# Randomized Privacy Budget Differential Privacy

Meisam Mohammady

(December 2018)

##  1 Abstract

While pursuing better utility by discovering knowledge from the data,
individual’s privacy may be compromised during an analysis. To that end,
differential privacy has been widely recognized as the state-of-the-art
privacy notion. By requiring the presence of any individual’s data in the
input to only marginally affect the distribution over the output, differential
privacy provides strong protection against adversaries in possession of
arbitrary background. However, the privacy constraints (e.g., the degree of
randomization) imposed by differential privacy may render the released data
less useful for analysis, the fundamental trade-off between privacy and
utility (i.e., analysis accuracy) has attracted significant attention in
various settings. In this report we present DP mechanisms with randomized
parameters, i.e., randomized privacy budget, and formally analyze its privacy
and utility and demonstrate that randomizing privacy budget in DP mechanisms
will boost the accuracy in a humongous scale.

##  2 Backgrounds

######  Definition 2.1.

Let $\epsilon,\delta\geq 0$. A mechanism
$M:\mathsf{D}\times\Omega\to\mathsf{R}$ is $(\epsilon,\delta)$-differentially
private for Adj if for all $d,d^{\prime}\in\mathsf{D}$ such that
$\text{Adj}(d,d^{\prime})$, we have

$\displaystyle\mathbb{P}(M(d)\in S)\leq
e^{\epsilon}\mathbb{P}(M(d^{\prime})\in S)+\delta,\;\;\forall
S\in\mathcal{M}.$ (1)

If $\delta=0$, the mechanism is said to be $\epsilon$-differentially private.

######  Definition 2.2.

(Usefulness Definition). A database mechanism $M_{q}$ is
($\zeta,\gamma(\zeta,u)$)-useful if with probability $1-\gamma(\zeta,u)$, for
every database $d\subseteq\mathsf{D}$, $|M_{q}(d)-q(d)|\leq\zeta$.

##  3 Randomized Parameter DP

Let $M_{q}(d,u)=q(d)\bigoplus\omega(u)$ be a randomized
($\epsilon(u),\delta(u)$)-differentially private mechanism where $\omega(u)$
is a random oracle with specified set of parameters $u$ and $\bigoplus$ stands
for the corresponding operator. Also suppose $M_{q}(d,u)$ is
($\zeta,\gamma(\zeta,u)$)-useful. Define by
$\mathcal{M}_{q}(d)=q(d)\bigoplus\omega(u)$, with $u\sim\mathcal{F}$, the
distribution of all possible randomized mechanism $M_{q}(d,u)$ where
$\mathcal{F}$ is a probability density function for all parameters in $u$. The
optimal utility achieved due to the application of optimal pdf $\mathcal{F}$
is shown in the following.

$U(\zeta)=Max\ \\{E\left[\mathbb{P}(|M_{q}(d,u)-q(d)|)<\zeta\right]\\}$ (2)

Accordingly, we say that $\mathcal{M}_{q}(d)$ improves the privacy-utility
trade-off if we have

  * •

Case I ($\delta=0$)

$E(e^{u})=e^{\epsilon(u_{0})}\Rightarrow U(\zeta)>1-\gamma(\zeta,u_{0})$ (3)

over

  * •

Case II ($\delta>0$)

$\displaystyle E(\delta(u))=\delta(u_{0})$ $\displaystyle\Rightarrow
U(\zeta)>1-\gamma(\zeta,u_{0})$ (4) $\displaystyle
E(e^{u})<e^{\epsilon(u_{0})}$ (5)

where, $E(.)$ denotes the expected value over distribution $\mathcal{F}$. We
now derive the corresponding conditions for two popular differentially private
mechanisms. In particular, a _Laplace Mechanism_ modifies an answer to a
numerical query by adding independent and identically distributed (i.i.d.)
zero-mean noise distributed [1], [3], [2] according to a Laplace distribution.
Recall that the Laplace distribution with mean zero and scale parameter $b$,
denoted $Lap(b)$, has density $p(x;b)=\frac{1}{2b}exp(-\frac{|x|}{b})$ and
variance $2b^{2}$ . Moreover, for $\omega\in\mathbb{R}^{k}$ with $\omega_{i}$
i.i.d. and $\omega_{i}\sim Lap(b)$ , denoted $\omega\sim Lap(b)^{k}$, we have
$p(\omega;b)=(\frac{1}{2b})^{k}exp(-\frac{\left\lVert\omega\right\rVert_{1}}{b})$,
$E(\left\lVert\omega\right\rVert_{1})=b$, and
$\mathbb{P}(\left\lVert\omega\right\rVert_{1}\geq tb)=e^{-t}$.

