=\,1715\,$ps. For
this reason, we sequentially varied the streak camera delay by multiples of
$300$ ps in order to obtain the entire fluorescence curve. For the measurement
in Fig. 1 (b), the permanent magnet of the MBS was 16.5 mm from the
interaction region. The results in (b) show beating imposed on exponential
decay, which clearly indicates interference in the fluorescence emission from
the Zeeman splitting of the excited state and is in agreement with calculated
quantum beating shown in Fig. 1 (a).

To understand the quantum beats observed in Fig. 1 (b), we consider an
isolated helium atom in its ground state, $1s^{2}$, placed in a static
magnetic field in the $z$-direction, $\mathbf{B}=B_{0}\cdot\mathbf{e}_{z}$.
The atom is excited by a FEL pulse propagating in the $y$-direction, which is
linearly polarized in the $x$-direction. The excitation pulse is resonant with
the $1s^{2}\,\rightarrow\,1s3p$ transition, and fluorescence is measured from
the $1s3p\,\rightarrow\,1s2s$ transition.

Due to the magnetic field of the MBS, the upper state transforms into a Zeeman
triplet state (see Fig. 2 (a)). The typical magnetic field strength of the
permanent magnet in a MBS is a few tesla [31], hence a rough estimation of the
energy gap gives $\hbar\Delta\approx 0.1$ meV, which is about two orders of
magnitude less than the FEL bandwidth ($\sim 30$ meV). Thus the magnetically
split states are coherently excited.

Transitions from the Zeeman triplet substates to the singlet state $1s2s$ have
different polarizations. The radial contributions to the matrix elements of
the dipole moment on the $1s3p\,\rightarrow\,1s2s$ transition are denoted by
$d$ and assumed to be real. Hence, the full matrix elements which take the
polarization effects into account are:

$\displaystyle\langle
m_{z}=-1|\widehat{\mathbf{d}}|1s2s\rangle=\frac{d}{\sqrt{2}}\left(\mathbf{e}_{x}+i\mathbf{e}_{y}\right),$
(1) $\displaystyle\langle
m_{z}=0|\widehat{\mathbf{d}}|1s2s\rangle=d\cdot\mathbf{e}_{z},$ (2)
$\displaystyle\langle
m_{z}=+1|\widehat{\mathbf{d}}|1s2s\rangle=\frac{d}{\sqrt{2}}\left(-\mathbf{e}_{x}+i\mathbf{e}_{y}\right).$
(3)

Since the FEL pulse has linear polarization in the $x$-direction, the
excitation process results in the population of the state

$|m_{x}=0\rangle=\frac{|m_{z}=-1\rangle-|m_{z}=+1\rangle}{\sqrt{2}},$ (4)

which is the eigenstate of spin-1 Pauli matrix $\sigma_{x}$ with zero
projection $m_{x}=0$. Hence, the level scheme is reduced to a four-level
system containing one ground state, two intermediate excited states, and one
final state (Fig. 2 (b)) with the initial state described by Eq. (4).

Figure 2: (a) Relevant excitation and fluorescence channels; (b) relevant
transitions in the presence of a magnetic field.

Since only single photon emission processes take place, the evolution of such
a system can be solved analytically using the Weisskopf-Wigner approximation.
For the case of ensemble of atoms, the single-atom contributions can be
incoherently summed up. The resulting fluorescence intensity to the $1s2s$
state for the energy level scheme depicted in Fig. 2 (b) is given by:

$I(t)\sim\int d\Delta\cdot p(\Delta)\cdot e^{-\Gamma t}\cdot(1+\cos\Delta\cdot
t),$ (5)

where $t$ is the retardation time, $\Gamma$ is the total spontaneous decay
rate of the upper states into the final states, and $p(\Delta)$ is the
distribution function, which accounts for the inhomogeneity of the magnetic
field strength within the interaction volume.

