# Why is zero spatial curvature special?

Raul Jimenez  Ali Rida Khalife  Daniel F. Litim  Sabino Matarrese  Benjamin D.
Wandelt

###### Abstract

Evidence for almost spatial flatness of the Universe has been provided from
several observational probes, including the Cosmic Microwave Background (CMB)
and Baryon Acoustic Oscillations (BAO) from galaxy clustering data. However,
other than inflation, and in this case only in the limit of infinite time,
there is no strong a priori motivation for a spatially flat Universe. Using
the renormalization group (RG) technique in curved spacetime, we present in
this work a theoretical motivation for spatial flatness. Starting from a
general spacetime, the first step of the RG, coarse-graining, gives a
Friedmann-Lemaître-Robertson-Walker (FLRW) metric with a set of parameters.
Then, we study the rescaling properties of the curvature parameter, and find
that zero spatial curvature of the FLRW metric is singled out as the unique
scale-free, non-singular background for cosmological perturbations.

##  1 Introduction

It has been known for a long time that the FLRW metric111We are using units in
which the speed of light $c$ and the reduced Planck constant $\hbar$ are 1.,

$ds^{2}=-dt^{2}+a^{2}(t)\bigg{(}\frac{dr^{2}}{1-kr^{2}}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}\bigg{)},$ (1.1)

is the one that maximizes the symmetries of space [1, 2]; FLRW maximizes the
symmetries of space, but only de-Sitter and Minkowski maximize symmetries of
space-time. In the above equation, $ds^{2}$ is the line element, $t$ is cosmic
time, $a(t)$ is the time dependent scale factor, $k$ is the curvature
parameter and $\\{r,\theta,\phi\\}$ are the spatial spherical coordinates.
Moreover, from the uniqueness theorem of maximally symmetric spaces, there is
no other metric with further symmetries (see part four in chapter 13 of [2]).
It is tempting to think this could be the reason why we live in a FLRW
Universe. However, a further question arises: is there anything special about
spatial flatness, i.e. $k=0$?

Observationally, analysis from the CMB [3, 4, 5], BAO [6, 7, 8] and the full
shape of the large scale structure power spectrum [9] have concluded that we
live in a Universe with $|\Omega_{k0}|\lesssim 0.001$, where
$\Omega_{k0}=-kc^{2}/H_{0}^{2}$ is the curvature density parameter today,
$H_{0}$ being the Hubble parameter today. Other interpretations of such data
read it as evidence for a closed universe, with $\Omega_{k0}<0$ [10, 11, 12,
13]. Intriguingly, the majority of observational analyses in the literature
assume flatness by setting $\Omega_{k0}=0$ from the outset. Is there any
compelling argument for the flat case to be singled out a priori in this way?

So far, inflation is considered as the primary theoretical motivation for
spatial flatness [14, 15, 16]. By describing a very early phase of accelerated
expansion, most inflationary models point towards $k\approx 0$ today being an
attractor solution for the metric of the Universe. In these models, deviations
from flatness are at the level of one part in $10^{5}$ due to initial
curvature fluctuations [17], which could be measured at that level with future
observations [18]. An inflationary phase is the most prominent explanation for
having a Universe consistent with spatial flatness today[19]. However, it
would be fruitful to have an additional theoretical motivation in favor of the
specialty of a spatially flat universe (for alternative explanations of why
the Universe is flat see e.g. [20]).

Stated in a different way, the curvature density parameter $\Omega_{k}\equiv-
kc^{2}/a^{2}H^{2}$ can a priori take any negative value in the closed Universe
case, any positive value in the open Universe case, but only 0 for the $k=0$
case. This means that the flat universe case corresponds to a set of measure
zero among all possible spatial geometries. Therefore, an exactly spatially
flat Universe has zero probability to be selected among all possible FRLW
models (a point that was recently argued in [21]). This raises the question
whether $\Lambda$CDM with $\Omega_{k}=0$ should be considered at all. It would
seem that this requires theoretical arguments that select $\Omega_{k}=0$ as an
a priori special, maximally symmetric background for cosmological
perturbations.

To that end, one might start by noting a crucial mathematical property of the
spatial part of the FLRW metric, which is its conformal flatness (see [22] and
references therein), i.e. its Weyl tensor vanishes. This means that the
spatial part of the FLRW metric can be transformed into the Minkowski one by
an appropriate combination of coordinate and Weyl transformations 222It should
be noted that these transformations are not technically diffeomorphisms, in
the sense that the global properties of Minkowski spacetime are not
recovered.. However, while the cases $k>0$ and $k<0$ involve relatively
complicated transformations to achieve that, as we will show below (see also
section 111 of [1]), the case $k=0$ becomes flat through a simple Weyl
rescaling of the metric by $a^{2}(\eta)$, where $\eta$ is the conformal time.
This simplicity could be hinting to a fundamental property of the case $k=0$
that makes it more favorable than the others. On the other hand, when studying
the properties of a system under rescaling of the coordinates, the first thing
that could come to mind is the renormalization group (RG) technique [23, 24,
25]. Although this technique is traditionally used to study critical phenomena
in condensed matter physics [26], nothing prevents its use in the context of
curved spacetime, as has been done before [27, 28, 29, 30].

It is therefore the purpose of this work to apply the RG technique to a
general spacetime, and show that $\Omega_{k}=0$ corresponds to a stable
critical fixed-point of the RG flow; and indeed the only fixed point that is
suitable as a background for cosmological perturbations. This provides model-
independent theoretical motivation for the case of a flat FLRW universe.

We start the discussion with a brief review of the RG, along with its steps
(coarse-graining, rescaling and renormalization) and terminology(fixed-points,
critical fixed-points, correlation length…) in section 2. Then, we apply the
coarse-graining step of the RG to a general spacetime in section 3.1 to get an
FLRW-like metric. In section 3.2, we recognise that $k$ has mass dimension 2
and take inspiration from the RG to find fixed points under rescaling. This
leads to examining a discrete number of fixed points. We then find that
amongst these, the only non-singular solution that remains and that is
suitable as a background for cosmological perturbations is $\Omega_{k}=0$. We
end the discussion with a conclusion and summary in section 4. We include in
appendix A some technical details.

##  2 A brief review of RG

The main realm of application of the RG technique is in Statistical mechanics
and Condensed Matter Physics, particularly to describe phase transitions [26].
These abrupt transformations occur when the parameters of a certain system
take particular values, known as the critical point of the system333It
corresponds to a point in the space of parameters, or equivalently the space
of Lagrangians or Hamiltonians., with a basic example being the ordered-
disordered transition of the Ising spin model [31, 32]. Prior to the
introduction of the RG method, the treatment of phase transitions was made
using the mean field approximation (MFA), in which one considers small
fluctuations around the average spin of the Ising model, for instance.
However, although the MFA worked well in spatial dimension $d\geq 4$, it broke
down for small dimensions, where one could not even see a phase transition for
$d=1$ [33, 34]. This motivated the community to check for alternatives to the
MFA, which brings us to the RG.

We will briefly review the RG formalism and direct the interested reader to
more detailed descriptions in [35, 34, 26, 36]. The basic notion upon which
the RG is based is that the physical description of a system changes with the
scale at which it is observed. Therefore, one can start by studying the system
at short distances, and the RG provides a step by step transition to larger
distances [36]. To understand how it works and its main outcome, let us start
with a simple example.