######  Theorem 3.1.

Let $q:\mathsf{D}\to\mathbb{R}^{k}$ be a query , $epsilon>0$. Then the
mechanism $M_{q}:\mathsf{D}\times\Omega\to\mathbb{R}^{k}$ defined by
$M_{q}(d)=q(d)+w$, with $w\sim Lap(b)^{k}$, where
$b\geq\frac{\Delta_{1}q}{\epsilon}$ is $\epsilon$-differentially private.

Hence, $\omega(u)$ is a Laplace distribution where $u=1/b=\epsilon$. Also,
$\gamma(\zeta,u_{0})=e^{-\zeta\epsilon}$. Thus, equations 2,7 can be re-
written as follows.

$U(\zeta)=Max\ E(1-e^{-\zeta\epsilon})$ (6)
$E(e^{\epsilon})=e^{\epsilon_{0}}\Rightarrow U(\zeta)>1-e^{-\zeta\epsilon}$
(7)

Similarly, for a Gaussian mechanism, we have

$Min\ E(e^{\epsilon})=\int^{\infty}_{-\infty}f(\epsilon)e^{\epsilon}d\epsilon$
(8)

over

$E(Q(\frac{\zeta}{\sqrt{\frac{2\zeta+1}{2\epsilon}}}))=\int^{\infty}_{-\infty}f(\epsilon)Q(\frac{\zeta}{\sqrt{\frac{2\zeta+1}{2\epsilon}}})d\epsilon\leq\delta$
(9)

##  4 Privacy and Utility Analysis

In this section, we formally characterize the privacy and the utility of the
Randomized DP mechanism.

###  4.1 Deriving PDF of Randomized DP

we can write the CDF of the output of an Randomized DP Laplace mechanism in
terms of the _Moment Generating Function (MGF)_ for the probability
distribution $f_{\frac{1}{b}}$. Recall that MGF of a random variable is an
alternative specification of its probability distribution, and hence provides
the basis of an alternative route to analytical results compared with working
directly with probability density functions or cumulative distribution
functions. In particular,

######  Definition 4.1.

(Moment Generating Function) The moment-generating function of a random
variable $x$ is $M_{X}(t):=\mathbb{E}\left[e^{tX}\right],t\in\mathbb{R}$
wherever this expectation exists. The moment-generating function is the
expectation of the random variable $e^{tX}$.

Accordingly, in the following, we give a general formula for the probability
of any measurable event originated from an Randomized DP Laplace Mechanism.

######  Theorem 4.1.

The search space of an Randomized DP Laplace mechanism is as large as the
space of all PDFs with non-negative support and existing MGF. Moreover,
generated PDFs are all log-convex.

Thus, for a PDF with non-negative support (scale parameter is always non-
negative), the Randomized DP Laplace mechanism outputs another PDF using the
MGF (CDF is the moment and PDF is its derivative) as shown in Equation 11 in
Appendix [3]. However, a challenge is that not all random variables have
moment generating functions (MGFs). Fortunately, MGFs possess an appealing
composability property between independent probability distributions, which
can be used to provide us with a search space of all linear combinations of a
set of popular distributions with known MGFs (infinite number of RVs).

######  Theorem 4.2 (MGF of Linear Combination of RVs).

If $x_{1},x_{2}$, $\cdots,x_{n}$ are $n$ independent RVs with respective MGFs
$M_{x_{i}}(t)=\mathbb{E}(e^{tx_{i}})$ for $i=1,2,\cdots,n$, then the MGF of
the linear combination $Y=\sum\limits_{i=1}^{n}a_{i}x_{i}$ is
$\prod\limits_{i=1}^{n}M_{x_{i}}(a_{i}t)$.