Figure 3: Fitted fluorescence curves for different positions of the permanent
magnet; (a) 6.5 mm from the interaction region with parameters
$\langle\Delta\rangle=18.6\cdot 10^{-3}$,
$\sigma_{\Delta}/\langle\Delta\rangle=0.13$; (b) 11.5 mm from the interaction
region with parameters $\langle\Delta\rangle=12.8\cdot 10^{-3}$,
$\sigma_{\Delta}/\langle\Delta\rangle=0.03$; (c) 16.5 mm from the interaction
region with parameters $\langle\Delta\rangle=8.01\cdot 10^{-3}$,
$\sigma_{\Delta}/\langle\Delta\rangle=0.012$. The value of $\Delta$ is given
in inverse picoseconds,
$\sigma_{\Delta}=\sqrt{\langle\Delta^{2}\rangle-\langle\Delta\rangle^{2}}$.

Fig. 3 shows the temporal profile of the fluorescence intensity for three
different positions for the permanent magnet; (a) 6.5 mm, (b) 11.5 mm, and (c)
16.5 mm from the interaction region. To model the experimental data presented
in Fig. 3 we have used Eq. (5) and convoluted the result with a $20$ ps
instrumental response function. For simplicity, within the interaction volume,
we have assumed a Gaussian variation of magnetic field in space, and have
recalculated the distribution function $p(\Delta)$ from it. The mean value and
variance of the magnetic field distribution were used as fitting parameters,
the spontaneous decay rate was taken fixed and equal to its theoretical value.

As expected, for the positions of the magnet closer to the interaction region,
the beating period is shorter and the mean value of the magnetic field is
larger. The inhomogeneity of the magnetic field strength within the
interaction volume results in a smearing of the beating pattern. As one can
see from the results of the fit, the magnetic field distribution is more
inhomogeneous for closer positions of the magnet. This is as expected since
the MBS is designed to confine electrons to helical orbits using a strongly
divergent magnetic field.

Figure 4: Fluorescence intensity as a function of different experimental
parameters in normalized units; (a) varied atomic target density with fixed
nozzle temperature $T=79$ K and pulse energy $W=66$ $\mu J$; (b) varied nozzle
temperature for fixed density $n$ and pulse energy $W=66$ $\mu J$; (c) varied
excitation pulse energy with fixed temperature $T=30$ K and concentration $n$.
For these measurements, the permanent magnet was 6.5 mm from the interaction
region.

In our investigation, we additionally varied different experimental
parameters, namely the target density and the nozzle temperature in the
supersonic jet and the energy of the FEL excitation pulse. The total radiated
energy may be estimated as the area under the fluorescence curve, which was
measured for the different sets of parameters. Fig. 4 (a) shows that the total
fluorescence intensity is linearly proportional to the atomic target density.
A similar result is observed in Fig. 4 (b), which shows linear dependence on
the nozzle temperature of the molecular beam source. Fig. 4 (c) shows an
increase of the total fluorescence intensity with pump pulse energy. The
linear scaling with respect to atomic concentration shows that no collective
effects – such as superfluorescence – have taken place. This is in contrast to
previous experimental results showing superfluorescent behavior [32] that were
found for the same atomic transition, but without an external magnetic field.
The primary reason is the target density from a supersonic gas jet used in
this experiment is much less than that of previous experiments using a gas
cell [32, 33, 34].

Overall, short wavelength, femtosecond pulses provided by XUV and X-ray FELs
offer a novel means to initiate fluorescent quantum beats. The general
requirement is that the pulses are i) short relative to the spontaneous decay
time and beating period and ii) intense and broad enough to excite multiple
excited states in order to produce the beat interference. Additionally, in
contrast to multidimensional spectroscopy, a high degree of pump coherence is
not necessary, as can be seen from fact that quantum beating was observed in
synchrotron pulses [22]. That said, FEL pulses offer a clear advantage of
being much shorter and have higher peak intensities compared to synchrotron
radiation.

