# Elucidating the local atomic and electronic structure of amorphous oxidized
superconducting niobium films

Thomas F. Harrelson  Materials Science Division, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  Molecular Foundry, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, USA.  Evan Sheridan  Materials
Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720,
USA  Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA
94720, USA  Theory and Simulation of Condensed Matter, Department of Physics,
King’s College London, The Strand, London WC2R 2LS, UK.  Ellis Kennedy
Department of Materials Science and Engineering, University of California,
Berkeley, CA 94720, USA  John Vinson  Material Measurement Laboratory,
National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
Alpha T. N’Diaye  Advanced Light Source, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  M. Virginia P. Altoé  Molecular Foundry,
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  Adam
Schwartzberg  Molecular Foundry, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA  Irfan Siddiqi  Materials Science Division, Lawrence
Berkeley National Laboratory, Berkeley, CA 94720, USA  Department of Physics,
University of California, Berkeley, CA 94720, USA  D. Frank Ogletree
Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA 94720,
USA  Mary C. Scott  Department of Materials Science and Engineering,
University of California, Berkeley, CA 94720, USA  NCEM, Molecular Foundry,
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  Sinéad M.
Griffin  Materials Science Division, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA  Molecular Foundry, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA

###### Abstract

Qubits made from superconducting materials are a mature platform for quantum
information science application such as quantum computing. However, materials-
based losses are now a limiting factor in reaching the coherence times needed
for applications. In particular, knowledge of the atomistic structure and
properties of the circuit materials is needed to identify, understand, and
mitigate materials-based decoherence channels. In this work we characterize
the atomic structure of the native oxide film formed on Nb resonators by
comparing fluctuation electron microscopy experiments to density functional
theory calculations, finding that an amorphous layer consistent with an Nb2O5
stoichiometry. Comparing X-ray absorption measurements at the Oxygen K edge
with first-principles calculations, we find evidence of d-type magnetic
impurities in our sample, known to cause impedance in proximal
superconductors. This work identifies the structural and chemical composition
of the oxide layer grown on Nb superconductors, and shows that soft X-ray
absorption can fingerprint magnetic impurities in these superconducting
systems.

Superconducting qubits are one of the leading solid-state platforms for QIS
(quantum information science) applications, with reported coherence times
reaching $\sim$100 microsecondsDevoret and Schoelkopf (2013); Kjaergaard  _et
al._ (2020). Despite this, materials-based decoherence channels contribute
significantly to microwave losses, and are now a central hurdle in device
coherence and scalingMcDermott (2009). In particular, the inevitable
inhomogeneities that are present from growth and fabrication, such as
interfaces, defects, and structural disorder, each contribute to the
decoherence in qubits amde from superconducting materialsOliver and Welander
(2013); de Leon  _et al._ (2021).

Precise knowledge of the atomistic structural and chemical makeup of
superconducting qubit materials is particularly necessary for understanding
materials-dependent decoherence processes. Intrinsic noise sources in
superconducting qubits are typically classified into two categories – two-
level system (TLS) noise, and non-TLS noiseMüller, Cole, and Lisenfeld (2019).
TLSs are fluctuating two-level states comprising local energy minima in the
atomic structural potential which were originally proposed to describe the
microstructure of amorphous materialsPhillips (1987). TLSs can couple to
electric and magnetic fields, reducing a qubit’s coherence time. Since the
amorphous materials present on superconducting qubit surfaces consist of a
variety of bonding environments, TLSs can host a range of barrier heights and
tunnelling rates, and correspondingly a distribution of fluctuation
frequencies even within a given materialBurnett, Faoro, and Lindström (2016).
Because of this, characterization of the local atomic arrangements is needed
to build any predictive description of TLS-related decoherence.

Non-TLS noise intrinsic in QIS materials includes the presence of
nonequilibrium quasiparticles (QP)Wilen  _et al._ (2020); Cardani  _et al._
(2020) and of magnetic impuritiesKharitonov  _et al._ (2012); Proslier  _et
al._ (2011); Sheridan  _et al._ (2021). While careful shielding can mitigate
some of these effects, the decay and control of QPs is materials dependent,
and can be crucially influenced by nanofabrication and materials’
controlVepsäläinen  _et al._ (2020); Wilen  _et al._ (2020); Martinis (2020).
Another key non-TLS loss mechanism is Cooper pair breaking induced by the
presence of magnetic impuritiesKharitonov  _et al._ (2012), which can occur
due to materials’ defects, interfaces, and surfaces, and cause impedance
losses in the superconductorKharitonov  _et al._ (2012); Proslier  _et al._
(2011); Sheridan  _et al._ (2021). Therefore, to understand the structure-
coherence relationships associated with the materials’ properties in
superconducting qubits, knowledge of the local structural and chemical
environment is needed, regardless of the origin (TLS, non-TLS) of the noise.

Superconducting qubits are typically comprised of Al/AlOx/Al Josephson
junctions with superconducting circuit elements commonly made from Al, Nb, Ta,
and alloys containing theseOliver and Welander (2013); Place  _et al._ (2021).
Of these, Nb has many advantages over other superconducting materials
including low kinetic inductance resulting in reduced variability, and a
higher superconducting gap making it less susceptible to QP poisoningKaplan
_et al._ (1976). Importantly, Nb forms a relatively clean surface, and is a
mature material for the advanced processing and lithographic patterning that
is required for contemporary qubit fabrication and for future scaling of
highly coherent superconducting architectures. However, Nb readily forms
surface oxides such as NbO, NbO2 and Nb2O5, which introduce both TLS and non-
TLS losses in the qubitDelheusy  _et al._ (2008); Altoé  _et al._ (2020).
Previous work has looked at the use of ultrahigh vacuum packing to reduce
surface contaminationMergenthaler  _et al._ (2021), in addition to an
understanding of the influence of both oxide surface removalAltoé  _et al._
(2020) and regrowthVerjauw  _et al._ (2021) on the performance of
superconducting resonators.

Despite extensive research on the variety of loss channels and their
mitigation through surface treatments and fabrication Romanenko and Schuster
(2017); Romanenko  _et al._ (2020), the precise microscopic origins of TLS and
non-TLS losses in superconducting systems is not known. This is primarily due
to the difficulty in accessing information about the local structural and
chemical environments which critically control the presence of these losses.
Since the native oxides formed on Nb are often amorphous, conventional
diffraction and computational techniques cannot be used for structural
information. Theoretical treatments often either rely on having crystalline
materials with periodic boundary conditionsHeinrich, Pascual, and Franke
(2018), or propose phenomenological models without incorporating nanoscale
structural information. Instead, in this work, we combine Fluctuation Electron
Microscopy (FEM), X-Ray Absorption Spectroscopy (XAS), and first-principles
calculations to investigate the structural and chemical composition of
amorphous oxides on superconducting Nb films. We classify the short- and mid-
range structural properties of our oxides by comparing our ab initio
calculations with experiments, identifying the structural and chemical makeup
of surface Nb oxides on superconducting resonators.

To characterize the mid-range atomic structure of the amorphous films we used
FEM, a 4-D scanning transmission electron microscopy technique that is
sensitive to medium-range atomic ordering in disordered materials Voyles and
Muller (2002). FEM experiments were performed using an FEI TitanX operated at
an acceleration voltage of 200 kV. Additionally, XAS measurements of the O
K-edge were performed at the bending magnet beamline 6.3.1 at the Advanced
Light Source at Lawrence Berkeley National Laboratory. We consider three
different Nb treatments: (1) unpatterned, oxidized Nb films without any
treatments, (2) Nb film from a chip patterned with qubits, and (3) Nb film
from a chip with resonators only (no Josephson junctions), which allows us to
potentially observe changes in the Nb oxides with these different fabrication
steps (Table 1). XAS was performed on all three samples. FEM was performed on
Sample 2 because it had the thickest oxide layer, which was required for
improved signal in FEM analysis. Further details of the FEM and XAS
measurements are given in the SI.

Electronic and magnetic properties were calculated using density functional
theory (DFT) as implemented in the Vienna Ab initio Simulation Package (VASP)
Kresse and Hafner (1993). We used Nb2O5 amorphous structures that were
generated previously with ab initio molecular dynamics as detailed in
Ref.Sheridan  _et al._ (2021), which are available on ZenodoHarrelson  _et
al._ (2021a). X-ray absorption calculations were carried out using the Bethe-
Salpeter equation (BSE) formalism as implemented within the ocean codeVinson
_et al._ (2011); *ocean2; *ocean0. The BSE calculations use a basis of
electron orbitals from DFT calculated with Quantum ESPRESSO, Giannozzi  _et
al._ (2017); *espresso0 with pseudopotentials from the PseudoDojo collection.
van Setten  _et al._ (2018); *pspdojo0; Hamann (2013); *oncvp More details on
the DFT and XAS calculations are given in the SI.