Suppose we are interested in studying the transition to the ferromagnetic
phase of a metal, which surely has a microscopic treatment. However, that
would be too complicated if we are merely concerned with this phase transition
due to thermal fluctuations. One can therefore define the average collection
of spins about a point $\mathbf{x}$ as the magnetization
$\mathbf{m}(\mathbf{x})$. In the presence of an external magnetic field
$\mathbf{h}$, one can write a general444This coarse-grained Hamiltonian is
based on the assumptions of locality, translational and rotational invariance.
Hamiltonian $H[\mathbf{m(x)}]$ as the Landau-Ginzburg one [37, 34]:

$\displaystyle\beta H[\mathbf{m(x)}]=\int d\mathbf{x}$
$\displaystyle\bigg{[}\frac{t}{2}\mathbf{m}^{2}+u\mathbf{m}^{4}+...+\frac{K}{2}(\nabla\mathbf{m})^{2}$
$\displaystyle+\frac{L}{2}(\nabla^{2}\mathbf{m})^{2}+\frac{N}{2}\mathbf{m}^{2}(\nabla\mathbf{m})^{2}+...$
$\displaystyle-\mathbf{h}.\mathbf{m}\bigg{]}$ (2.1)

where $\beta=1/(k_{B}T)$, with $k_{B}$ the Boltzmann constant and $T$ the
temperature. Plus, the coefficients $t,u,K...$ are phenomenological parameters
for their respective terms, and they define the axes of the theory space. Each
point in this space corresponds to a different Hamiltonian, and thus a
different theory. From here, one can then apply the recipe of the RG as
follows:

  1. 1.

Coarse-Graining: As mentioned previously, the study of phase transitions is
more relevant on distances larger than the microscopic scale $a$, i.e. the
lattice spacing. Therefore, we first must decrease the resolution by averaging
out fluctuations in $\mathbf{m}(\mathbf{x})$ that are smaller than $ba$,

$\bar{\mathbf{m}}(\mathbf{x})=\frac{1}{(ba)^{d}}\int_{C}d\mathbf{y}\
\mathbf{m}(\mathbf{y})$ (2.2)

where $b>1$ is the rescaling parameter, $d$ is the number of spatial
dimensions and the integral is over a cell $C$ of size $(ba)^{d}$ centered on
$\mathbf{x}$. Note that this is equivalent to integrating out large momentum
modes.

  2. 2.

Rescaling: By changing the resolution, one now has a “grainier” picture than
the original one, and therefore one cannot directly link the two. In order to
restore that, one simply rescales the coordinates $\mathbf{x}\rightarrow
b\mathbf{x}$.

  3. 3.

Finally, due to the above two steps, the transformed theory will end up with
some additional coefficient in front of its kinetic term, call it $\zeta^{2}$.
This could be interpreted as a change in the contrast between the two
theories. In order to return it to its canonical form, one can simply rescale
the magnetization
$\mathbf{m}(\mathbf{x})\rightarrow\zeta\bar{\mathbf{m}}(b\mathbf{x)}$.

This operation can be compactly stated as
$\mathbf{S}\rightarrow\mathbf{S^{\prime}}=\mathbf{R}_{b}\mathbf{S}$, where
$\mathbf{R}_{b}$ is the RG transformation from one point in theory
space($\mathbf{S}$) to another($\mathbf{S^{\prime}}$) by a scaling parameter
$b$. Note that symmetries are automatically preserved under
$\mathbf{R}_{b}$555That doesn’t mean that there’s no symmetry enhancement at
the critical point, as we will see shortly., and so the new $H_{\zeta}$ has
the same form as the old $H$ but in terms of the renormalized variables.
Moreover, Hamiltonians that are statistically self-similar correspond to
“fixed points” in parameter space. They are invariant under the RG: if $S^{*}$
is a fixed point then $S^{*}=\mathbf{R}_{b}S^{*}$. These points are crucial
because they are identified with the phase transition points; this is where
the link between RG and thermodynamics appears. More specifically, since under
RG the correlation length666The correlation length is the distance over which
field fluctuations are correlated. It thus defines the length scale up to
which we have self-similarity. $\xi\rightarrow b\xi$, this means that at the
fixed point, $\xi$ can be either zero or infinite. The latter corresponds to
critical phase transitions.

A seminal example for critical points is given by the $3d$ Ising model [38],
where the parameters of the effective Hamiltonian (2.1) may take fixed points
under the RG transformation. Concretely, in terms of the effective mass term
$\sim t$, the model is known to display the free Gaussian fixed point
$(t_{*}=0$), the infinite Gaussian fixed point ($1/t_{*}=0$), and the seminal
Wilson-Fisher fixed point ($t_{*}=$ finite) characterising a second order
phase transition with a diverging correlation length [39].

Having identified the fixed points of the theory space, one can then study its
stability by linearizing deviations about it:

$S_{i}^{*}+\delta S_{i}^{{}^{\prime}}=S_{i}^{*}+(\mathbf{R}_{b})_{ij}\delta
S^{j}$ (2.3)

where $(\mathbf{R}_{b})_{ij}=\frac{\partial S_{i}^{{}^{\prime}}}{\partial
S^{j}}$. Since $\mathbf{R}_{b}$ is a transformation, one can find its
eigenvalues and eigenvectors $\lambda_{i}(b)=b^{y_{i}}$ and
$\mathbf{\Theta}_{i}$, respectively, where $y_{i}$ is called the anomalous
dimension of operator $\mathbf{\Theta}_{i}$. For $y_{i}>0,<0$ or 0, the
operator $\mathbf{\Theta}_{i}$ is said to be a relevant (RO), irrelevant (IO)
or marginal operator (MO), respectively. If we imagine moving away from a
fixed point in theory space according to a flow dictated by these operators,
the IOs are those that will take us back to the original fixed point. That is
why IOs are crucial to identify stable fixed points which define a subspace in
theory space called the “basin of attraction".

In many areas of physics, the RG approach has unveiled fundamental properties
of the laws of nature. In this context, it is easy to understand what a
critical component of a phase transition is as the critical fixed point of the
renormalization group. This is universal no matter what the nature of the
material undergoing the phase transition is. In particular, one can imagine
applying this formalism to other, more abstract physical entities, such as
spacetime and its metric, which we will now discuss.

##  3 RG in Curved Spacetime

After the brief review of the RG technique, we will now apply it to the metric
of spacetime. Our theory space will be that of the metric components, in
analogy to the Hamiltonian of Sec. 2, and the metric. As a starting point, we
use completely general metrics in four space-time dimensions without assuming
any form of symmetry. We further assume that General Relativity (GR) is the
underlying theory of gravity. We then argue that FLRW metrics (1.1) with
vanishing spatial curvature are distinguished, much like critical points of
the RG.

###  3.1 Coarse Graining

In the present context, coarse-graining amounts to averaging over small
distances, and then to look at how this modifies the large distance behaviour.
To do this for curved spacetime metrics, we adopt the machinery presented in
[40, 41, 42]. The latter assumes spacetime to be filled with matter in the
form of dust or a perfect fluid, which is a good description for all known
components of our universe – baryons, dark matter, dark energy, and radiation.
We will not go through the mathematical details here, but rather we focus on
the main principles and results needed for our purpose.