The quantum beat fluorescence approach with XUV/X-ray FEL pulses can be used
to address excited states which have an energy difference $\Delta E$ such that
beating with period $\hbar/\Delta E$ can be resolved with a streak camera. For
a commercially available camera, this corresponds to about 100 fs,
corresponding to $\Delta E$ below tens of meV. If the excited state decays
more slowly than the beating period, the temporal oscillation of fluorescent
radiation could be directly observed. A possible example would be valence
double ionization of Xe. In this case, the branching ratio of this process –
leading to excited Xe2+ state – is relatively large ($\sim\,30\%$ [35, 36]).
The lifetime of the valence-excited state levels is quite long, about 1 ns,
hence if the excited state has a multiplet sub-structure with level spacing
smaller than tens of meV, the fluorescence emitted on transition to the ground
state of Xe2+ could show quantum beating.

An extension to quantum beat fluorescence would be to increase the number of
emitting atoms in order to drive the fluorescent radiation into the
collective, superflourescent regime. As was previously shown in optical regime
[19], under the conditions of superfluorescence, the beating could originate
not only from the splitting of the upper levels ($V$-level systems), but also
from the splitting of the lower levels ($\Lambda$-level systems) where
spontaneous fluorescence quantum beating is fundamentally impossible [7].

##  4 Conclusions

In summary, we have used XUV radiation from a seeded FEL to resonantly excite
the $1s3p$ state in atomic He. Using a femtosecond streak camera, we have
observed time-resolved quantum beats in the fluorescence emission from the
$1s3p\,\rightarrow\,1s2s$ transition due to the Zeemann splitting of the
$1s3p$ state. To systematically understand the phenomenon, we have
investigated the effect of the magnetic field, FEL pulse energy, and atomic
source conditions on the fluorescence emission and quantum beating. The
results demonstrate the feasibility of studying fluorescent quantum beating
phenomena using ultrashort FEL pulses in the XUV and X-ray regime.

##  5 Acknowledgements

This work was funded by the Chemical Sciences, Geosciences, and Biosciences
Division, Office of Basic Energy Sciences, Office of Science, US Department of
Energy, Grant No. DESC0012376. R.W.F. acknowledges support from the Department
of Energy, National Nuclear Security Administration Award DE-NA0003842 and
Department of Energy, Office of Science, Office of Fusion Energy Sciences
Award DESC0018298. R.M. and F.S. acknowledge support from the Deutsche
Forschungsgemeinschaft under grant STI 125/19-2. The research leading to this
result has been supported by the project CALIPSOplus under grant agreement
730872 from the EU Framework Programme for Research and Innovation HORIZON
2020.