Figure 1: (a) Representative ab initio molecular dynamics generated amorphous
structure of Nb2O5. (b) Averaged speckle pattern of Nb2O5 using FEM over many
diffraction patterns. (c) Radial Distribution Function for Nb2O5 obtained from
averaging over nine amorphous stoichiometric configurations generated using ab
initio molecular dynamics, (d) Annular mean of normalized variances of the FEM
data measuring the average interatomic spacing between Nb centers in the
Nb2O5.

We first describe our FEM diffraction results of a representative oxidized Nb
sample with the largest oxide thickness (Sample 2), and compare the short-
range structural description to ab initio generated structures. In contrast to
other diffraction techniques, which generally identify long-range ordering,
FEM is uniquely sensitive to the medium-range ordering on the size scale of
the electron beam probe Voyles and Abelson (2003); Daulton, Bondi, and Kenneth
(2010). FEM data is acquired by rastering a small electron probe over a sample
and capturing a diffraction pattern at each probe location. The diffraction
patterns are digitally preprocessed to remove imaging distortions, and the
variance of the measured intensity as a function of scattering vector is
calculated Kennedy  _et al._ (2020). As Bragg scattering in the diffraction
patterns creates large variance in intensity, the calculated variance is a
metric for ordering in the amorphous material on the length scale of the
electron probe Hwang and Voyles (2011). Full details of the FEM method and
data analysis are given in the SI. In Figure 1b, we show the average speckle
pattern of many nanodiffraction patterns taken over the Nb oxide region of the
film cross-section (see SI). The brighter spots in the speckled halo primarily
represent Nb-Nb distances because electron scattering from Nb atoms dominates
over scattering from O atoms. The broad diffuse halo present in the average
nanodifraction pattern suggests that the Nb oxide film is amorphous. In Figure
1(d) we show an average spatial variance computed from six regions of the
sample, where each region differs in its thickness, as shown in Figure S1 of
the Supplementary material. The broad peak centered at the wavevector
$\approx$3 nm-1 is a measure of the average interatomic spacing between Nb
centres, corresponding to an average Nb-Nb distance of 3.37 Å.

We next analyze ab initio-generated amorphous structures to investigate the
short- and medium-range structural order across a sample of stoichiometric
Nb2O5 amorphous configurations. Figure 1(a) illustrates a representative
stoichiometric amorphous configuration of Nb2O5 containing 105 atoms in the
unit cell. The solid line indicates the Nb-Nb distance for edge sharing Nb
sites in Nb2O5, while the dashed line highlights the longer Nb-Nb distances
for corner sharing Nb sites. These features are also present in Figure 1(c),
where we show the averaged radial distribution function (RDF) obtained from
nine stoichiometric amorphous configurations of Nb2O5 whose volume and
internal coordinates were optimized using DFT. We see from the first peak that
the shorter edge sharing Nb sites are typically 3.15 Å apart, and the longer
corner sharing Nb sites are 3.8 Å apart as indicated by the second peak. The
immediate dip of the RDF at 4 Å suggests that the edge- and corner-sharing
environments shown in Figure 1(a) are the primary structural motifs present in
our amorphous Nb2O5. Given the reasonable comparison between ab initio-
generated amorphous structures and FEM analysis of our Nb oxide thin films, we
can conclude that indeed our films are amorphous, lacking any long-range
order, and that our generated structures can be used for further analysis.
Additionally, we find the average Nb-Nb distance in the FEM measurement to be
3.37 Å, which is between the average corner- and edge-shared Nb-Nb distances
in the ab initio structures, suggesting our amorphous films comprise a mix of
corner- and edge-sharing polyhedra.

Figure 2: Measured XAS spectra of O K-edge for the three different samples
described in Table 1. Inset: Sketch of the electronic structure of
octahedrally coordinated Nb forming $t_{2g}$ and $e_{g}$ split orbital sets.
These hybridize with the unoccupied O orbitals that are excited upon X-ray
absorption, creating the observed splitting between the peaks in the data.
Spectra were normalized by matching the baselines, and dividing by the maximum
value in the 525 eV to 550 eV window.

We next use X-ray absorption spectroscopy (XAS) to obtain information about
the local morphology, electronic structure, and potential magnetism. We focus
on O K-edge spectra for three different NbOx samples, which are described in
Table 1. While XAS of the O K edge probes unoccupied p-type states surrounding
the oxygen atoms, these states are hybridized with the neighboring Nb, and so
provide information on both the Nb and O species. In Figure 2 we plot the
measured XAS for the three samples, and find the XAS is similar for all three.
As is typical in transition metal oxides, we identify the two peaks at 533 eV
and 537 eV as hybridized with the empty Nb 4d orbitals, split by the crystal
field splitting $\Delta$ into lower-energy $t_{2g}$ and higher-energy $e_{g}$
states. The broad feature near 544 eV reflects hybridization with Nb 5sp-like
states. Changes in the relative intensities of the $t_{2g}$ and $e_{g}$, and
splitting between them $\Delta$, and (less reliably) position of the edge
onset, reflect changes in the Nb d-manifold occupation, strength of the Nb-O
bonding, and oxidation state of the metal ion, respectivelyFrati, Hunault, and
de Groot (2020).

Sample |  NbOx Thickness |  Description
---|---|---
1 |  3 nm |  Unpatterned, oxidized Nb film.
2 |  15 nm |  Nb film fabricated with qubits including AlOx Josephson junctions.
3 |  5 nm |  Nb film fabricated with resonators only (no Josephson junctions).
Table 1: Summary of sample details used in experiments. XAS measurements were
performed on all three samples, whereas FEM measurements were performed on
Sample 2.

Comparing the XAS spectra of the three measured samples shows that the
unpatterned film (Sample 1) and the resonator chip (Sample 3) are the most
similar. We observe a slight increase in energy of the peak near 537 eV, and
the slight increase in intensity of the broad feature near 544 eV for the
resonator sample (Sample 3) compared to the unpatterned sample (Sample 1). The
qubit sample (Sample 2) has the largest NbOx thickness ($\sim 15$ nm), and
largest increase in energy of the 537 eV peak. The observed increase in the
energy of the 537 eV peak in the patterned samples suggests a greater crystal-
field splitting hence more crystalline character compared to the unpatterned
films.

We use a combination of DFT and BSE calculations to further analyze the XAS
spectra. We calculate spectra for fifteen different Nb2O5 amorphous
configurations (both stoichiometric and non-stoichiometric), and five
different crystalline phases of Nb-oxides. In Figure 3(a), we plot the
calculated crystalline spectra for NbO ($Pm3m$), NbO2 ($P4_{2}/mnm$), and the
average of 3 different Nb2O5 phases (N-phase ($C_{2}/m$), M-phase ($I4/mmm$),
and B-phase ($C_{2}/c$) and compare to the experimental spectrum of Sample 1.
We find that the measured XAS spectra are best described by Nb2O5. The
splitting between the two dominant peaks is larger in the crystalline
reference samples, while the relative heights of the two dominant peaks is
qualitatively described by Nb2O5, suggesting amorphous structures with Nb2O5
stoichiometry. We further find that as the oxidation state of the Nb atom
increases ($+2$ in NbO to $+5$ in Nb2O5), both the intensity of the first peak
increases, and the ratio of the intensity of the first peak to the second peak
increases. This is partially explained by considering the resulting filling of
the $t_{2g}$ and $e_{g}$ states of an octahedrally coordinated Nb atom (see
inset of Figure 2); NbO deviates slightly from the trend because the
coordination of the Nb atoms is square planar.

Figure 3: (a) Calculated XAS spectra for crystalline NbO, NbO2 and Nb2O5
(averaged over all three calculated phases) and XAS measurements of the O K
edge of Sample 1. (b) Calculated XAS spectra for crystalline Nb2O5 in the M-,
B-, and B- phases, a representative ab initio generated amorphous structure,
and XAS measurements of Sample 1. Experimental data is normalized by rigidly
shifting the spectrum to the relative scale, removing the background signal,
and normalizing the heights to be comparable to our XAS calculations.

In Figure 3(b), we compare the calculated XAS spectra for three different
crystalline polymorphs and an amorphous structure of stoichiometric Nb2O5 to
the measured XAS of Sample 1. We choose Sample 1 since we anticipate the
oxidized film with no additional fabrication steps is most similar to a
completely amorphous phase. We find that both the average amorphous spectrum
and the crystalline N-phase spectrum are most similar to the experimental
spectrum from Sample 1. Of the crystalline phases, we find that the N-phase
best agrees with the XAS measurements, but the calculation shows a larger
splitting between the two dominant peaks than the measured spectrum. This is
the case for all of the considered crystalline phases of Nb2O5 (Figure 3(b)),
which is caused by the crystalline order increasing the crystal field
splitting.