We then consider an irrotational cosmic fluid and foliate spacetime into
hypersurfaces orthogonal to its 4-velocity.777Note that the irrotational
assumption can be relaxed and the final results do not depend on the
particular foliation, see Sec. 2.1 of [41]. This allows us to write the line
element as

$ds^{2}=-N^{2}dt^{2}+g_{ij}dX^{i}dX^{j},$ (3.1)

where $N$ is the lapse function, $g_{ij}$ are the spatial components of the
metric and $X^{i}$ are called Lagrangian coordinates. Next, we consider a
compact and simply-connected domain888A compact domain is one that is closed
and bounded,i.e. it has no holes or missing limiting points. For e.g. the set
$[0,1]$ is compact while $[0,1]-\\{0.2\\}$ is not since it has a hole (it’s
missing the 0.2). A simply connected domain is one without any holes in it
[43]. $\mathcal{D}$ contained within spatial hypersurfaces of constant time to
define the spatial average of a quantity $\Psi$ as

$\langle\Psi\rangle_{\mathcal{D}}\equiv\frac{1}{V_{\mathcal{D}}}\int_{\mathcal{D}}\Psi\,\sqrt{\det
g_{ij}}\,d^{3}X,$ (3.2)

where the domain’s volume $V_{\mathcal{D}}$ is given by

$V_{\mathcal{D}}(t)\equiv\int_{\mathcal{D}}\,\sqrt{\det g_{ij}}\,d^{3}X\,.$
(3.3)

Further, using the definition of $V_{\mathcal{D}}(t)$, one can then extract a
dimensionless scale factor,

$a_{\mathcal{D}}(t)\equiv\bigg{(}\frac{V_{\mathcal{D}}(t)}{V_{\mathcal{D}_{0}}(t)}\bigg{)}^{1/3},$
(3.4)

normalized to the volume of the initial domain $V_{\mathcal{D}_{0}}$. With
this scale factor, we are simply concerned with the effective dynamics over
the domain. Intriguingly, continuing the process of averaging the equations of
motion over increasingly larger volumes invariably leads to modified Friedmann
equations (see eqs.(13a,b) in [41]). These include a curvature term $k$, in
addition to kinematical and dynamical backreaction terms $Q$ and $P$,
respectively [41, 44]. We conclude that maximally symmetric FLRW-like metrics
(1.1) arise from averaging general spacetimes such as (3.1), with the
curvature parameter $k$ and the backreaction parameters $Q$ and $P$ becoming
part of the theory space of parameters.

###  3.2 Scaling analysis

We are now in a position to perform a scaling analysis of (1.1) in the sense
of the RG. On dimensional grounds, starting with (1.1), we observe that the
product $k\,r^{2}$ must be dimensionless. Given that the canonical mass
dimension of a length is $[r]=-1$, it follows that the canonical mass
dimension of curvature is $[k]=2$, much like the mass term $t$ in the model
Hamiltonian (2.1). We also have that $[a]=0$, by definition of the metric. It
then follows that the (classical) running of curvature with coarse-graining
scale takes the form

$\ell\partial_{\ell}\,\hat{k}=2\hat{k}\,,$ (3.5)

with $\hat{k}=k\ell^{2}$ the dimensionless version of the curvature parameter.
Hence, we observe two scale-free classical fixed points, the Gaussian fixed
point $(\hat{k}_{*}=0)$, and the infinite Gaussian fixed point
$(1/\hat{k}_{*}=0)$ [39]. While the former is self-evident, the latter becomes
visible using the mapping $\hat{k}\rightarrow 1/\hat{k}$. These classical
fixed points can be viewed as the counterparts of the Gaussian and infinite
Gaussian fixed point of the mass term in Ising-type models indicated earlier,
and characterise the asymptotic limits where spatial curvature is either
absent, or fully dominant.

This also reflects the well-known results that, firstly, in general
relativity, the flat solution is unstable under IR perturbations (also known
as the “flatness” problem), and, secondly, that the curvature of the universe
is of no relevance when we run particle detectors (i.e. in the UV limit).

As an aside, we note that the analogue of a Wilson-Fisher fixed point may
arise if spatial curvature itself, by virtue of the successive coarse graining
steps (3.2), depends on the coarse graining scale, $k=k(\ell)$. The latter
would generate an additional term $\sim\ell^{3}\partial_{\ell}k(\ell)$ on the
right-hand side of (3.5), which may offer non-trivial fixed points. We defer
further exploration of such non-trivial fixed points to future work.

Next, we relate our discussion to the conformal flatness of FLRW space-times.
Applying the coordinate transformation

$r\rightarrow\frac{r}{1+k\,r^{2}/4}$ (3.6)

to (1.1) entails that the spatial metric (or line element $ds_{3}^{2}$) can be
written as

$ds^{2}_{3}=a^{2}\left(1+\frac{kr^{2}}{4}\right)^{-2}\left[dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}\right]$ (3.7)

(see the Appendix for derivation). The conformal flatness of the spatial part
of the FLRW is now manifest as (3.7) coincides with the flat space metric,
except for an overall $r$-dependent factor. Moreover, notice that a
characteristic length scale

$r_{k}\equiv\frac{2}{\sqrt{|k|}}$ (3.8)

automatically comes out of this transformation. It states that space looks
effectively flat for distances $r\ll r_{k}$, while it appears significantly
curved for distances $r\gg r_{k}$. We also note that $r_{k}\to\infty$ from the
outset if and only if spatial curvature vanishes $k=0$. For the purpose of
this study, we are mainly interested in the range $r\leq\ell_{D}$, where
$\ell_{D}$ is the scale up to which spatial sections of the initial (foliated)
space-time have been spatially averaged. It follows that for $k\neq 0$, FLRW
space times will look spatially curved rather than flat at asymptotic
distances. However, the transition from flat to curved disappears in the limit
$k\rightarrow 0$, in which $r_{k}\rightarrow\infty$, whence vanishing spatial
curvature can be interpreted as a critical point.

###  3.3 A background for cosmological perturbations

Having shown that the mass dimension of $k$ naturally leads to a scale of the
FLRW universe, the curvature radius $r_{k}$, we will now show that removing
this scale by rescaling uniquely leads to a scale-free, symmetric spacetime
suitable as a background for cosmological perturbations. In the following we
will use scaling arguments inspired by the RG approach described above.

However, it should be noted that the application differs in the following way.
Usually, the volume average is done to define a coarse grained field and the
relevant equations for perturbations which lead to predictions for coarse-
grained observables. In contrast, here we are attempting to define a metric
that can serve as a background for cosmological perturbation theory. Buchert’s
approach [40, 41] shows that using a volume average starting from a general
space-time metric achieves an FLRW effective “background" geometry.

Let us now present our argument. The case of vanishing spatial curvature $k=0$
is already scale-free, so we focus on $k\neq 0$. In either cases we have two
options to remove the curvature scale by achieving $r_{k}\rightarrow 0$ or
$r\rightarrow\infty$. This leads to the following cases, as can be seen from
(3.7). For either $k>0$ or $k<0$, the limit of $r_{k}\rightarrow\infty$
recovers $\Omega_{k}=0$ except with non-trivial global topology. Moreover, for
positive spatial curvature $k>0$, the limit $r_{k}\rightarrow 0$ leads to a
singular solution with zero volume of the spatial sections. On the other hand,
negative spatial curvature $k<0$ corresponds to a FLRW universe with
hyperbolic spatial sections of infinite volume. Further, the limit
$r_{k}\rightarrow 0$ here is clearly singular and, as such, unsuitable as a
background for cosmological perturbations999An hyperbolic spatial section with
infinite curvature evokes the notion of a fractal universe that is nowhere
smooth, but we will leave a detailed mathematical exploration of this limit to
future work.. In consequence, the unique scale-free symmetric background is
one that is locally indistinguishable from a flat FLRW universe with
$\Omega_{k}=0$, i.e. one that is identical to the latter on any finite patch,
however large.