Thus, our search space is given as all possible linear combinations of a set
of independent RVs with existing MGF (Section 4.5 demonstrates on how to
choose the set of independent RVs).

Figure 1: Illustrating the performance of Randomized DP Laplace mechanism. The
left, middle and right figures in each of the configurations are respectively
the usefulness for the Laplace mechanism, the Randomized DP Laplace mechanism
and the optimal noise, i.e., Laplace distribution in high privacy regime and
Staircase shape distribution in low privacy regime Figure 2: Validating the
effectiveness of Randomized DP for small $\epsilon$ Figure 3: Comparing the
performance of (a) the baseline Laplace mechanisms, for count queries and (b)
Randomized DP, when varying $\gamma$ and $\epsilon$

####  4.1.1 Determining the Optimal PDF

After giving the differential privacy guarantee and characterizing the utility
of the Randomized DP Laplace mechanism (see Section 4), we will show that the
Randomized DP framework can unify two parallel concepts, i.e., privacy and
utility, into one optimization problem defined over the defined search space
of RVs.

###  4.2 Numerical Analysis

We now present numerical results to fine tune the Randomized DP parameters
under more general settings. In particular, Figure 1 depicts the corresponding
performance of Laplace mechanism, Randomized DP Laplace mechanism and
Staircase mechanism. Figure 1 clearly demonstrates the fact that Randomized DP
can achieve both objectives mentioned earlier, i.e., approaching the optimal
mechanism and improving Laplace mechanisms for larger $\epsilon$. We now
analyze the improvements provided by Randomized DP under two different
settings. First, we discuss the performance of Randomized DP under a stronger
privacy guarantee (e.g., $\epsilon<2$). Next, we study the improvement for
counting queries ($\Delta q=1$) while varying the error bound $\gamma$.

###  4.3 Privacy Analysis

We now show the Randomized DP Laplace mechanism provides differential privacy
guarantee. Using theorem 4.1, the DP bound is

$\displaystyle e^{\epsilon}=\max\limits_{\forall
S\in\mathsf{R}}\left\\{\frac{-M_{\frac{1}{b}}(-|x-q(d)|)|_{S_{\geq
q(d)}}+M_{\frac{1}{b}}(-|x-q(d)|)|_{S_{<q(d)}}}{-M_{\frac{1}{b}}(-|x-q(d^{\prime})|)|_{S_{\geq
q(d^{\prime})}}+M_{\frac{1}{b}}(-|x-q(d^{\prime})|)|_{S_{<q(d^{\prime})}}}\right\\}$

Hence, the value of $e^{\epsilon}$ only depends on the distribution of
reciprocal of the scale parameter $b$, i.e., $f_{\frac{1}{b}}$. Moreover, an
MGF is positive and log-convex where the latter property is desirable in
defining various natural logarithm upper-bounds, e.g., DP bound. In the
following theorem, we demonstrate the fact that our MGF-based formula for the
probability $\mathbb{P}(\\{q(d)+Lap(b)\\}\in S)$ in Equation
LABEL:lRandomized_DPlap1 can be easily applied to calculate the differential
privacy guarantee.

######  Theorem 4.3.

The Randomized DP mechanism $\mathcal{M}_{q}(d,b)$ is

$\ln\left[\cfrac{\mathbb{E}(\frac{1}{b})}{\diff{M_{\frac{1}{b}}(t)}{t}|_{t=-\Delta
q}}\right]\text{-differentially private.}$ (10)

Finally, Theorem 4.2 can be directly applied to calculate the differential
privacy guarantee of any RV from our defined search space (all linear
combinations of a set of independent RVs with known MGFs).

######  Corollary 4.4 (differential privacy of combined PDFs).

If $x_{1},x_{2},\cdots,x_{n}$ are $n$ independent random variables with
respective MGFs $M_{x_{i}}(t)=\mathbb{E}(e^{tx_{i}})$ for $i=1,2,\cdots,n$,
then the Randomized DP mechanism $\mathcal{M}_{q}(d,b)$ where $\frac{1}{b}$ is
defined as the linear combination
$\frac{1}{b}=\sum\limits_{i=1}^{n}a_{i}x_{i}$ is

$\displaystyle\scriptsize ln\left[\cfrac{\sum\limits_{j=1}^{n}a_{j}\cdot
E_{x_{j}}(\frac{1}{b})}{\sum\limits_{j=1}^{n}a_{j}\cdot
M^{\prime}_{x_{j}}(-a_{j}\cdot\Delta
q)\cdot\prod\limits_{\begin{subarray}{c}i=1\\\ i\neq
j\end{subarray}}^{n}M_{x_{i}}(-a_{i}\cdot\Delta q)}\right]$ (11)

-differentially private.