Figure 4: (a) Comparison between stoichiometric, oxygen-deficient, and oxygen-
rich amorphous calculated spectra versus the experimental spectrum of sample
1. (b) Statistical analysis of the expected relative changes in structural
descriptors given the three experimental XAS spectra. The highest variance
descriptors are on the left and lowest variance descriptors are on the right.

Previous works suggest magnetic impurities contribute to impedance-based
losses in superconducting qubitsKharitonov  _et al._ (2012); Sheridan  _et
al._ (2021). In particular, $d$-type magnetic impurities on Nb atoms were
found to be more detrimental than $p$-type impurities on O atoms in Nb
oxidesSheridan  _et al._ (2021). To investigate if our XAS measurements can
identify a low density of magnetic impurities, we compare our calculated XAS
spectra with those measured. We divide our calculations into three groups;
stoichiometric Nb2O5, oxygen rich Nb2O5 (includes oxygen interstitials or Nb
vacancies), and oxygen poor Nb2O5 (includes oxygen vacancies) with the results
given in Figure 4. As expected, we find a pre-edge feature in the oxygen rich
calculations coming from O-O dangling bonds, and resulting in the p-type
magnetic impurities. However, such a pre-peak feature is not observed in any
of the measured XAS spectra, so we can conclude that there is not a
significant density of p-type magnetic impurities in our measured samples. We
find a slightly better agreement between the oxygen-poor calculated spectra
and the measured spectra with shape of the second peak at $\approx 13$ eV a
closer match in this case. This suggests the presence of d-type magnetic
impurities associated with oxygen poor (and Nb rich) samples. To further
quantify this, we perform statistical analysis on the 1250 distinct calculated
atomic spectra and compare them to the measured XAS to correlate spectral
changes with structural and magnetic changes (details in the SI). These
results suggest that small densities of d-type magnetic impurities are present
in all three measured samples with an estimated density of 1.8$\times
10^{22}$, 1.7$\times 10^{22}$, and 1.5$\times 10^{22}$ d-type impurities per
mol formed from oxygen vacancies found in Samples 1, 2, and 3, respectively.
This density of magnetic moments is higher than previous magnetic measurements
on bulk crystalline T\- and TT-Nb2O5 where they estimated a density of
$10^{21}$ – $10^{22}$ effective magnetic moments per molHerval  _et al._
(2015). However, the greater density of magnetic moments in our samples is
consistent with the increased disorder and off-stoichiometry that we expect in
our amorphous surface oxides. To elucidate the structural changes amongst the
different samples that are correlated with changes in the XAS spectra, we
calculate the conditional mean for each structural and magnetic descriptor
given to the experimental spectrum. The relative expected changes for the most
and least varying descriptors are plotted in Figure 4b. The shape descriptors
(volume, area, etc.) refer to Voronoi polyhedra constructed around each atom,
$\sigma$ values refer to distortions within those descriptors (full
descriptions are in the SI). We find that the fabrication procedure has a
reasonably large effect on the p-type impurity density along with the shape
descriptors of both Nb and O atoms. Bond length and coordination
characteristics along with d-type impurity density showed less variation
amongst the samples.

In summary,our FEM measurements confirm the lack of long-range order in a
representative Nb-oxide film, observed from the broad halo in the average
speckle pattern of the FEM image. Comparing the calculated RDF of our ab
initio-generated stoichiometric Nb2O5 amorphous structures to the angular
average of the FEM pattern indicates that our generated amorphous
configurations are a good representation of the distribution of structures
observed in real Nb-oxide films, containing a mix of edge and corner-sharing
polyhedra motifs. We next compared our measured XAS spectra for a selection of
Nb-oxide samples (Table 1) to first-principles calculations of both
crystalline and amorphous Nb-oxide compounds, of which the amorphous phase
most closely matched the data, which isconsistent with our FEM results, and
prior elemental analysis of Nb-oxide filmsAltoé  _et al._ (2020). Finally, we
analyze our first-principles predictions for signatures of magnetic impurities
in the amorphous configurations to identify experimental markers of these
magnetic impurities in the XAS spectra. We find a better fit of the XAS
spectra for Nb2O5 configurations with oxygen vacancies, suggesting the
presence of d-type magnetic impurities. We find no evidence for pre-edge
impurity states associated with p-type magnetic impurities. Our results give
an estimate of the density of decoherence-inducing local magnetic moments, and
suggest experimental fingerprints for the characterization of superconducting
thin films using spectroscopic approaches.

## Data Availability

The data that support the findings of this study are openly available in
Zenodo at Ref Harrelson  _et al._ (2021b).

## Acknowledgments

We thank John Clarke and David Santiago for useful discussions. Specific
software and hardware is identified for information purposes only and is not
intended to imply recommendation or endorsement by NIST. This work was funded
by the U.S. Department of Energy, Office of Science, Office of Basic Energy
Sciences, Materials Sciences and Engineering Division under Contract No. DE-
AC02-05-CH11231 “High-Coherence Multilayer Superconducting Structures for
Large Scale Qubit Integration and Photonic Transduction program (QIS-LBNL)”.
This research used resources of the National Energy Research Scientific
Computing Center (NERSC), a U.S. Department of Energy Office of Science User
Facility located at Lawrence Berkeley National Laboratory, operated under
Contract No. DE-AC02-05CH11231. E.S. acknowledges support from the US-Irish
Fulbright Commission, the Air Force Office of Scientific Research under award
number FA9550-18-1-0480 and the EPSRC Centre for Doctoral Training in Cross-
Disciplinary Approaches to Non-Equilibrium Systems (EP/L015854/1). This work
also used the Extreme Science and Engineering Discovery Environment (XSEDE),
which is supported by National Science Foundation grant number ACI-1548562.
Electron microscopy data acquisition for this work was supported by National
Science Foundation STROBE Grant No. DMR-1548924. Work at the Molecular Foundry
was supported by the Office of Science, Office of Basic Energy Sciences, of
the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. This
research used resources of the Advanced Light Source, which is a DOE Office of
Science User Facility under Contract no. DE-AC02-05CH11231.

## Author Declarations

The authors have no conflicts to disclose.