##  4 Summary

Observationally, it will be impossible to prove conclusively that spatial
curvature of the universe vanishes exactly, since any constraint on
$\Omega_{k0}=0$ is limited by the cosmic variance value of $10^{-5}$.
Theoretically, however, one might wonder whether arguments can be found that
single-out spatial flatness ($k=0$).

Here, we have demonstrated that amongst all maximally symmetric and scale free
background metrics that emerge from renormalization group arguments, the
$\Omega_{k}=0$ FLRW universe is the only one that is suitable as a background
for cosmological perturbations.

Let us clearly state what we do not claim. We do not propose a mechanism that
dynamically generates such background solutions based on physical assumptions;
in this sense the claim in this paper should not be seen as proposing an
alternative to models of the very early universe, such as inflation, that aim
to motivate a nearly flat and perturbed FLRW background. We did not consider
any effect of possible fluctuations as we were concerned only with pure
background effects; the effect of fluctuations on the background has been done
by one of us (SM). In [42] they considered the RG to investigate how super and
sub-horizon fluctuations behave. Their eq. (64) shows that super-horizon
fluctuations are just curvature; this is well establsihed. For the sub-horizon
perturbations the situation is more complicated, the recent findings from
relativistic numerical simulations is that probably the effect is small but
even the calculation in their section 3.2 shows that this effect will be small
at best and in no case will affect the fixed points of the RG.

However, our result does indicate that it is admissible to accord flat
solutions an a priori special status even though they are a set of measure
zero amongst all homogeneous and isotropic solutions to Einstein’s equations.
This may include explicitly focusing on flat models when exploring
cosmological parameter constraints.

## Acknowledgments

We thank Cesar Gomez and Roy Maartens for very useful feedback on drafts of
this paper. Funding for this work was partially provided by project
PGC2018-098866-B-I00 MCIN/AEI/10.13039/501100011033 y FEDER “Una manera de
hacer Europa”, and the “Center of Excellence Maria de Maeztu 2020-2023” award
to the ICCUB (CEX2019-000918-M funded by MCIN/AEI/10.13039/501100011033). This
work was performed in part at Aspen Center for Physics, which is supported by
National Science Foundation grant PHY-1607611, and was partially supported by
a grant from the Simons Foundation. ARK acknowledges receiving funding from
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program (grant agreement No 101001897). DFL is
supported by the Science and Technology Research Council (STFC) under the
Consolidated Grant ST/T00102X/1. SM acknowledges financial support from the
COSMOS network (www.cosmosnet.it) through the ASI (Italian Space Agency)
Grants 2016-24-H.0, 2016-24-H.1-2018 and 2019-9-HH.0. BDW acknowledges support
by the ANR BIG4 project, grant ANR-16-CE23-0002 of the French Agence Nationale
de la Recherche; and the Labex ILP (reference ANR-10-LABX-63) part of the Idex
SUPER, and received financial state aid managed by the Agence Nationale de la
Recherche, as part of the programme Investissements d’avenir under the
reference ANR-11- IDEX-0004-02. The Flatiron Institute is supported by the
Simons Foundation.

##  Appendix A Derivation of transformation (3.7)

The standard form of FLRW metric (spatial part) in spherical coordinates reads

$ds^{2}=a^{2}\bigg{(}\frac{dr^{2}}{1-kr^{2}}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}\bigg{)}.$ (A.1)

Consider $r={r_{1}}/({1+\frac{kr_{1}^{2}}{4}})$, leading to

$\displaystyle
dr=\frac{1-\frac{kr_{1}^{2}}{4}}{\bigg{(}1+\frac{kr_{1}^{2}}{4}\bigg{)}^{2}}dr_{1}.$
(A.2)

Plug into (A.1)

$\displaystyle ds^{2}=$ $\displaystyle
a^{2}\bigg{[}\frac{1}{1-\frac{kr_{1}^{2}}{\bigg{(}1+\frac{kr_{1}^{2}}{4}\bigg{)}^{2}}}\times\frac{\bigg{(}1-\frac{kr_{1}^{2}}{4}\bigg{)}^{2}}{\bigg{(}1+\frac{kr_{1}^{2}}{4}\bigg{)}^{4}}dr_{1}^{2}+\frac{r_{1}^{2}}{\bigg{(}1+\frac{kr_{1}^{2}}{4}\bigg{)}^{2}}d\theta^{2}$
$\displaystyle+\frac{r_{1}^{2}}{\bigg{(}1+\frac{kr_{1}^{2}}{4}\bigg{)}^{2}}\sin^{2}\theta
d\phi^{2}\bigg{]}$
$\displaystyle=a^{2}\bigg{(}1+\frac{kr_{1}^{2}}{4}\bigg{)}^{-2}\bigg{[}\frac{\bigg{(}1-\frac{kr_{1}^{2}}{4}\bigg{)}^{2}}{\bigg{(}1+\frac{kr_{1}^{2}}{4}\bigg{)}^{2}-kr_{1}^{2}}dr_{1}^{2}+...\bigg{]}$
$\displaystyle=a^{2}\bigg{(}1+\frac{kr_{1}^{2}}{4}\bigg{)}^{-2}\bigg{[}dr_{1}^{2}+r_{1}^{2}d\theta^{2}+r_{1}^{2}\sin^{2}\theta
d\phi^{2}\bigg{]}$ (A.3)

which is (3.7) above.

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# Low frequency correlated charge noise measurements across multiple energy
transitions in a tantalum transmon

Daniel M. Tennant<EMAIL_ADDRESS>Lawrence Livermore National Laboratory,
Livermore, California 94550, USA  Luis A. Martinez  Lawrence Livermore
National Laboratory, Livermore, California 94550, USA  Kristin M. Beck
Lawrence Livermore National Laboratory, Livermore, California 94550, USA  Sean
R. O’Kelley  Lawrence Livermore National Laboratory, Livermore, California
94550, USA  Christopher D. Wilen  Department of Physics, University of
Wisconsin-Madison, Madison, Wisconsin 53706, USA  R. McDermott  Department of
Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
Jonathan L DuBois  Lawrence Livermore National Laboratory, Livermore,
California 94550, USA  Yaniv J. Rosen  Lawrence Livermore National Laboratory,
Livermore, California 94550, USA

###### Abstract

Transmon qubits fabricated with tantalum metal have been shown to possess
energy relaxation times greater than 300 $\muup$s and, as such, present an
attractive platform for high precision, correlated noise studies across
multiple higher energy transitions. Tracking the multi-level fluctuating qudit
frequencies with a precision enabled by the high coherence of the device
allows us to extract the charge offset and quasi-particle dynamics. We observe
qualitatively different charge offset behavior in the tantalum device than
those measured in previous low frequency charge noise studies. In particular,
we find the charge offset dynamics are dominated by rare, discrete jumps
between a finite number of quasi-stationary charge configurations, a
previously unobserved charge noise process in superconducting qubits.