Therefore, we have established a search space of probability distributions
with a universal formulation for their differential privacy guarantees, which
is the key enabler for the universality of Randomized DP. Next, we
characterize the utility of Randomized DP Laplace mechanisms.

###  4.4 Characterizing the Utility

We now characterize the utility of the Randomized DP Laplace mechanism. To
make concrete discussions, we first focus our discussion on the usefulness
metric (see Section LABEL:sec:back), then discuss how a similar logic applies
to other metrics. Denote by $U(\epsilon,\Delta q,\gamma)$ the usefulness of an
Randomized DP Laplace mechanism for all $\epsilon>0$, sensitivity $\Delta q$
and error bound $\gamma$. The optimal usefulness is then given as the answer
of the following optimization problem over the search space of PDFs.

$\displaystyle\max\limits_{f_{\frac{1}{b}}\in F}\big{\\{}U(\epsilon,\Delta
q,\gamma)\big{\\}}=\max\limits_{f_{\frac{1}{b}}\in
F}\bigg{\\{}\frac{1}{2}\cdot\Big{[}-M_{\frac{1}{b}}(-|x-q(d)|)|_{q(d)}^{q(d)+\gamma}$
$\displaystyle\hskip
99.58464pt+M_{\frac{1}{b}}(-|x-q(d)|)|_{q(d)-\gamma}^{q(d)}\Big{]}\bigg{\\}},$
$\displaystyle\text{subject to }\ \ \
\epsilon=\ln\left[\cfrac{\mathbb{E}(\frac{1}{b})}{\diff{M_{\frac{1}{b}}(t)}{t}|_{t=-\Delta
q}}\right]$

Note that $\epsilon$ and $\Delta q$ do not directly impact the usefulness but
they do so indirectly through the differential privacy constraint.
Furthermore, as shown in Theorem 4.3, the differential privacy guarantee
$\epsilon$ over the established search space $\mathcal{F}$ is a unique
function of the parameters of the second fold distribution.

######  Corollary 4.5.

Denote by $u$, the set of parameters for a probability distribution
$f_{\frac{1}{b}}$, and by $M_{f(u)}$ its MGF. Then, the optimal usefulness of
an Randomized DP mechanism utilizing $f_{\frac{1}{b}}$, at each triplet
$(\epsilon,\Delta q,\gamma)$ is

$\displaystyle U_{f}(\epsilon,\Delta
q,\gamma)=\max\limits_{u\in\mathbb{R}^{|u|}}\bigg{\\{}\frac{1}{2}\cdot\Big{[}-M_{f(u)}(-|x-q(d)|)|_{q(d)}^{q(d)+\gamma}$
$\displaystyle\hskip
99.58464pt+M_{f(u)}(-|x-q(d)|)|_{q(d)-\gamma}^{q(d)}\Big{]}\bigg{\\}},$
$\displaystyle\text{subject to }\ \ \
\epsilon=\ln\left[\cfrac{\mathbb{E}(\frac{1}{b})}{\diff{M_{\frac{1}{b}}(t)}{t}|_{t=-\Delta
q}}\right]$

However, MGFs are positive and log-convex, with $M(0)=1$ and hence,
$U_{f}(\epsilon,\Delta
q,\gamma)=1-\min\limits_{u\in\mathbb{R}^{|u|}}M_{f(u)}(-\gamma)$. Therefore,
for usefulness metric, the best distribution for $\epsilon$ is the one with
minimum MGF evaluated at $\gamma$. In particular, for a set of privacy and
utility parameters, one can find the optimal point using the Lagrange
multiplier method. i.e.,

$\displaystyle\mathcal{L}(u,\lambda)=M_{f(u)}(-\gamma)+\lambda\cdot(\ln\left[\cfrac{\mathbb{E}(\frac{1}{b})}{\diff{M_{\frac{1}{b}}(t)}{t}|_{t=-\Delta
q}}\right]-\epsilon)$ (12)

Next, under the DP guarantee of several probability distributions, we will
apply Equation 12 to find the optimal trade-off.