## References

  * Devoret and Schoelkopf (2013) M. H. Devoret and R. J. Schoelkopf, “Superconducting circuits for quantum information: an outlook,” Science 339, 1169–1174 (2013).
  * Kjaergaard  _et al._ (2020) M. Kjaergaard, M. E. Schwartz, J. Braumüller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver, “Superconducting qubits: Current state of play,” Annual Review of Condensed Matter Physics 11, 369–395 (2020).
  * McDermott (2009) R. McDermott, “Materials origins of decoherence in superconducting qubits,” IEEE Trans. Appl. Supercond. 19, 2–13 (2009).
  * Oliver and Welander (2013) W. D. Oliver and P. B. Welander, “Materials in superconducting quantum bits,” MRS Bull. 38 (2013), 10.1557/mrs.2013.229.
  * de Leon  _et al._ (2021) N. P. de Leon, K. M. Itoh, D. Kim, K. K. Mehta, T. E. Northup, H. Paik, B. S. Palmer, N. Samarth, S. Sangtawesin, and D. W. Steuerman, “Materials challenges and opportunities for quantum computing hardware,” Science 372 (2021), 10.1126/science.abb2823.
  * Müller, Cole, and Lisenfeld (2019) C. Müller, J. H. Cole, and J. Lisenfeld, “Towards understanding two-level-systems in amorphous solids: Insights from quantum circuits,” Reports on Progress in Physics 82, 1–34 (2019).
  * Phillips (1987) W. A. Phillips, “Two-level states in glasses,” Rep. Prog. Phys. 50, 1657–1708 (1987).
  * Burnett, Faoro, and Lindström (2016) J. Burnett, L. Faoro, and T. Lindström, “Analysis of high quality superconducting resonators: consequences for TLS properties in amorphous oxides,” Supercond. Sci. Technol. 29, 044008 (2016).
  * Wilen  _et al._ (2020) C. D. Wilen, S. Abdullah, N. A. Kurinsky, C. Stanford, L. Cardani, G. D’Imperio, C. Tomei, L. Faoro, L. B. Ioffe, C. H. Liu, A. Opremcak, B. G. Christensen, J. L. DuBois, and R. McDermott, “Correlated charge noise and relaxation errors in superconducting qubits,”  (2020), arXiv:2012.06029 [quant-ph] .
  * Cardani  _et al._ (2020) L. Cardani, F. Valenti, N. Casali, G. Catelani, T. Charpentier, M. Clemenza, I. Colantoni, A. Cruciani, L. Gironi, L. Grünhaupt, D. Gusenkova, F. Henriques, M. Lagoin, M. Martinez, G. Pettinari, C. Rusconi, O. Sander, A. V. Ustinov, M. Weber, W. Wernsdorfer, M. Vignati, S. Pirro, and I. M. Pop, “Reducing the impact of radioactivity on quantum circuits in a deep-underground facility,”  (2020), arXiv:2005.02286 [cond-mat.supr-con] .
  * Kharitonov  _et al._ (2012) M. Kharitonov, T. Proslier, A. Glatz, and M. J. Pellin, “Surface impedance of superconductors with magnetic impurities,” Phys. Rev. B Condens. Matter 86, 024514 (2012).
  * Proslier  _et al._ (2011) T. Proslier, M. Kharitonov, M. Pellin, J. Zasadzinski, and Ciovati, “Evidence of surface paramagnetism in niobium and consequences for the superconducting cavity surface impedance,” IEEE Trans. Appl. Supercond. 21, 2619–2622 (2011).
  * Sheridan  _et al._ (2021) E. Sheridan, T. F. Harrelson, E. Sivonxay, K. A. Persson, M. V. P. Altoe, I. Siddiqi, D. F. Ogletree, D. I. Santiago, and S. M. Griffin, “Microscopic theory of magnetic Disorder-Induced decoherence in superconducting nb films,” (2021).
  * Vepsäläinen  _et al._ (2020) A. P. Vepsäläinen, A. H. Karamlou, J. L. Orrell, A. S. Dogra, B. Loer, F. Vasconcelos, D. K. Kim, A. J. Melville, B. M. Niedzielski, J. L. Yoder, S. Gustavsson, J. A. Formaggio, B. A. VanDevender, and W. D. Oliver, “Impact of ionizing radiation on superconducting qubit coherence,” Nature 584, 551–556 (2020).
  * Martinis (2020) J. M. Martinis, “Saving superconducting quantum processors from qubit decay and correlated errors generated by gamma and cosmic rays,”  (2020), arXiv:2012.06137 [quant-ph] .
  * Place  _et al._ (2021) A. P. M. Place, L. V. H. Rodgers, P. Mundada, B. M. Smitham, M. Fitzpatrick, Z. Leng, A. Premkumar, J. Bryon, A. Vrajitoarea, S. Sussman, G. Cheng, T. Madhavan, H. K. Babla, X. H. Le, Y. Gang, B. Jäck, A. Gyenis, N. Yao, R. J. Cava, N. P. de Leon, and A. A. Houck, “New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds,” Nat. Commun. 12, 1779 (2021), arXiv:2003.00024 [quant-ph] .
  * Kaplan  _et al._ (1976) S. B. Kaplan, C. Chi, D. Langenberg, J.-J. Chang, S. Jafarey, and D. Scalapino, “Quasiparticle and phonon lifetimes in superconductors,” Physical Review B 14, 4854 (1976).
  * Delheusy  _et al._ (2008) M. Delheusy, A. Stierle, N. Kasper, R. Kurta, A. Vlad, H. Dosch, C. Antoine, A. Resta, E. Lundgren, and J. Andersen, “X-ray investigation of subsurface interstitial oxygen at nb/oxide interfaces,” Applied Physics Letters 92, 101911 (2008).
  * Altoé  _et al._ (2020) M. V. P. Altoé, A. Banerjee, C. Berk, A. Hajr, A. Schwartzberg, C. Song, M. A. Ghadeer, S. Aloni, M. J. Elowson, J. M. Kreikebaum, E. K. Wong, S. Griffin, S. Rao, A. Weber-Bargioni, A. M. Minor, D. I. Santiago, S. Cabrini, I. Siddiqi, and D. F. Ogletree, “Localization and reduction of superconducting quantum coherent circuit losses,”  (2020), arXiv:2012.07604 .
  * Mergenthaler  _et al._ (2021) M. Mergenthaler, S. Paredes, P. Müller, C. Müller, S. Filipp, M. Sandberg, J. Hertzberg, V. P. Adiga, M. Brink, and A. Fuhrer, “Ultrahigh vacuum packaging and surface cleaning for quantum devices,” Review of Scientific Instruments 92, 025121 (2021).
  * Verjauw  _et al._ (2021) J. Verjauw, A. Potočnik, M. Mongillo, R. Acharya, F. Mohiyaddin, G. Simion, A. Pacco, T. Ivanov, D. Wan, A. Vanleenhove, _et al._ , “Investigation of microwave loss induced by oxide regrowth in high-q niobium resonators,” Physical Review Applied 16, 014018 (2021).
  * Romanenko and Schuster (2017) A. Romanenko and D. I. Schuster, “Understanding quality factor degradation in superconducting niobium cavities at low microwave field amplitudes,” Phys. Rev. Lett. 119, 264801 (2017).
  * Romanenko  _et al._ (2020) A. Romanenko, R. Pilipenko, S. Zorzetti, D. Frolov, M. Awida, S. Belomestnykh, S. Posen, and A. Grassellino, “Three-Dimensional superconducting resonators at $T<20$ mk with photon lifetimes up to $\\{\tau\\}=2$ s,” Phys. Rev. Applied 13, 034032 (2020).
  * Heinrich, Pascual, and Franke (2018) B. W. Heinrich, J. I. Pascual, and K. J. Franke, “Single magnetic adsorbates on s-wave superconductors,” Progress in Surface Science 93, 1–19 (2018).
  * Voyles and Muller (2002) P. Voyles and D. Muller, “Fluctuation microscopy in the STEM,” Ultramicroscopy 93, 147–159 (2002).
  * Kresse and Hafner (1993) G. Kresse and J. Hafner, “Ab initio molecular dynamics for liquid metals,” Physical Review B 47, 558–561 (1993).
  * Harrelson  _et al._ (2021a) T. F. Harrelson, E. Sheridan, E. Sivonxay, K. A. Persson, and S. M. Griffin, “Amorphous Niobium Oxide Structures Calculated from First Principles using Density Functional Theory and Molecular Dynamics,”  (2021a), 10.5281/ZENODO.5139270.
  * Vinson  _et al._ (2011) J. Vinson, J. J. Rehr, J. J. Kas, and E. L. Shirley, “Bethe-salpeter equation calculations of core excitation spectra,” Phys. Rev. B 83, 115106 (2011).
  * Gilmore  _et al._ (2015) K. Gilmore, J. Vinson, E. Shirley, D. Prendergast, C. Pemmaraju, J. Kas, F. Vila, and J. Rehr, “Efficient implementation of core-excitation bethe-salpeter equation calculations,” Comput. Phys. Comm. 197, 109 – 117 (2015).
  * (30) www.ocean-code.com v 2.9.7.
  * Giannozzi  _et al._ (2017) P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. D. Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer, U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawamura, H.-Y. Ko, A. Kokalj, E. Küçükbenli, M. Lazzeri, M. Marsili, N. Marzari, F. Mauri, N. L. Nguyen, H.-V. Nguyen, A. O. de-la Roza, L. Paulatto, S. Poncé, D. Rocca, R. Sabatini, B. Santra, M. Schlipf, A. P. Seitsonen, A. Smogunov, I. Timrov, T. Thonhauser, P. Umari, N. Vast, X. Wu, and S. Baroni, “Advanced capabilities for materials modelling with quantum ESPRESSO,” Journal of Physics: Condensed Matter 29, 465901 (2017).
  * (32) www.quantum-espresso.org v 6.7.
  * van Setten  _et al._ (2018) M. van Setten, M. Giantomassi, E. Bousquet, M. Verstraete, D. Hamann, X. Gonze, and G.-M. Rignanese, “The pseudodojo: Training and grading a 85 element optimized norm-conserving pseudopotential table,” Computer Physics Communications 226, 39 – 54 (2018).
  * (34) http://www.pseudo-dojo.org Scalar-relativstic v. 0.4.
  * Hamann (2013) D. R. Hamann, “Optimized norm-conserving vanderbilt pseudopotentials,” Phys. Rev. B 88, 085117 (2013).
  * (36) The open-source code oncvpsp is avaiable at http://www.mat-simresearch.com v. 3.3.1.
  * Voyles and Abelson (2003) P. M. Voyles and J. R. Abelson, “Medium-range order in amorphous silicon measured by fluctuation electron microscopy,” Solar energy materials and solar cells 78, 85–113 (2003).
  * Daulton, Bondi, and Kenneth (2010) T. Daulton, K. Bondi, and K. Kenneth, “Nanobeam diffraction fluctuation electron microscopy technique for structural characterization of disordered materials — application to Al${}_{8}8-x$Y7Fe5Tix metallic glasses,” Ultramicroscopy 110, 1279–1289 (2010).
  * Kennedy  _et al._ (2020) E. Kennedy, N. Reynolds, L. Rangel DaCosta, F. Hellman, C. Ophus, and M. Scott, “Tilted fluctuation electron microscopy,” Applied Physics Letters 117, 091903 (2020).
  * Hwang and Voyles (2011) J. Hwang and P. Voyles, “Variable resolution fluctuation electron microscopy on cu-zr metallic glass using a wide range of coherent stem probe size,” Microscopy and Microanalysis 17, 67–74 (2011).
  * Frati, Hunault, and de Groot (2020) F. Frati, M. O. Hunault, and F. M. de Groot, “Oxygen k-edge x-ray absorption spectra,” Chemical reviews 120, 4056–4110 (2020).
  * Herval  _et al._ (2015) L. K. Herval, D. Von Dreifus, A. C. Rabelo, A. D. Rodrigues, E. C. Pereira, Y. G. Gobato, A. J. De Oliveira, and M. P. De Godoy, “The role of defects on the structural and magnetic properties of Nb2O5,” Journal of Alloys and Compounds 653, 358–362 (2015).
  * Harrelson  _et al._ (2021b) T. F. Harrelson, J. Vinson, E. Sheridan, and S. M. Griffin, “Calculated O K-edge XAS Spectra of Niobium Oxide Phases using Bethe-Salpeter equation,”  (2021b), 10.5281/ZENODO.5156863.