††preprint: APS/123-QED

##  I Introduction

Superconducting qubits are one of the most promising hardware platforms for
near future quantum computing processors [1]. Recent demonstrations of high
fidelity two qubit gates [2, 3, 4], quantum simulations [5, 6, 7], quantum
algorithms [8, 9], and quantum error correction [10, 11, 12, 13] all indicate
the potentiality of superconducting devices for quantum information systems.
Devices based on superconducting circuits, cavities, and resonators leverage
well established fabrication techniques to create a wide variety of structures
capable of the array of functionalities listed above. While superconducting
quantum devices are capable of a wide range of behavior, it is precisely this
flexibility that introduces unwanted sources of noise. Fabrication
imperfections, material defects, and environmental interactions all conspire
to limit the coherence properties necessary for universal quantum computation.

Much recent effort has focused on seeking material solutions to decoherence in
superconducting qubits. Energy relaxation times in these devices are believed
to be limited by Two Level Systems (TLS) hosted in the superconducting metal
oxide and substrate interfaces [14, 15, 16]. Superconducting qubits have
traditionally been fabricated with either aluminum or niobium sputtered onto
silicon or sapphire substrates. Niobium, in particular, generates multiple
species of oxide on its surface with significant populations extending up to
10 nm into the metal [17], creating an unwanted environment for TLS to reside
[18]. Alternative superconducting metals might exist that produce more uniform
oxides or create cleaner interfaces between the metal and substrate.

Tantalum has recently emerged as such a candidate superconducting metal [19].
Energy relaxation times in a transmon device, fabricated with tantalum metal
on a sapphire substrate, have been shown to consistently exceed 300 $\muup$s
in select devices, a state of the art coherence time for the transmon design.
This indicates a reduced TLS density of states fluctuating at gigahertz
frequencies in these devices. X-ray photoelectron spectroscopy (XPS) and
transmission electron microscopy (TEM) measurements of the surface reveal 2-3
nm thick layer of predominantly Ta2O5 which, unlike niobium, only extends 1 or
2 atomic sites into the bulk [20]. Additionally, when compared to aluminum
resonators and qubits with similar surface participation ratios, tantalum
devices exhibit loss tangents reduced by a factor of 3-4 [14, 20].

To further understand the coherence properties of the tantalum qubit, we
perform low frequency charge noise measurements on a tantalum transmon.
Specifically, we measure both the quasi-particle tunneling rate across the
Josephson junction and the low frequency charge offset dynamics of the
transmon environment. While the contribution of quasi-particle fluctuations to
qubit decoherence is not believed to be a dominant factor, it does limit
superconducting qubit operations on a fundamental level and thus motivates our
study [21, 22, 23]. The low frequency superconducting qubit charge environment
is comprised of multiple processes hosted both in the device substrate and
superconducting metal surface [24]. These processes are not well understood,
especially in the case of tantalum at cryogenic temperatures.

Low frequency charge noise measurements have been performed on 3D transmons
[25, 26, 27] and single island planar transmons [28, 37]. These devices
spanned a number of material and design combinations: a 3D aluminum transmon
on sapphire substrate in a copper [25] and aluminum [26, 27] cavity, and a
planar, single island niobium transmon, with aluminum Josephson junctions, on
a silicon substrate [28, 37]. These previous studies all employed charge
sensitive transmons such that their $f_{01}$ transition frequency was
sensitive to the device’s charge offset. They all found quasi-particle dwell
times of 1-6 ms [29], confirming their coherence properties are not limited by
quasi-particle fluctuations. Additionally, all previous studies observed
diffusive charge environment dynamics, uniformly sampling the device offset
charge, mod 2$e$. This behavior was interrupted by large ($>$ 0.1 $e$),
discrete charge jumps separated by 5-7 minutes [25, 26, 27, 28] and 12 minutes
[37] on average. These discrete charge events have been reasonably modeled as
ionizing radiation, predominantly $\gamma$-rays, colliding with the device
substrate and producing liberated charges and energetic phonons [28, 37].

For our measurements, we make use of the multi-level, or qudit, nature of the
transmon. The improved coherence of the tantalum transmon, which persists in
the excited states, allows a high precision frequency measurement across
multiple levels of the device. The improved stability of the excited states
enables the use of long time, narrow bandwidth excitation pulses capable of
resolving the two charge-parity bands of the higher transitions. Unlike
similar noise studies previously performed [25, 26, 27, 28, 37], this device
is fully in the transmon regime, $E_{J}\gg E_{C}$, with the charge dispersion
of the 0-1 transition much less than the natural linewidth of the
corresponding transition. This is not the case for the higher transitions, and
with the narrow bandwidth pulses allowed by the high coherence of the device,
it is possible to resolve the two separate frequencies of the two charge-
parity states down to a separation of approximately 25 kHz. With this
capability, we monitor both the quasi-particle tunneling rate across the
Josephson junction as well as the charge offset dynamics affecting the qubit.
In addition, we perform these measurements in an interwoven manner across
multiple transitions to detect correlated charge noise behavior.

We find that while the quasi-particle tunneling rates observed in the tantalum
transmon are similar to those measured in aluminum and niobium devices, the
charge offset dynamics are qualitatively different. The differential tantalum
transmon charge offset dynamics are dominated by fluctuations between a few
meta-stable configurations of the charge environment. This leads to remarkably
stable charge offsets, often lasting hours, punctuated by large discrete
charge jumps. By repeating our measurements over a range of cryogenic
temperatures, it is possible to discern two categorical sources of noise. One
type of charge noise event is temperature dependent and causes predominantly
nearest-neighbor charge offset transitions. The other is temperature
independent and causes large charge offsets. The temperature independent noise
compares favorably to ionizing radiation events while temperature dependent
noise is likely on-chip charge fluctuations which may be a function of
tantalum’s uniquely uniform surface chemistry.

Figure 1: Planar tantalum transmon: a. False color optical image of
differential transmon device, courtesy of [19]. In blue are the tantalum
capacitor pads, ground plane and coupling to the readout resonator. The
Josephson junction connecting the two capacitor paddles is aluminum. b. For
this particular device we consistently measure energy relaxation times in the
qubit subspace of 247 $\pm$ 9 $\muup$s. c. Shown are spectroscopy measurements
of the qudit levels we will make use of in this work. d. Ramsey measurement in
the qubit subspace yield $T_{2}$* of approximately 30 $\muup$s, depending on
cooldown.

##  II Device

The tantalum transmon used in this study was designed and fabricated by the
Houck group at Princeton [19]. To produce the device, tantalum was sputtered
on a sapphire substrate heated to 500 C to ensure growth of tantalum’s
$\alpha$ phase. Photolithography and wet etching were then applied to define
the capacitor and resonator structures. Finally, electron beam lithography and
an in situ ion etch were applied followed by electron beam evaporation to
create the Al/AlOx/Al Josephson junction. Between most fabrication steps,
optimized solvent and piranha cleaning were used to reduce contaminants
introduced during the fabrication steps.

The Hamiltonian for the transmon in the charge basis is

$\begin{split}H=4\,E_{C}\,&(\hat{n}-n_{g})^{2}\ket{n}\bra{n}\\\
-&\frac{E_{J}}{2}(\ket{n+1}\bra{n}+\ket{n-1}\bra{n}),\end{split}$ (1)

where $\hat{n}$ is the operator representing the number of Cooper pairs on the
transmon island and $n_{g}$ is the dimensionless gate charge in units of 2$e$,
experienced by the transmon [30]. It is the fluctuating gate charge, $n_{g}$,
or charge offset, that we are primarily measuring in this work. The transmon
is designed to be in the strong transmon regime, with $E_{J}/h=6.3366$ GHz and
$E_{C}/h=208.3$ MHz, ensuring the 0-1 transition to be completely insensitive
to charge offset. In these measurements, we take advantage of the device
spectrum up to the third excited state, treating the transmon as a slightly
anharmonic oscillator as opposed to a two level qubit [31, 32].