Utility under Other Metrics. We derive the utility of the Randomized DP
Laplace mechanism under some well-known utility metrics. Due to space
limitation, we present only the final results in Table 1.

Table 1: Utility of the Randomized DP (Laplace) under different metrics

$\ell_{1}$ $\ell_{2}$ Entropy Usefulness
$\int\limits_{0}^{\infty}M_{\frac{1}{b}}(-x)dx$
$\sqrt{2\iint\limits_{0}^{\infty}M_{\frac{1}{b}}(-u)dudx}$
$\displaystyle\int_{0}^{\infty}-M^{\prime}_{\frac{1}{b}}(-x)\cdot\ln
M^{\prime}_{\frac{1}{b}}(-x)dx$ $1-M_{\frac{1}{b}}(-\gamma)$

The results in Table 1 can be easily applied to optimize each measure in
different applications.

Necessary Condition on Selected Distributions. Not all second fold probability
distributions can boost the utility of the baseline Laplace mechanism.
Accordingly, in the following theorem, we derive a necessary condition on the
differential privacy guarantee of an Randomized DP Laplace mechanism to boost
the utility of the baseline Laplace mechanism (refer to Appendix in [3] for
the proof). Using this necessary condition, we can easily filter out those
probability distributions that cannot deliver any utility improvement.

######  Theorem 4.6.

The utility of an Randomized DP Laplace mechanism with
$\epsilon\geq\ln\Big{[}\mathbb{E}_{\frac{1}{b}}\big{(}e^{\epsilon(b)}\big{)}\Big{]}$
is always upper bounded by the utility of the $\epsilon$-differentially
private baseline Laplace mechanism. Equivalently, for an Randomized DP Laplace
mechanism to boost the utility, the following relation is necessarily true.

$e^{\epsilon}=\frac{\mathbb{E}(\frac{1}{b})}{M^{\prime}_{\frac{1}{b}}(-\Delta
q)}<M_{\frac{1}{b}}(\Delta q)$ (13)

We note that
$\epsilon=\ln\Big{[}\mathbb{E}_{\frac{1}{b}}\big{(}e^{\epsilon(b)}\big{)}\Big{]}$
provides a tight upper bound as it gives the overall $e^{\epsilon}$ of an
Randomized DP Laplace mechanism as the average of differential privacy
leakages.

###  4.5 Finding Utility-Maximizing Probability Distributions

We now examine a set of well-known probability distributions to establish the
required search space by selecting those offer a significantly improved
$\epsilon$ compared with the bound given in Theorem 4.6. Promisingly, our
analytic evaluations for _three_ of these distributions, i.e., Gamma, Uniform
and truncated Gaussian distributions demonstrates such a payoff. Finally, we
note that those chosen distributions are general enough to cover a large
family of other probability distributions. For instance, since Exponential
distribution, Erlang distribution, and Chi-squared distribution are special
cases of Gamma distribution, we will only consider Gamma distribution.

####  4.5.1 Discrete Probability Distributions

First, we consider two different mixture Laplace distributions that can be
applied for constructing Randomized DP Laplace mechanisms with discrete
probability distribution $f_{b}$.

(1) Degenerate distribution. A degenerate distribution is a probability
distribution in a (discrete or continuous) space with support only in a space
of lower dimension. If the degenerate distribution is uni-variate (involving
only a single random variable) it will be a deterministic distribution and
takes only a single value. Therefore, the degenerate distribution is identical
to the baseline Laplace mechanism as it also assigns the mechanism one single
scale parameter $b_{0}$. Specifically, the probability mass function of the
uni-variate degenerate distribution is:

$f_{\delta,k_{0}}(x)=\begin{cases}1&x=k_{0}\\\ 0&x\neq k_{0}\end{cases}$

The MGF for the degenerate distribution $\delta_{k_{0}}$ is given by
$M_{k}(t)=e^{t\cdot k_{0}}$. Using Equation 10, Theorem 4.7 gives the same DP
guarantee as the baseline Laplace mechanism.