# A Minimal Approach to Baryogenesis via Affleck-Dine and Inflaton Mass Terms

Amy Lloyd-Stubbs and John McDonald<EMAIL_ADDRESS><EMAIL_ADDRESS>Dept. of Physics, Lancaster University, Lancaster
LA1 4YB, UK

###### Abstract

We present a minimal approach to the generation of the baryon ($B$) asymmetry
of the Universe, in which the asymmetry is generated in a complex inflaton
condensate via $B$-violating quadratic inflaton potential terms and the
Affleck-Dine (AD) mechanism. We show that the $B$-violating quadratic mass
terms create an oscillating asymmetry in the complex inflaton condensate at
late times. The final asymmetry transferred to the Standard Model sector at
reheating is naturally reduced to the magnitude of the observed $B$ asymmetry
by the effect of averaging over the $B$ oscillations. This approach to
baryogenesis can easily be realised in a wide range of inflation models.

##  I Introduction

The Affleck-Dine (AD) mechanism [1, 2] provides a remarkably simple and
elegant explanation for the baryon ($B$) asymmetry of the Universe. A complex
scalar with a $U(1)$ global symmetry, corresponding to conserved baryon
number, evolves into a coherently oscillating condensate. $B$ violating terms
in the potential act on the field, pushing it into an elliptical trajectory in
the complex field plane, which is equivalent to a $B$ asymmetry in the scalar
field.

The conventional AD mechanism is based on a complex scalar field $\Phi$ with a
potential which at late times is dominated by a $|\Phi|^{2}$ mass term.
Higher-order operators that violate baryon number cause the real ($\phi_{1}$)
and imaginary ($\phi_{2}$) parts of $\Phi$ to evolve differently when the
$|\Phi|^{2}$ term comes to dominate the potential, pushing the trajectory into
an ellipse in the $(\phi_{1},\phi_{2})$ plane. The higher-order operators
become less important as the magnitude of $\Phi$ decreases due to expansion,
effectively switching off the $B$ violation and leaving a conserved baryon
asymmetry in the complex field at late times.

Here we will present a new and unconventional implementation of AD
baryogenesis, in which $B$-violating $\Phi^{2}$ terms in the potential of a
complex inflaton $\Phi$ generate the asymmetry111The same model can also be
used to generate a lepton asymmetry which is subsequently processed via
sphalerons into a baryon asymmetry.. (Applications of the conventional AD
mechanism to a complex inflaton have been considered in [3, 4, 5, 6, 7].) We
will show that these terms generate a $B$ asymmetry in the $\Phi$ condensate
which oscillates about zero. When the condensate asymmetry is transferred to
the Standard Model (SM) sector by $\Phi$ decay, a net asymmetry is left in the
SM sector. The oscillating baryon asymmetry initially generated in the $\Phi$
condensate is typically much larger than that required to explain the observed
baryon-to-entropy ratio. The asymmetry transferred to the SM is subsequently
suppressed by averaging over the condensate asymmetry oscillations, reducing
the asymmetry to the observed value222AD baryogenesis via mass terms has
previously been considered in the context of a different class of model in
[8]. The analysis of [8] assumes that the averaging over of asymmetry
oscillations washes out the final asymmetry. We will show that although the
asymmetry is suppressed, it is significantly non-zero. This suppression plays
an important role in the model described here.. The resulting model is
dynamically quite different from existing inflaton-based AD baryogenesis
models, with the inflaton asymmetry being generated at late times during
inflaton oscillations rather than during or shortly after inflation.

The paper is organised as follows. In Section 2 we discuss the generation of
the asymmetry via quadratic B-violating potential terms. In Section 3 we
consider possible washout of the asymmetry via inflaton exchange operators. In
Section 4 we discuss the validity of the classical calculation of the
asymmetry. In Section 5 we present our conclusions.

##  II Affleck-Dine Baryogenesis via Quadratic Potential Terms

We will consider a renormalisable $B$ symmetric inflaton potential together
with $B$-violating $\Phi^{2}$ terms,

$\mbox{$$}V(\Phi)=m_{\phi}^{2}|\Phi|^{2}+\lambda_{\Phi}|\Phi|^{4}-(A\Phi^{2}+{\rm
h.\,c.})~{},\vspace{0.1cm}$ (1)

where $A$ is real and positive. Such potentials are naturally compatible with
inflation models which are non-minimally coupled to gravity [9]. More
generally, they represent the leading order terms of an inflaton potential
during post-inflation evolution333Whilst the inflaton is the natural candidate
for the field responsible for reheating, we note that the model can apply to
any coherently oscillating complex scalar that is responsible for reheating..
$\Phi$ is initially coherently oscillating, with the potential dominated by
the $|\Phi|^{4}$ term and with no asymmetry in the field. In terms of
$\Phi=(\phi_{1}+i\phi_{2})/\sqrt{2}$, the potential becomes

$\mbox{$$}V(\Phi)=\frac{1}{2}(m_{\Phi}^{2}-2A)\phi_{1}^{2}+\frac{1}{2}(m_{\Phi}^{2}+2A)\phi_{2}^{2}+\frac{\lambda_{\Phi}}{4}(\phi_{1}^{2}+\phi_{2}^{2})^{2}~{}.\vspace{0.1cm}$
(2)

The field equations are

$\mbox{$$}\ddot{\phi}_{1}+3H\dot{\phi}_{1}=-m_{1}^{2}\phi_{1}-\lambda_{\Phi}(\phi_{1}^{2}+\phi_{2}^{2})\phi_{1}~{}\vspace{0.1cm}$
(3)

and

$\mbox{$$}\ddot{\phi}_{2}+3H\dot{\phi}_{2}=-m_{2}^{2}\phi_{2}-\lambda_{\Phi}(\phi_{1}^{2}+\phi_{2}^{2})\phi_{2}~{},\vspace{0.1cm}$
(4)

where

$\mbox{$$}m_{1}^{2}=m_{\Phi}^{2}-2A\;\;\;;\;\;\;m_{2}^{2}=m_{\Phi}^{2}+2A~{}.\vspace{0.1cm}$
(5)

In the limit $\lambda_{\Phi}\rightarrow 0$ the equations for $\phi_{1}$ and
$\phi_{2}$ are decoupled from each other, with coherently oscillating
solutions for $\phi_{1}$ and $\phi_{2}$ which have angular frequencies $m_{1}$
and $m_{2}$, respectively.

We first derive an analytical expression for the asymmetry using a threshold
approximation, which we then compare to a complete numerical solution. In the
threshold approximation we consider the potential to be approximated by

$V(\Phi)=\lambda_{\Phi}|\Phi|^{4}\;\;\;;\;\;\;\phi>\phi_{*}$
$\mbox{$$}V(\Phi)=m_{\Phi}^{2}|\Phi|^{2}-(A\Phi^{2}+{\rm
h.c.})\;\;\;;\;\;\;\phi<\phi_{*}~{},\vspace{0.1cm}$ (6)

where $\phi_{*}=m_{\Phi}/\sqrt{\lambda_{\Phi}}$ is the value of $\phi$ at
which $V^{\prime}(\phi)$ becomes dominated by the $|\Phi|^{4}$ term (here
$\Phi=\phi e^{i\theta}/\sqrt{2}$ and we have set $A=0$ when determining
$\phi_{*}$). The potential is initially strongly dominated by the $|\Phi|^{4}$
term, with $\phi_{i}\gg\phi_{*}$, and the field is initially at rest with
initial values $(\phi_{1,\;i},\phi_{2,\;i})$. Assuming rapid coherent
oscillations, the field amplitude will initially evolve as $\phi\propto 1/a$
when $\phi>\phi_{*}$. Therefore the field amplitudes at $a_{*}$ are