Figure 2: Comparison between qubit spectroscopy and Ramsey measurements in
higher levels. a. Ramsey measurements in the 1-2 qudit subspace shows a beat
frequency in its decay. A fast Fourier transform is applied to the measurement
trace and the two frequencies corresponding to the two charge-parity state
frequencies are revealed. b. The high precision qubit spectroscopy measurement
in the 1-2 level uses a 40 $\muup$s excitation pulse after being prepared in
the qudit’s first excited state. The narrow bandwidth 1-2 excitation pulse
allows the two charge-parity state frequencies to be resolved. c. The energy
level diagram of the qudit’s excited states as a function of charge offset
illustrates how the the two frequencies present in these measurements are due
to the qudit being in either its even or odd charge-parity state. d. Taking
interwoven qubit spectroscopy and Ramsey measurements over the course of
twelve hours show clear agreement between the two methods.

As shown in Fig. 1, the transmon used in these measurements possesses a
$f_{01}$ = 3.4578 GHz with an anharmonicity of -208.3 MHz. Population of the
higher states is achieved by both single photon sequential state population
transfer between adjacent qudit states [33] and two photon state population
transfer between next-nearest qudit states [34, 35]. Similar to previous
measurements on 3D transmons, energy relaxation predominantly occurs through a
ladder of adjacent qudit states [33], and due to the high coherence of the
tantlaum device, even these higher levels have extended lifetimes [36].

As previously mentioned, the charge dispersion of the 0-1 transition is too
small to be detectable, but the higher transitions, starting with the 1-2
transition with a maximum charge dispersion of 120 kHz, have large enough
charge dispersions as to make their two charge-parity bands distinguishable.
Quasi-particles tunnel across the Josephson junction [21, 22] causing a change
in offset charge of 1$e$ at a rate much faster than the total Ramsey or
spectroscopy measurement time. As illustrated in Fig. 2c, this leads to two
distinguishable frequency bands per qudit energy transition. The separation of
these two frequencies is a measure of the charge offset experienced by the
qudit via

$f^{\pm}_{ij}=\bar{f}_{ij}\pm\epsilon_{ij}\cos(2\pi n_{g}),$ (2)

where $\epsilon_{ij}$ is the difference in charge dispersion between the two
levels being probed, $\bar{f}_{ij}$ is the average frequency of the
transition, $n_{g}$ is the dimensionless gate charge counted in units of 2$e$,
and the $\pm$ refers to whether there is a difference of an even or odd number
of quasi-particles across the Josephson junction.

Figure 3: Charge-parity fluctuations measured in the 1-2 transition. a. A mid-
band qubit excitation pulse of 10 $\muup$s was applied to one charge-parity
qubit state frequency followed by the opposite charge parity qubit state
frequency. These measurements followed one another with a total duty cycle of
approximately 50 $\muup$s. The left vertical side of Panel a indicates the
quadrature channel demodulated voltages in faded color, phase tuned for high
resolution between the ground and excited states, of the single shot data
points. As a guide for the eye, the 10-point running average of the output
voltages are shown in bold. The Hidden Markov model results, applied to the
single shot data, are displayed on the right vertical axis. b. The quasi-
particle dynamics, fit to a Lorentzian distribution on a white noise
background, reveal a typical dwell time of $1/\Gamma$ = 5.9 ms.

##  III Measurements

The exceptional lifetimes of the higher levels, combined with the exponential
increase in charge dispersion with respect to qudit energy level [30], make
the higher levels of this device a particularly well suited platform for
studying low frequency charge noise. The coherence properties of the higher
levels of a 3D transmon have been previously explored using Ramsey
spectroscopy [33]. By applying a ladder of $\pi$ pulses, it is possible to
populate the higher states with high fidelity. Following this ladder upwards,
a Ramsey measurement can be performed in a qudit subspace spanned by two
adjacent levels. As shown in Fig. 2a, by performing a fast Fourier transform
on the resulting trace, the two charge-parity qudit frequencies of that
subspace can then be extracted.

In our measurements, as well as in previously reported higher level Ramsey
measurements, there are typically two separate frequencies present in the
Ramsey decay. These two frequencies always fall within the expected charge
dispersion of the investigated level transition. Hence, as the total
measurement time is much longer than the quasi-particle fluctuation time, this
is interpreted as the experiment sampling both charge-parity states of the
transmon.

This conjecture is confirmed by high precision spectroscopy measurements of
the higher levels. The lifetime of the $\ket{1}$, state, for example, is
approximately 247 $\muup$s. This allows us, after populating the $\ket{1}$
state with a short, high fidelity $\pi_{01}$ pulse, to apply longer $\pi_{12}$
pulses, up to the 10’s of $\muup$s, without significant relaxation out of the
$\ket{1}$ state. As illustrated in Fig. 2b, measurements reported in this
article utilized high precision $\pi_{ij}$ pulses with a typical width of 40
$\muup$s allowing a spectral resolution of approximately 25 kHz. Performing
concurrent Ramsey and spectroscopy measurements yields the two charge-parity
state frequencies in agreement within experimental uncertainties.

Furthermore, high precision qudit spectroscopy and Ramsey measurements in the
1-2 qudit transition were taken repeatedly in an interwoven manner over the
course of 12 hours. As shown in Fig. 2d, the two sets of extracted qudit
transition frequencies, corresponding to the two charge-parity states, agree
well with one another over the course of many hours. These measurements
provide a confirmation to the conjecture posed in [33] that the two
frequencies extracted by Ramsey measurements in the higher qudit levels are
caused by an interplay between low frequency charge environment dynamics and
millisecond scale quasi-particle tunneling.

A similar spectroscopy measurement protocol can be employed to extract the
charge-parity state of the qudit. Quasi-particle fluctuations have been
measured to occur on timescales from 0.8 - 6 ms [25, 27, 28]. Therefore,
single shot measurements and appropriate signal analysis must be employed to
extract the quasi-particle dynamics. Similar to the high precision, higher
level qudit spectroscopy measurements described in the previous paragraph, we
prepare the $\ket{1}$ state with a short time, high fidelity $\pi_{01}$ pulse.
Then, instead of implementing a high precision, 40 $\muup$s duration pulse, we
enact a moderately precise $\pi_{12}$ pulse of 10 $\muup$s. This pulse has a
spectral content of approximately 100 kHz, approximately the size of the
charge dispersion of one of the charge-parity bands. The $\pi_{12}$ qudit
excitation pulse is executed with a center frequency just inside the maximum
charge dispersion frequency, followed by the measurement sequence repeated
with a center frequency targeted at the opposite charge-parity band. Each
measurement sequence is punctuated by an active qubit reset protocol, enabled
by the digitizer’s on-board field-programmable gate array (FPGA), allowing a
total duty cycle of approximately 50 $\muup$s [36]. These measurements are
then repeated, in an interwoven fashion, for a minute.