## References

  * [1] Ahmed H. Zewail.  Femtochemistry:atomic-scale dynamics of the chemical bond.  J. Phys. Chem. A, 104(24):5660–5694, 2000.
  * [2] Shaul Mukamel.  Principles of nonlinear optical spectroscopy.  Number 6 in Oxford series in optical and imaging sciences. Oxford Univ. Press, New York, 2009.
  * [3] Eleftherios Goulielmakis, Zhi-Heng Loh, Adrian Wirth, Robin Santra, Nina Rohringer, Vladislav S. Yakovlev, Sergey Zherebtsov, Thomas Pfeifer, Abdallah M. Azzeer, Matthias F. Kling, Stephen R. Leone, and Ferenc Krausz.  Real-time observation of valence electron motion.  Nature, 466(7307):739–743, August 2010.
  * [4] Andreas Wituschek, Lukas Bruder, Enrico Allaria, Ulrich Bangert, Marcel Binz, Roberto Borghes, Carlo Callegari, Giulio Cerullo, Paolo Cinquegrana, Luca Giannessi, Miltcho Danailov, Alexander Demidovich, Michele Di Fraia, Marcel Drabbels, Raimund Feifel, Tim Laarmann, Rupert Michiels, Najmeh Sadat Mirian, Marcel Mudrich, Ivaylo Nikolov, Finn H. O’Shea, Giuseppe Penco, Paolo Piseri, Oksana Plekan, Kevin Charles Prince, Andreas Przystawik, Primož Rebernik Ribič, Giuseppe Sansone, Paolo Sigalotti, Simone Spampinati, Carlo Spezzani, Richard James Squibb, Stefano Stranges, Daniel Uhl, and Frank Stienkemeier.  Tracking attosecond electronic coherences using phase-manipulated extreme ultraviolet pulses.  Nat. Commun., 11(1):883, December 2020.
  * [5] Evgenii B Aleksandrov.  Soviet Physics Uspekhi, 15(4):436–451, apr 1973.
  * [6] S. Haroche.  Quantum beats and time-resolved fluorescence spectroscopy.  In Koichi Shimoda, editor, High-Resolution Laser Spectroscopy, Topics in Applied Physics, pages 253–313. Springer, Berlin, Heidelberg, 1976.
  * [7] Marlan O Scully.  On quantum beat phenomena and the internal consistency of semiclassical radiation theories.  In A. O. Barut, editor, Foundations of Radiation Theory and Quantum Electrodynamics, pages 45–48. Springer US, Boston, MA, 1980.
  * [8] Evgenii B Aleksandrov.  Luminescence beats induced by pulse excitation of coherent states.  Opt. Spectrosc., 17:522, dec 1964.
  * [9] J N Dodd, R D Kaul, and D M Warrington.  The modulation of resonance fluorescence excited by pulsed light.  Proceedings of the Physical Society, 84(1):176–178, jul 1964.
  * [10] Evgenii B Aleksandrov, N I Kaliteevskiĭ, and M P Chaĭka.  Superhigh-resolution spectroscopy based on interference of states.  Soviet Physics Uspekhi, 22(9):760–767, sep 1979.
  * [11] Peter Schenck, Robert C Hilborn, and Harold Metcalf.  Phys. Rev. Lett., 31(4):189, 1973.
  * [12] S Haroche, JA Paisner, and AL Schawlow.  Phys. Rev. Lett., 30(20):948, 1973.
  * [13] R Wallenstein, JA Paisner, and AL Schawlow.  Phys. Rev. Lett., 32(24):1333, 1974.
  * [14] Joseph Chaiken, M Gurnick, and JD McDonald.  J. Chem. Phys., 74(1):106–116, 1981.
  * [15] H Bitto and J Robert Huber.  Opt. Commun., 80(2):184–198, 1990.
  * [16] Z Ficek, R Tanaś, and S Kielich.  Physica A, 146(3):452–482, 1987.
  * [17] Zbigniew Ficek and Barry C Sanders.  Phys. Rev. A, 41(1):359, 1990.
  * [18] EO Göbel, K Leo, TC Damen, J Shah, S Schmitt-Rink, W Schäfer, JF Müller, and K Köhler.  Phys. Rev. Lett., 64(15):1801, 1990.
  * [19] Q. H. F. Vrehen, H. M. J. Hikspoors, and H. M. Gibbs.  Quantum beats in superfluorescence in atomic cesium.  Phys. Rev. Lett., 38:764–767, Apr 1977.
  * [20] D Bartholdtsen and R-H Rinkleff.  Z. Phys. D Atom. Mol. Cl., 30(4):265–273, 1994.
  * [21] E Boursey, J Meziane, and A Topouzkhanian.  Phys. Rev. A, 48(4):3047, 1993.
  * [22] Yasumasa Hikosaka, Hiroshi Iwayama, and Tatsuo Kaneyasu.  Zeeman quantum beats of helium Rydberg states excited by synchrotron radiation.  Journal of Synchrotron Radiation, 27(3):675–680, May 2020.