$\mbox{$$}\phi_{1,\;*}=\left(\frac{a_{i}}{a_{*}}\right)\phi_{1,\;i}=\left(\frac{\phi_{*}}{\phi_{i}}\right)\phi_{1,\;i}\;\;\;;\;\;\;\phi_{2,\;*}=\left(\frac{a_{i}}{a_{*}}\right)\phi_{2,\;i}=\left(\frac{\phi_{*}}{\phi_{i}}\right)\phi_{2,\;i}~{},\vspace{0.1cm}$
(7)

where $\phi_{i}=\left(\phi_{1,\;i}^{2}+\phi_{2,\;i}^{2}\right)^{1/2}$. The
field evolves purely due to the mass squared terms once $a>a_{*}$. We assume
that $m_{1,2}\gg H$, so that we can neglect the effect of expansion on the
rapid $\phi_{1,2}$ oscillations and simply factor in the effect of expansion
by damping the oscillation amplitude. The solution for $\phi_{1}$ and
$\phi_{2}$ is then

$\mbox{$$}\phi_{1}=\phi_{1,\;*}\left(\frac{a_{*}}{a}\right)^{3/2}\cos(m_{1}(t-t_{*}))\;\;\;;\;\;\;\phi_{2}=\phi_{2,\;*}\left(\frac{a_{*}}{a}\right)^{3/2}\cos(m_{2}(t-t_{*}))~{}.\vspace{0.1cm}$
(8)

The baryon asymmetry in the $\Phi$ condensate is

$\mbox{$$}n(t)=i\left(\Phi^{\dagger}\dot{\Phi}-\dot{\Phi}^{\dagger}\Phi\right)=\dot{\phi}_{1}\phi_{2}-\dot{\phi}_{2}\phi_{1}~{}.\vspace{0.1cm}$
(9)

Therefore

$\mbox{$$}n(t)=\phi_{1,\;*}\phi_{2,\;*}\left(\frac{a_{*}}{a}\right)^{3}\left[m_{2}\sin(m_{2}(t-t_{*}))\cos(m_{1}(t-t_{*}))-m_{1}\sin(m_{1}(t-t_{*}))\cos(m_{2}(t-t_{*}))\right]~{}.\vspace{0.1cm}$
(10)

We will assume that $2A\ll m_{\Phi}^{2}$. In this limit, to leading order in
$A/m_{\Phi}^{2}$, the condensate baryon asymmetry becomes

$\mbox{$$}n(t)=\phi_{1,\;*}\phi_{2,\;*}\left(\frac{a_{*}}{a}\right)^{3}\left[m_{\Phi}\sin\left(\frac{2A(t-t_{*})}{m_{\Phi}}\right)+\frac{A}{m_{\Phi}}\sin\left(2m_{\Phi}(t-t_{*})\right)\right]~{}.\vspace{0.1cm}$
(11)

During averaging over the $\phi_{1\,,2}$ coherent field oscillations, we can
consider the scale factor to be constant since $H\ll m_{\Phi}$. The second
term in Eq. (11) then averages to zero. The condensate asymmetry at $t>t_{*}$,
in terms of the initial field values, is then

$\mbox{$$}n(t)=\phi_{1,\;i}\phi_{2,\;i}\left(\frac{\phi_{i}}{\phi_{*}}\right)\left(\frac{a_{i}}{a}\right)^{3}m_{\Phi}\sin\left(\frac{2A(t-t_{*})}{m_{\Phi}}\right)~{}.\vspace{0.1cm}$
(12)

Thus the baryon asymmetry in the $\Phi$ condensate oscillates about zero with
period $T_{asy}=\pi m_{\Phi}/A$.

It is useful to define a comoving asymmetry
$n_{c}(t)\equiv(a(t)/a_{i})^{3}n(t)$, which is constant when there is no
production or decay of the asymmetry. For the threshold model at $t>t_{*}$

$\mbox{$$}n_{c}(t)=\phi_{1,\;i}\phi_{2,\;i}\left(\frac{\phi_{i}}{\phi_{*}}\right)m_{\Phi}\sin\left(\frac{2A(t-t_{*})}{m_{\Phi}}\right)~{},\vspace{0.1cm}$
(13)

with $n_{c}(t)=0$ at $t<t_{*}$. The $\Phi$ condensate asymmetry is assumed to
transfer to a conserved SM baryon asymmetry via $B$-conserving $\Phi$ decays
to SM particles444A specific implementation of the model to baryogenesis from
AD leptogenesis via a decaying inflaton will be presented in a future work
[10]. Here we focus on the general features of inflaton mass term AD
baryogenesis.,555It is also possible for the inflaton to decay via gravity
mediated modes [11]. The importance of this process will depend upon the
coupling of the inflaton to the Ricci curvature in a given inflation model.
The condensate will decay away completely after a time
$t_{R}\approx\Gamma_{\Phi}^{-1}$, where $R$ denotes reheating, with continuous
production of SM baryon asymmetry due to decay of the condensate asymmetry
from $t_{*}$ to $t_{R}$. Neglecting any reduction of the $\Phi$ field due to
decays at $t<t_{R}$, the comoving baryon asymmetry transferred to the SM
sector, which we denote by $\hat{n}_{c}$(t), is

$\mbox{$$}\hat{n}_{c}(t)=\int_{t_{i}}^{t}\Gamma_{\Phi}n_{c}(t)dt~{}.\vspace{0.1cm}$
(14)

Thus the comoving baryon asymmetry transferred out of the $\Phi$ condensate as
a function of $t$ is

$\mbox{$$}\hat{n}_{c}(t)=\frac{\Gamma_{\Phi}\phi_{1,\;i}\phi_{2,\;i}m_{\Phi}^{2}}{2A}\left(\frac{\phi_{i}}{\phi_{*}}\right)\left[1-\cos\left(\frac{2A(t-t_{*})}{m_{\Phi}}\right)\right]~{}.\vspace{0.1cm}$
(15)

$\hat{n}_{c}(t)$ increases linearly with $t-t_{*}$ until $t-t_{*}\approx\pi
m_{\Phi}/4A$. On longer timescales, $\hat{n}_{c}(t)$ oscillates between a
maximum value and zero with period $T_{asy}$. The maximum possible asymmetry
is obtained when $A=A_{max}=\pi m_{\Phi}\Gamma_{\Phi}/2$.

The $\Phi$ condensate decays away completely once
$t-t_{*}\;^{>}{}_{\sim}\;\Gamma_{\Phi}^{-1}$. To take into account the
B-conserving decay of the condensate asymmetry, we include in Eq. (14) an
exponential decay factor,

$\mbox{$$}\hat{n}_{c}(t)=\int_{t_{*}}^{t}\Gamma_{\Phi}n_{c}(t)e^{-\Gamma_{\Phi}(t-t_{*})}dt~{}.\vspace{0.1cm}$
(16)

The total comoving asymmetry transferred to the SM sector as
$t\rightarrow\infty$ is then

$\mbox{$$}\hat{n}_{c,\;tot}=\frac{\Gamma_{\Phi}\phi_{1,\;i}\phi_{2,\;i}m_{\Phi}^{2}}{2A}\left(\frac{\phi_{i}}{\phi_{*}}\right)\left(1+\left(\frac{\Gamma_{\Phi}m_{\Phi}}{2A}\right)^{2}\right)^{-1}~{}.\vspace{0.1cm}$
(17)

The transferred asymmetry is proportional to $A$ until
$A>\Gamma_{\Phi}m_{\Phi}/2$, in which case $\tau_{\Phi}>T_{asy}$ and the
transferred asymmetry decreases as $A^{-1}$ and $\tau_{\Phi}^{-1}$, where
$\tau_{\Phi}=\Gamma_{\Phi}^{-1}$ is the lifetime of the $\Phi$ scalars. This
can be understood as due to the effect of averaging condensate oscillations
over the time taken for the condensate to decay. When $\tau_{\phi}\gg
T_{asy}$, the asymmetry in the condensate will undergo many oscillations from
positive to negative values during the decay of the condensate. Therefore the
asymmetry produced during a positive half-cycle will almost cancel against
that produced during the following negative half-cycle, up to the effect of
the small decrease in the condensate asymmetry amplitude due to the decay of
the condensate during $\Delta t\sim T_{asy}$. Therefore only a small net
asymmetry is produced during each condensate oscillation cycle as compared to
the case with $T_{asy}\;^{>}{}_{\sim}\;\tau_{\Phi}$, where there is no
averaging over oscillations.