##  IV Results

Shown in Fig. 3 are the results of the quasi-particle tunneling measurement. A
25 ms portion of the entire minute long trace is displayed in Panel a. In
faded color is the raw single shot demodulated quadrature voltage data, phased
adjusted to be most sensitive to the transition. A 3-state general mixture
model is then applied to the single shot data to convert the demodulated I/Q
voltages to qubit state classification probabilities. Finally, a Hidden Markov
model (HMM) was then applied to the state classified traces to produce the
charge-parity transitions show on the right vertical axis of Fig. 3a [36]. As
a guide to the eye as well as further validating our HMM analysis, the ten-
point running average of the quadrature voltage is also displayed.

The Power Spectral Density (PSD) is taken of the charge-parity state
occupation trace. This PSD is well fit by a Lorentzian curve with a
characteristic frequency of 169 Hz. This corresponds to a typical quasi-
particle dwell time of $1/\Gamma$ = 5.9 ms, consistent with other reported
results on devices with aluminum Josephson junctions [25, 27, 28]. As noted by
previous authors, we take this observation of quasi-particle dwell time being
greater than the energy relaxation time of the qudit to be evidence that
quasi-particle tunneling is not the dominant mechanism limiting relaxation
times in the tantalum device [23, 26, 27].

In order to characterize the transmon’s charge environment dynamics,
interwoven qudit spectroscopy measurements were performed across the 1-2
single-photon and 1-3 two-photon qudit transitions repeatedly over the course
of approximately 70 hours. Both transitions have charge dispersions that agree
well with simulations of the device. Taking into account the full charge
dispersions of the two transitions, and the relationship in Eq. 2, both
frequency traces can be converted to a charge offset. Due to the lack of
charge offset control in this device, we can only project the measured
frequencies onto the 0-0.5 $e$ range. A strong agreement in the charge offsets
extracted from the 1-2 and 1-3 transitions, illustrated in Fig. 4a, confirms
that both transitions are influenced by a common charge environment.

Looking closer at the charge offset distribution of this long time trace
reveals some unexpected features. Previous measurements of the charge offset
distribution yield near uniform charge distributions [25]. The charge
environments sampled all possible offset values evenly, not preferring one
charge offset over others. The local charge environment of the tantalum
transmon behaves quite differently. As shown in Fig. 4b, there are charge
offsets the system prefers and regions of charge offset the system only very
rarely samples. The charge offset dynamics are dominated by long periods, up
to hours, of stability. These long stable periods are punctuated by large
discrete charge jumps between these quasi-stable charge configurations. The
fact that the frequency traces across multiple transitions, when converted to
a common charge offset, behave quantitatively similarly, indicate that this
non-uniformity is not due to a resonant or dispersive frequency interaction,
but rather is a characteristic of the transmon’s charge environment.

To further investigate the low frequency charge dynamics, we collected the
charge offset values into the separate time bins shown in Fig. 4c. The
different colors represent different 17 hour consecutive sets of data. If the
system is somehow not in equilibrium and we allowed it to evolve for a long
enough time, the charge offset distribution would approach uniformity. If this
were so, the different colored data sets in Fig. 4c would be quantitatively
distinct from each other as the charge environment further explored its
configuration space. This is not the case. The different colored data sets
appear fairly self-similar with no new favored locations appearing at later
times. This leads us to believe the charge environment visible to the tantalum
transmon has fully sampled its configuration space in the first 17 hours.

Figure 4: Low frequency dynamics of the qubit charge environment. a. High
precision qubit spectroscopy measurements in the 1-2 and 1-3 transitions were
taken interwoven with each other over the course of 70 hours. Displayed are
ordered pairs of the offset charge extracted from adjacent measurements of the
1-2 and 1-3 transitions showing a clear correlation between the two data sets.
b. The charge offsets form a highly nonuniform distribution. c. The same data
displayed above in Panel b is binned such that the different colors in the
histogram correspond to each 17 hours of data. The similarity between the
different time bins suggests the charge environment prefers certain discrete
configurations as opposed to simply not having the opportunity to fully
explore its configuration space in the measurements time. d. The power
spectral density of the charge offset dynamics at base temperature shows an
$1/f^{\alpha}$ noise with $\alpha$=1.94 and magnitude at 1 Hz of $1.11\times
10^{-6}\,e^{2}/\rm{Hz}$.

Displayed in Fig. 4d is the PSD of the low frequency charge dynamics at base
temperature. It yields a $1/f^{\alpha}$ spectrum with $\alpha_{\rm{e}}$ = 1.94
and magnitude at 1 Hz of $1.11\times 10^{-6}\,e^{2}$/Hz. Choosing instead to
view this as frequency noise, by analyzing the PSD of the original frequency
trace, allows us to make a direct comparison to previous charge noise
measurements. By normalizing our charge dispersion to a common frequency with
previous measurements, we find a similar character of noise spectrum with
$\alpha_{\rm{f}}=2.06$. However, our measurements yield a charge noise
magnitude at 1 Hz two orders of magnitude smaller, $A=7.4\times 10^{5}$
Hz2/Hz, than previous measurements on aluminum and niobium devices, which
consistently fall in the $10^{7}$ Hz2/Hz range [25, 26, 28].

We then repeated the long time series of measurements of 1-3 two photon qudit
spectroscopy measurements at 50 mK, 100 mK, and 150 mK. For each temperature,
we employed a HMM to identify the finite number of quasi-stable states and
their occupation time. We repeated this analysis forcing the stable charge
configurations to be fixed across temperatures and, alternatively, allowing
them to be independently determined at each temperature. The two methods yield
similar results for the number and location of quasi-stable charge offsets
[36]. For the remainder of the paper we will use the charge offsets identified
by analyzing the data sets from all temperatures simultaneously. The states
displayed in Fig. 5a-d are labeled sequentially 1-8 ranging from charge offset
0-0.5 $e$. Shown in Fig. 5e are the average stable charge configuration times
for each temperature. This shows a clear decrease of stable charge time with
increasing temperature.

Figure 5: The transition matrices between quasi-stable charge configurations
at various temperatures. A HMM was employed to identify the finite number of
quasi-stable charge configurations consistently across temperatures. a-d. The
diagonal elements of the matrices represent the probability for the charge
environment to remain in that same configuration. The off-diagonal elements
represent the probability for the charge environment to transition between
configurations. The region denoting nearest and next-nearest neighbor
transitions are outlined in blue. For each temperature, the majority of
transitions happen between adjacent charge configurations. e. Shown are the
average stable charge times for each temperature. They show a decrease in
stable charge time with increasing temperature. f. The transitions can be
neatly divided into nearest and next-nearest neighbor hopping and the
remaining transitions. The nearest and next-nearest hopping shows a increasing
dependence on temperature while the remaining transitions are temperature
independent.

The transitions between states, or off-diagonal elements in Fig. 5a-d, can
divided neatly into temperature dependent and independent subgroups. As shown
in Fig. 5e, the nearest and next-nearest state transitions increase with
temperature while the remaining transitions are temperature independent. This
suggests separate physical mechanisms for the two subgroups of transitions,
i.e. thermally assisted tunneling between similar charge configurations being
responsible for the neighboring transitions and some external noise source,
independent of mixing chamber temperature, being responsible for the charge
configuration scrambling transitions.

To summarize our results, we have measured an approximate $1/f^{2}$ low
frequency charge noise PSD with amplitude two orders of magnitude smaller than
previous measurements on aluminum and niobium superconducting qubits. The
previous studies were based on devices fabricated in both planar and 3-D
geometries as well as having both silicon and sapphire substrates. Also
distinct in our results is that the low frequency charge noise study on the
tantalum qubit revealed preferred charge offset values indicating a finite
number of quasi-stable environmental charge configurations. This is different
from all previously reported low frequency charge noise measurements performed
on superconducting qubits regardless of their geometry or substrate.