######  Theorem 4.7.

The Randomized DP Laplace mechanism $M_{q}(d,\epsilon)$, $\epsilon\sim
f_{\delta,\frac{1}{b_{0}}}(\epsilon)$, is $\frac{\Delta
q}{b_{0}}$-differentially private.

Obviously, this distribution does not improve the bound in Theorem 4.6 but
shows the soundness of our findings.

(2) Bernoulli distribution. The probability mass function of this
distribution, over possible outcomes $k$, is

$f_{B}(k;p)=\begin{cases}p&{\text{if }}k=1,\\\ q=1-p&{\text{if
}}k=0.\end{cases}$

Note that the binary outcomes $k=0$ and $k=1$ can be mapped to any two
outcomes $X_{0}$ and $X_{1}$, respectively. Therefore, we consider the
following Bernoulli outcomes

$f_{B,X_{0},X_{1}}(X;p)=\begin{cases}p&{\text{if }}X=X_{1},\\\ q=1-p&{\text{if
}}X=X_{0}.\end{cases}$

The MGF for Bernoulli distribution $f_{B,X_{0},X_{1}}(X;p)$ is
$M_{X}(t)=p\cdot e^{t\cdot X_{0}}+(1-p)\cdot e^{t\cdot X_{1}}$. We now derive
the precise differential privacy guarantee of an Randomized DP Laplace
mechanism with its scale parameter randomized according to a Bernoulli
distribution.

######  Theorem 4.8.

The Randomized DP Laplace mechanism $M_{q}(d,\epsilon)$, $\epsilon\sim
f_{B,\frac{1}{b_{0}},\frac{1}{b_{1}}}(\epsilon;p)$, satisfies $\ln[p\cdot
e^{\frac{\Delta q}{b_{0}}}+(1-p)\cdot e^{\frac{\Delta q}{b_{1}}}]$-DP.

However, this bound is exactly the mean value of $e^{\epsilon(b)}$ and
therefore, this distribution does not improve the bound given in Theorem 4.6,
either.

####  4.5.2 Continuous Probability Distributions

We now investigate three compound Laplace distributions.

(1) Gamma distribution. The gamma distribution is a two-parameter family of
continuous probability distributions with a shape parameter $k>0$ and a scale
parameter $\theta$. Besides the generality, the gamma distribution is the
maximum entropy probability distribution (both w.r.t. a uniform base measure
and w.r.t. a $1/x$ base measure) for a random variable $X$ for which
$\mathbb{E}(X)=k\theta=\alpha/\beta$ is fixed and greater than zero, and
$\mathbb{E}[\ln(X)]=\psi(k)+\ln(\theta)=\psi(\alpha)-\ln(\beta)$ is fixed
($\psi$ is the digamma function). Therefore, it may provide a relatively
higher privacy-utility trade-off in comparison to the other candidates. A
random variable $X$ that is gamma-distributed with shape $\alpha$ and rate
$\beta$ is denoted by $X\sim\Gamma(k,\theta)$ and the corresponding PDF is

$f_{\Gamma}(X;k,\theta){\displaystyle={\frac{x^{k-1}e^{-\frac{x}{\theta}}}{\Gamma(k)\cdot\theta^{k}}}\quad{\text{
for }}X>0{\text{ and }}k,\theta>0,}$

where $\Gamma(\alpha)$ is the gamma function. We now investigate the
differential privacy guarantee provided by assuming that the reciprocal of the
scale parameter $b$ in Laplace mechanism is distributed according to the gamma
distribution (see Appendix [3] for the proof).

######  Theorem 4.9.

The Randomized DP Laplace mechanism $M_{q}(d,\epsilon)$, $\epsilon\sim
f_{\Gamma}(\epsilon;k,\theta)$, satisfies $\big{(}(k+1)\cdot\ln(1+\Delta
q\cdot\theta)\big{)}$-DP.

We now apply the necessary condition given in Equation 13.

######  Lemma 4.10.

Randomized DP using Gamma distribution can satisfy the necessary condition in
Equation 13.