We first consider the case where the lifetime of $\Phi$ is much longer than
$T_{asy}$, such that $2A/m_{\Phi}\Gamma_{\Phi}\gg 1$. $\hat{n}_{c\;tot}$ can
then be expressed as

$\mbox{$$}\hat{n}_{c,\;tot}=\frac{\Gamma_{\Phi}\phi_{i}^{2}m_{\Phi}^{2}\sin\left(2\theta\right)}{4A}\left(\frac{\phi_{i}}{\phi_{*}}\right)~{},\vspace{0.1cm}$
(18)

where $\theta$ is the initial phase of $\Phi$. The total baryon asymmetry
transferred to the SM, $\hat{n}_{tot}$, is then

$\mbox{$$}\hat{n}_{tot}=\left(\frac{a_{i}}{a_{R}}\right)^{3}\hat{n}_{c,\;tot}=\frac{3M_{Pl}^{2}\Gamma_{\Phi}^{3}\sin\left(2\theta\right)}{2A}~{},\vspace{0.1cm}$
(19)

where we have used $a\propto H^{-2/3}$ when $a>a_{R}$ and $a\propto 1/\phi$
when $a<a_{R}$ to obtain the final expression. This can also be expressed in
terms of the baryon-to-entropy ratio, $n_{B}/s$. Using $s=4k_{T}^{2}T^{3}$ and
$\Gamma_{\Phi}=H_{R}=k_{T_{R}}T_{R}^{2}/M_{Pl}$, where $T_{R}$ is the
reheating temperature and $k_{T}=(\pi^{2}g(T)/90)^{1/2}$, the baryon-to-
entropy ratio is

$\mbox{$$}\frac{n_{B}}{s}\equiv\frac{\hat{n}_{tot}}{s}=\frac{3}{8}\frac{k_{T_{R}}T_{R}^{3}\sin\left(2\theta\right)}{AM_{Pl}}=5.2\times
10^{-21}\frac{m_{\Phi}^{2}}{A}\left(\frac{T_{R}}{10^{8}{\rm\
GeV}}\right)^{3}\left(\frac{10^{13}{\rm\
GeV}}{m_{\Phi}}\right)^{2}\sin\left(2\theta\right)~{},\vspace{0.1cm}$ (20)

where we have normalised the expression to some representative
values666$T_{R}=10^{8}{\rm\ GeV}$ is within the range of reheating
temperatures that may be detectable in the spectrum of primordial
gravitational waves [12]. of $T_{R}$ and $m_{\Phi}$. The observed baryon-to-
entropy ratio is $(n_{B}/s)_{obs}=0.861\pm 0.005\times 10^{-10}$. In order to
account for the observed asymmetry, we require that

$\mbox{$$}\frac{A^{1/2}}{m_{\Phi}}=7.8\times
10^{-6}\sin^{1/2}\left(2\theta\right)\left(\frac{10^{13}{\rm\
GeV}}{m_{\Phi}}\right)\left(\frac{T_{R}}{10^{8}{\rm\
GeV}}\right)^{3/2}~{}.\vspace{0.1cm}$ (21)

The maximum possible asymmetry, which corresponds to
$A=\Gamma_{\Phi}m_{\Phi}/2$ in Eq. (17), is

$\mbox{$$}\frac{n_{B,\;max}}{s}=\frac{3T_{R}\sin\left(2\theta\right)}{8m_{\Phi}}=3.8\times
10^{-6}\,\left(\frac{T_{R}}{10^{8}{\rm\ GeV}}\right)\left(\frac{10^{13}{\rm\
GeV}}{m_{\Phi}}\right)\sin\left(2\theta\right)~{}.\vspace{0.1cm}$ (22)

This can easily be much larger than the observed baryon asymmetry. Therefore
the suppression of the asymmetry by averaging over oscillations plays an
important role in this model.

In the case where $\Phi$ decays before any condensate asymmetry oscillations
can occur, corresponding to $\Gamma_{\Phi}m_{\Phi}/2A\gg 1$ in Eq. (17), the
total transferred asymmetry obtains an additional factor
$(2A/\Gamma_{\Phi}m_{\Phi})^{2}$ compared to Eq. (20). Therefore

$\mbox{$$}\frac{n_{B}}{s}=\frac{3}{2}\frac{AM_{Pl}\sin(2\theta)}{k_{T_{R}}T_{R}m_{\phi}^{2}}~{}\vspace{0.1cm}$
(23)

and we find that the required value of $A^{1/2}/m_{\Phi}$ is

$\mbox{$$}\frac{A^{1/2}}{m_{\Phi}}=8.9\times
10^{-11}\,\left(\frac{T_{R}}{10^{8}{\rm\
GeV}}\right)^{1/2}\left(\frac{1}{\sin\left(2\theta\right)}\right)^{1/2}~{}.\vspace{0.1cm}$
(24)

This is typically much smaller than in the case with asymmetry oscillations,
due to the lack of additional suppression of the baryon asymmetry from
averaging over condensate oscillations.

The threshold asymmetry is a good approximation if the $B$-violating mass
terms do not cause the field to significantly evolve until the potential is
$|\Phi|^{2}$ dominated. The condition for this to be true, which we have
confirmed in our numerical solutions, is that the mass of the angular field
perturbations about the minimum of the potential as a function of $\theta$,
$m_{\delta\theta}=2A^{1/2}$, is less than $H$ when $\phi=\phi_{*}$. This is
satisfied if

$\mbox{$$}\frac{A^{1/2}}{m_{\Phi}}\;^{<}_{\sim}\;\frac{A^{1/2}_{th}}{m_{\Phi}}=\frac{m_{\Phi}}{4\sqrt{\lambda_{\Phi}}M_{Pl}}\equiv
1.0\times 10^{-6}\lambda_{\Phi}^{-1/2}\left(\frac{m_{\Phi}}{10^{13}\;{\rm\
GeV}}\right)~{}.\vspace{0.1cm}$ (25)

We finally compare the threshold approximation to the complete numerical
solution for the case $\Gamma_{\Phi}(t-t_{*})\ll 1$ 777Further details of the
numerical analysis will be presented in [10]. As an example, we show in Figure
1 the numerical results for the case $m_{\Phi}=10^{16}{\rm\ GeV}$ and
$\lambda_{\Phi}=0.1$ for a range of values of $A^{1/2}/m_{\Phi}$. The
analytical approximation in left-hand figure is given by Eq. (13) and in the
right-handed figure by Eq. (15), with $\Gamma_{\Phi}$ corresponding to
$T_{R}=10^{8}{\rm\ GeV}$. For this case, the upper limit for the threshold
approximation to be valid is $A_{th}^{1/2}/m_{\Phi}\approx 3\times 10^{-3}$.
We find that the threshold approximation is in perfect agreement with the
numerical solution for both the condensate and transferred asymmetries when
Eq. (25) is satisfied. For larger $A^{1/2}/m_{\Phi}$, the evolution during the
$|\Phi|^{4}$ dominated era modifies the asymmetries. The amplitude of the
transferred asymmetry $A\hat{n}_{c}$ rapidly decreases with increasing
$A>A_{th}$ down to an approximately constant value, which is suppressed
relative to the threshold value of $A\hat{n}_{c}$ by a factor that numerically
is approximately $m_{\Phi}/10^{17}{\rm GeV}$. The transferred asymmetry
$A\hat{n}_{c}$ oscillates between zero and a maximum when the threshold
approximation is valid, but for larger $A^{1/2}/m_{\Phi}$ it oscillates about
zero. However, since the transferred asymmetry is the total asymmetry
transferred to the SM sector as a function of time after averaging over
condensate asymmetry oscillations, the oscillation of the transferred
asymmetry about zero has no impact on the typical magnitude of the baryon
asymmetry transferred to the SM.

Figure 1: The condensate asymmetry (left) and transferred asymmetry (right)
for the case $m_{\Phi}=10^{16}{\rm\ GeV}$, $\lambda_{\Phi}=0.1$ and
$T_{R}=10^{8}{\rm\ GeV}$. The threshold asymmetry and the numerical results
for $A^{1/2}/m_{\Phi}=0.001,0.005,0.007,0.01$ and 0.05 are shown. (The
numerical result for $A^{1/2}/m_{\Phi}=0.001$ coincides with the threshold
result, in agreement with Eq. (25).)

##  III Baryon Washout due to Inflaton Exchange

In application to a specific model, the possible washout of the asymmetry must
be considered. The interaction which allows the decay of the inflaton will
generally result in a B-violating operator via $\Phi$ exchange. Dimensionally,
the rate of B-violating scattering processes at reheating due to $\phi_{1}$
and $\phi_{2}$ exchange is

$\mbox{$$}\Gamma_{\Delta
B}\sim\frac{\lambda_{\psi}^{2}A^{2}T_{R}^{5}}{m_{\Phi}^{8}}~{},\vspace{0.1cm}$
(26)

where $\lambda_{\psi}$ is the coupling responsible for $\Phi$ decay and $A$ is
necessary in the scattering amplitude in order to have B-violation. Washout
due to $\Phi$ exchange will be negligible if $\Gamma_{\Delta B}<H(T_{R})$,
which is satisfied if

$\mbox{$$}\lambda_{\psi}\lesssim\frac{m_{\Phi}^{4}}{M_{Pl}^{1/2}T_{R}^{3/2}A}=6\times
10^{4}\left(\frac{m_{\Phi}^{2}}{A}\right)\left(\frac{m_{\Phi}}{10^{13}{\rm\
GeV}}\right)^{2}\left(\frac{10^{8}{\rm\
GeV}}{T_{R}}\right)^{3/2}~{}.\vspace{0.1cm}$ (27)

The inflaton decay rate is
$\Gamma_{\Phi}\approx\lambda_{\psi}^{2}m_{\Phi}/4\pi$, therefore the reheating
temperature from $H(T_{R})=\Gamma_{\Phi}$ is
$T_{R}\approx\lambda_{\psi}(m_{\Phi}M_{Pl})^{1/2}$. Thus Eq. (27) is satisfied
if

$\mbox{$$}T_{R}\;^{<}{}_{\sim}\;\left(\frac{m_{\Phi}^{2}}{A}\right)^{2/5}m_{\Phi}~{},\vspace{0.1cm}$
(28)

where $A<m_{\Phi}^{2}$. Therefore washout due to B-violating $\Phi$ exchange
is negligible if $T_{R}\;^{<}{}_{\sim}\;m_{\Phi}$ and so it is unlikely
present a serious obstacle to this class of model. A complete analysis of
washout will depend upon the specific model for the decay of the inflaton and
the transfer of the baryon asymmetry.