Large discrete charge offset jumps have been recently reported in niobium
devices with aluminum Josepshon junctions fabricated on a silicon substrate
with typical occurrence times of approximately 12 minutes [37]. The discrete
charge jumps measured in these niobium transmons were reasonably modeled as
gamma ray collisions with the device substrate. The average time between
charge offset jumps in the tantalum device at 10 mK is almost twice as long,
approximately 22 minutes, with some stable charge times reaching over 4 hours.

To further investigate whether our data can be explained in terms of these
ionizing particle tracks, we performed finite element electromagnetic
simulations of both the niobium transmons used in this previous study and the
tantalum transmon to determine the volume of the substrate that the transmons
are sensitive to charge events such as gamma ray collisions [36]. While the
differential design of the tantalum does reduce its charge sensitive region
into its substrate, the overall much larger size of the tantalum transmon
compared to the single island niobium device actually makes the tantalum
transmon’s charge sensitive region larger. This would suggest, all other
things being equal, that the tantalum device’s charge environment would
fluctuate at a faster rate.

This discrepancy becomes even more acute if we compare the previously reported
rate of charge events occurring every 0.2 hours with our temperature
independent charge rate of an event every 1.6 hours. Some of this difference
can be explained by the greater bandgap of sapphire, 8.3 eV, compared to that
of silicon, 1.2 eV. However, if the scrambling charge transitions observed in
the tantalum transmon’s charge environment are due to particle collisions with
the sapphire substrate, the tantalum-sapphire device has a much reduced
characteristic radiation trapping and charge production rate.

The temperature dependent behavior of the neighboring charge configuration
transitions point to these transitions being caused by jumps between local
charge configurations in the immediate vicinity of the transmon. The fact that
the temperature dependent transitions are between neighboring configurations
also seems to rule out a collection of Two Level Systems (TLS) as the basis of
these quasi-stable charge offsets. TLS have been identified as a primary
source of charge noise in Single Electron Transistors (SET), devices typically
fabricated out of aluminum on silicon substrates [38, 39, 40, 41]. While the
amplitude of charge noise found in these studies is of the same order of
magnitude measured in our device, there are important qualitative differences.
The low frequency charge noise typically measured in SETs is $1/f$ in
character, points to a collection of TLS with a broad range of frequencies as
the underlying mechanism. Additionally, if TLSs were primarily responsible for
the temperature dependent transitions measured in our study, the transitions
would be distributed across the matrices shown in Fig. 5a-d as opposed to
being concentrated around the diagonal. For example, if the eight quasi-stable
charge offsets were due to three ($2^{3}=8$) independently fluctuating TLS,
there would be three populated transitions per row in Fig. 5a-d evenly
distributed across the transition matrices. To further test this hypothesis,
it could be possible to design superconducting devices capable of spatially
detecting local charge fluctuations. In fact, local charge dynamics have
recently been tracked in the vicinity of a multi-mode superconducting qubit
[42].

One possible explanation of the qualitatively different behavior of the
tantalum transmon’s charge environment is the different surface chemistries
found in the superconducting device metals. Tantalum has been shown to produce
a highly uniform Ta2O5 oxide layer which could greatly reduce the density of
surface defects and suppress charge tunneling between these localized states.
High quality interfaces have also been demonstrated in the tantalum device
[19] which could play a role in the reduction of surface defects. Niobium and
aluminum host a much more irregular oxide layer, possibly allowing a high
density of closely packed defect sites with low tunneling barriers to
facilitate charge transport across its surface.

All of this discussion points to the need of systematic comparisons between
superconducting qubits of different designs, materials, and substrates as well
as first principle calculations to better understand and mitigate charge noise
processes in these devices. For example, there is a growing appreciation of
the detrimental effects of radiation on qubit coherence properties [43, 37,
44, 45, 46]. If the sapphire substrate does, in fact, effectively reduce the
charge production rate due to ionizing particle collisions, it might be
possible to accommodate the stringent requirements of surface error correction
codes [47].

##  V Conclusions

Taking advantage of the long coherence times of a tantalum based
superconducting qudit, we have performed high precision frequency measurements
of the higher levels of this device. Interweaving both Ramsey and spectroscopy
measurements across multiple transitions, we have explicitly confirmed
previous conjectures on the charge noise behavior in the transmon’s higher
levels. In particular, by performing interwoven Ramsey and spectroscopy
measurements in the higher qudit levels over the course of many hours, we have
confirmed that the fluctuating Ramsey beat frequency found in this and
previous measurements [33] is due to a sampling of the two charge-parity state
frequencies. Also, by tracking the two fluctuating frequencies in multiple
transitions confirms the assumption that multiple transitions do experience a
common charge environment.

Most striking of our findings however is the novel behavior of the tantalum
transmon’s charge environment. There appear to be a finite number of quasi-
stable configurations the charge environment samples. This is at odds with all
previous low frequency charge noise studies on superconducting qubits.
Additionally, the transitions between charge configurations happen at a much
lower rate than discrete jumps in charge offsets observed in previous
measurements on aluminum and niobium transmons. If our measured discrete
charge offset jumps are in fact due to ionizing particles colliding with the
sample substrate, this implies that the tantalum qubit-sapphire substrate
device experiences either a reduced charge production rate or a reduced
released charge trap length. However, the temperature dependence of the stable
charge configurations, and their transition behavior, further complicates this
picture. The increased rate of transitions between charge configurations as a
function of temperature, as well as the fact that they happen primarily
between adjacent configurations, suggest that the local chemistry of the
device is responsible for the majority of the behavior we observe.

The long coherence times of the superconducting tantalum offer exciting
technological possibilities for utilizing higher qudit levels. With coherence
times in the hundreds of microseconds, tantalum based qudit devices could
offer the first platforms to realize recently proposed computational tasks
that take advantage of the qudit connectivity such as more compact error
correction codes [48], efficient cryptology protocols [49, 50], and many body
quantum chaos simulations [51]. In addition, the use of highly coherent
excited states offer the possibility of performing in situ studies of the
charge noise environment in a quantum processor constructed out of these
devices and eliminates the need for specialty charge-sensitive transmons,
specifically designed for charge noise studies. In closing, we believe that
performing high precision frequency measurements across multiple qudit
transitions can be a unique and valuable tool for distinguishing between
different material sources of decoherence in superconducting devices.

##  VI Acknowledgments

The authors wish to thank Andrew Houck for providing the tantalum transmon
used in this study and Kevin Osborn for helpful discussions on early charge
noise measurements. We thank MIT-Lincoln Laboratories and IARPA for providing
the Traveling Wave Parametric Amplifier used in this study. This work was
supported by the U.S. Department of Energy, Office of Science, Office of Basic
Energy Sciences under Award DE-SC0020313. Additionally, this work was
performed under the auspices of the U. S. Department of Energy by Lawrence
Livermore National Laboratory under Contract No. DE-AC52-07NA27344.
Experimental characterization was made possible by the National Nuclear
Security Administration Advanced Simulation and Computing Beyond Moore’s Law
(NA-ASC-127R-16) program support of the LLNL Quantum Design and Integration
Testbed (QuDIT).

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