  * [23] E. Gerdau, R. Rüffer, R. Hollatz, and J. P. Hannon.  Quantum Beats from Nuclei Excited by Synchrotron Radiation.  Phys. Rev. Lett., 57(9):1141–1144, September 1986.
  * [24] E. Gerdau and H. DeWaard.  Nuclear resonant scattering of synchrotron radiation.  Hyperfine Interactions, 123(1):0–0, March 1999.
  * [25] Yu.V. Shvyd’ko and U. van Bürck.  Hybrid forms of beat phenomena in nuclear forward scattering of synchrotron radiation.  Hyperfine Interactions, 123(1):511–527, March 1999.
  * [26] Viktor Lyamayev, Yevheniy Ovcharenko, Raphael Katzy, Michele Devetta, Lukas Bruder, Aaron LaForge, Marcel Mudrich, Ulrich Person, Frank Stienkemeier, Maria Krikunova, and et al.  J. Phys. B, 46(16):164007, 2013.
  * [27] E Allaria, Roberto Appio, L Badano, WA Barletta, S Bassanese, SG Biedron, A Borga, E Busetto, D Castronovo, P Cinquegrana, and et al.  Nat. Photonics, 6(10):699, 2012.
  * [28] E Allaria, A Battistoni, F Bencivenga, R Borghes, C Callegari, F Capotondi, D Castronovo, P Cinquegrana, D Cocco, M Coreno, and et al.  New J. Phys., 14(11):113009, 2012.
  * [29] Cristian Svetina, Cesare Grazioli, Nicola Mahne, Lorenzo Raimondi, Claudio Fava, Marco Zangrando, Simone Gerusina, Michele Alagia, Lorenzo Avaldi, Giuseppe Cautero, et al.  The low density matter (ldm) beamline at fermi: optical layout and first commissioning.  J. Synchrotron Radiat., 22(3):538–543, 2015.
  * [30] Akitaka Matsuda, Mizuho Fushitani, Chien-Ming Tseng, Yasumasa Hikosaka, John HD Eland, and Akiyoshi Hishikawa.  Rev. Sci. Instrum., 82(10):103105, 2011.
  * [31] Toshio Tsuboi, Emily Y Xu, Young K Bae, and Keith T Gillen.  Rev. Sci. Instrum., 59(8):1357–1362, 1988.
  * [32] James R Harries, Hiroshi Iwayama, Mitsuru Nagasono, Tadashi Togashi, Makina Yabashi, Susumu Kuma, Kyo Nakajima, Yuki Miyamoto, Chiaki Ohae, Noboru Sasao, and Eiji Shigemasa.  A streak camera study of superfluorescence at multiple wavelengths from helium atoms excited using free electron laser pulses.  J. Phys. B: At., Mol. Opt. Phys., 48(10):105002, apr 2015.
  * [33] Kyo Nakajima, James R. Harries, Hiroshi Iwayama, Susumu Kuma, Yuki Miyamoto, Mitsuru Nagasono, Chiaki Ohae, Tadashi Togashi, Makina Yabashi, Eiji Shigemasa, and Noboru Sasao.  Simultaneous measurements of super-radiance at multiple wavelengths from helium excited states: I. experiment.  J. Phys. Soc. Jpn., 84(5):054301, 2015.
  * [34] Chiaki Ohae, James R. Harries, Hiroshi Iwayama, Kentarou Kawaguchi, Susumu Kuma, Yuki Miyamoto, Mitsuru Nagasono, Kyo Nakajima, Itsuo Nakano, Eiji Shigemasa, Noboru Sasao, Satoshi Uetake, Tomonari Wakabayashi, Akihiro Yoshimi, Koji Yoshimura, and Motohiko Yoshimura.  Simultaneous measurements of superradiance at multiple wavelength from helium excited states: Ii. analysis.  J. Phys. Soc. Jpn., 85(3):034301, 2016.
  * [35] M. A. Khalal, P. Lablanquie, L. Andric, J. Palaudoux, F. Penent, K. Bučar, M. Žitnik, R. Püttner, K. Jänkälä, D. Cubaynes, S. Guilbaud, and J.-M. Bizau.  $4d$-inner-shell ionization of $\mathrm{X}{\mathrm{e}}^{+}$ ions and subsequent auger decay.  Phys. Rev. A, 96:013412, Jul 2017.
  * [36] L. Mercadier, A. Benediktovitch, C. Weninger, M. A. Blessenohl, S. Bernitt, H. Bekker, S. Dobrodey, A. Sanchez-Gonzalez, B. Erk, C. Bomme, R. Boll, Z. Yin, V. P. Majety, R. Steinbrügge, M. A. Khalal, F. Penent, J. Palaudoux, P. Lablanquie, A. Rudenko, D. Rolles, J. R. Crespo López-Urrutia, and N. Rohringer.  Evidence of extreme ultraviolet superfluorescence in xenon.  Phys. Rev. Lett., 123:023201, Jul 2019.