##  IV Validity of the Classical Analysis of the Baryon Asymmetry

Throughout our analysis we have assumed that classical fields can be used to
calculate the baryon asymmetry. When the potential is dominated by quadratic
terms, the $\phi_{1}$ and $\phi_{2}$ fields evolve as independent non-
interacting coherently oscillating scalars. In general, a classical
oscillating scalar field corresponds to a quantum coherent state in the limit
where the occupation number of the state is large compared to one [13, 14].
The condition for this to be true is that $\phi_{i}>m_{\Phi}$ ($i=1,\,2$).
However, this is typically not satisfied at inflaton decay in the present
model. Nevertheless, the classical calculation of the baryon asymmetry remains
correct. This is because it is the coherent state corresponding to the
classical field that is important for AD baryogenesis.

By construction, the expectation value of the field operator $\hat{\phi}_{i}$
in the coherent state $|\phi_{i}(t)>$ is equal to the classical field
$\phi_{i,\;cl}(t)$

$\mbox{$$}<\phi_{i}(t)|\hat{\phi}_{i}|\phi_{i}(t)>=\phi_{i,\;cl}(t)~{}.\vspace{0.1cm}$
(29)

We have included a time dependence in the coherent state to take into account
the dilution of the number density by expansion. Since the scalar fields
$\phi_{1}$ and $\phi_{2}$ are independent fields, the coherent state of the
complex field is a product of the coherent states for $\phi_{1}$ and
$\phi_{2}$, $|\Phi(t)>=|\phi_{1}(t)>|\phi_{2}(t)>$. Therefore, with the baryon
number density operator given by
$\hat{n}=\hat{\dot{\phi}}_{1}\hat{\phi}_{2}-\hat{\dot{\phi}}_{2}\hat{\phi}_{1}$,
the expectation value of the baryon asymmetry in the coherent state is given
by

$\mbox{$$}<\Phi(t)|\hat{n}|\Phi(t)>=<\Phi(t)|\hat{\dot{\phi}}_{1}\hat{\phi}_{2}-\hat{\dot{\phi}}_{2}\hat{\phi}_{1}|\Phi(t)>=\dot{\phi}_{1,\;cl}\phi_{2,\;cl}-\dot{\phi}_{1,\;cl}\phi_{1,\;cl}\equiv
n_{cl}~{}.\vspace{0.1cm}$ (30)

Therefore the expectation value of the baryon number density operator is equal
to the baryon number density $n_{cl}$ calculated using the classical fields.
When $\phi_{i}<m_{\Phi}$, the variance of the field in the coherent state will
become large compared to the squared classical field. Therefore there will be
large quantum fluctuations of the fields about their expectation values and so
the field cannot be considered classical. However, the correlation length of
the quantum fluctuations cannot be larger than the horizon at inflaton decay.
Since the volume that evolves into the presently observed Universe will be
very much larger than the horizon volume at inflaton decay, the observed
baryon asymmetry will be given by its spatial average value and so will equal
the expectation value of the baryon asymmetry. Therefore the baryon asymmetry
will equal its classical value even when $\phi_{i}<m_{\Phi}$. This shows that
it is the coherent state describing the scalar field, rather than its
classical nature, that is essential for AD baryogenesis.

In reaching this conclusion we have assumed that the mean asymmetry
transferred from the condensate by decay is equal to the mean asymmetry in the
coherent state of the condensate and that there is no additional washout
effect due to the decay process. Condensate decay in this model occurs when
the occupation number is less than one, therefore the conventional classical
analysis based on production of particles due to a time-dependent classical
field is no longer valid. Whilst there is no obvious reason to expect an
additional source of washout due to the decay process when the coherent state
is no longer in the classical limit, this should be confirmed by a full
quantum field theory analysis.

##  V Conclusions

We have presented a new minimal approach to baryogenesis which is based on
$B$-violating mass terms for the inflaton. The resulting model requires only
the addition of $B$-violating mass terms to an existing inflaton potential and
therefore can easily be realised in a wide range of inflation models. The
asymmetry is generated at late times during inflaton oscillations, in contrast
to existing inflaton-based AD baryogenesis models which generate the asymmetry
during or shortly after inflation. The model also provides exact analytical
expressions for the resulting baryon asymmetry.

In this analysis we have not addressed the question of baryon isocurvature
perturbations. We note that these can easily be controlled by including a
$\Phi^{4}+\Phi^{\dagger\;4}$ term in the potential which is significant during
inflation and becomes negligible after inflation, whilst leaving open the
possibility of observable isocurvature perturbations. A detailed
implementation of the mechanism to baryogenesis from AD leptogenesis via
inflaton decay, including a discussion of isocurvature perturbations, will be
presented in a future work [10]. The model also raises new questions regarding
the Affleck-Dine mechanism in the limit where the classical approximation is
no longer valid, which requires a dedicated analysis.

## Acknowledgements

The work of ALS is supported by STFC.

## References

  * [1] I. Affleck and M. Dine, Nucl. Phys. B 249 (1985), 361-380 doi:10.1016/0550-3213(85)90021-5
  * [2] M. Dine, L. Randall and S. D. Thomas, Nucl. Phys. B 458 (1996), 291-326 doi:10.1016/0550-3213(95)00538-2 [arXiv:hep-ph/9507453 [hep-ph]].
  * [3] J. M. Cline, M. Puel and T. Toma, Phys. Rev. D 101 (2020) no.4, 043014 doi:10.1103/PhysRevD.101.043014 [arXiv:1909.12300 [hep-ph]]; J. M. Cline, M. Puel and T. Toma, JHEP 05 (2020), 039 doi:10.1007/JHEP05(2020)039 [arXiv:2001.11505 [hep-ph]].
  * [4] Y. Y. Charng, D. S. Lee, C. N. Leung and K. W. Ng, Phys. Rev. D 80 (2009), 063519 doi:10.1103/PhysRevD.80.063519 [arXiv:0802.1328 [hep-ph]].
  * [5] M. P. Hertzberg and J. Karouby, Phys. Lett. B 737 (2014), 34-38 doi:10.1016/j.physletb.2014.08.021 [arXiv:1309.0007 [hep-ph]]; M. P. Hertzberg and J. Karouby, Phys. Rev. D 89 (2014) no.6, 063523 doi:10.1103/PhysRevD.89.063523 [arXiv:1309.0010 [hep-ph]].
  * [6] N. Takeda, Phys. Lett. B 746 (2015), 368-371 doi:10.1016/j.physletb.2015.05.039 [arXiv:1405.1959 [astro-ph.CO]].
  * [7] C. M. Lin and K. Kohri, [arXiv:2003.13963 [hep-ph]].
  * [8] E. Babichev, D. Gorbunov and S. Ramazanov, Phys. Lett. B 792 (2019), 228-232 doi:10.1016/j.physletb.2019.03.046 [arXiv:1809.08108 [astro-ph.CO]].
  * [9] D. S. Salopek, J. R. Bond and J. M. Bardeen, Phys. Rev. D 40 (1989), 1753 doi:10.1103/PhysRevD.40.1753
  * [10] A.Lloyd-Stubbs and J.McDonald, In progress.
  * [11] Y. Ema, R. Jinno, K. Mukaida and K. Nakayama, JCAP 05 (2015), 038 doi:10.1088/1475-7516/2015/05/038 [arXiv:1502.02475 [hep-ph]].
  * [12] K. Nakayama, S. Saito, Y. Suwa and J. Yokoyama, JCAP 06 (2008), 020 doi:10.1088/1475-7516/2008/06/020 [arXiv:0804.1827 [astro-ph]].
  * [13] S. Davidson, Astropart. Phys. 65 (2015), 101-107 doi:10.1016/j.astropartphys.2014.12.007 [arXiv:1405.1139 [hep-ph]].
  * [14] K. D. Lozanov, [arXiv:1907.04402 [astro-ph.CO]].