###### Proof.

We need to show that there exist $k$ and $\theta$ such that
$(k+1)\cdot\ln(1+\Delta q\cdot\theta)<-k\cdot\ln(1-\Delta q\cdot\theta)$ ,
$\theta<\frac{1}{\Delta q}$. Given $\theta=\frac{1}{2\Delta q}$, we need to
show that $\exists k,k\cdot\ln(2)>(k+1)\cdot\ln(1.5)$, which always holds for
all $k>1.4094$. ∎

Therefore, Gamma distribution may improve over the baseline, and this can be
computed by optimizing the privacy-utility trade-off using the Lagrange
multiplier function in Equation 12. Also, our optimization shows that, this
distribution is more effective for large $\epsilon$ (weaker privacy
guarantees).

(2) Uniform distribution. In probability theory and statistics, the continuous
uniform distribution or rectangular distribution is a family of symmetric
probability distributions such that for each member of the family, all
intervals of the same length on the support of the distribution are equally
probable. The support is defined by the two parameters, $a$ and $b$, which are
the minimum and maximum values. The distribution is often abbreviated as
$U(a,b)$, which is the maximum entropy probability distribution for a random
variable $X$ under no constraint; other than that, it is contained in the
distribution’s support. The MGF for $U(a,b)$ is

$M_{X}(t)=\begin{cases}\frac{e^{tb}-e^{ta}}{t(b-a)}&{\text{for }}t\neq 0,\\\
1&{\text{for }}{\text{for }}t=0.\end{cases}$

Using Theorem 4.3, we now drive the precise differential privacy guarantee of
an Randomized DP Laplace mechanism for uniform distribution $U(a,b)$.

######  Theorem 4.11.

The Randomized DP Laplace mechanism $M_{q}(d,\epsilon)$, $\epsilon\sim
f_{U(a,b)}(\epsilon)$, is
$\ln\big{[}\frac{\alpha^{2}-\beta^{2}}{2((1+\beta)e^{-\beta}-(1+\alpha)e^{-\alpha})}\big{]}$-differentially
private, where $\alpha=a\cdot\Delta q$ and $\beta=b\cdot\Delta q$.

We now apply the necessary condition given in Equation 13. One can easily
verify that the inequality holds for infinite number of settings, e.g.,
$a=0.5$, $b=9$ and $\Delta q=1.2$.

######  Lemma 4.12.

Randomized DP using uniform distribution can satisfy the necessary condition
in Equation 13.

Therefore, Randomized DP using uniform distribution may improve over the
baseline, and this can be computed by optimizing the privacy-utility trade-off
using the Lagrange multiplier function in Equation 12. Also, our simulation
shows that, this distribution can also be effective for both small and large
$\epsilon$.

(3) Truncated Gaussian distribution. The last distribution we consider is the
Truncated Gaussian distribution. This distribution is derived from that of a
normally distributed random variable by bounding the random variable from
either below or above (or both). Therefore, we can benefit from the numerous
useful properties of Gaussian distribution, by truncating the negative region
of the Gaussian distribution. Suppose $X\sim\mathcal{N}(\mu,\sigma^{2})$ has a
Gaussian distribution and lies within the interval $X\in(a,b),\;-\infty\leq
a<b\leq\infty$. Then, $X$ conditional on $a<X<b$ has a truncated Gaussian
distribution with the following probability density function.

$f_{\mathcal{N}^{T}}(X;\mu,\sigma,a,b){\displaystyle={\frac{\phi(\frac{X-\mu}{\sigma})}{\sigma\cdot\big{(}\Phi(\frac{b-\mu}{\sigma})-\Phi(\frac{a-\mu}{\sigma})\big{)}}}\quad{\text{for
}}a\leq x\leq b}$

and by $f_{\mathcal{N}^{T}}=0$ otherwise. Here,
$\phi(x)=\frac{1}{\sqrt{2\pi}\cdot}e^{-\frac{x^{2}}{2}}$ and $\Phi(x)=1-Q(x)$
are PDF and CDF of the standard Gaussian distribution, respectively. Next,
using Theorem 4.3, we give the differential privacy guarantee provided by the
mechanism assuming that the reciprocal of $b$ is distributed according to the
truncated Gaussian distribution.

######  Theorem 4.13.

The Randomized DP Laplace mechanism $\mathcal{M}_{q}(d,\epsilon)$, and
$\epsilon\sim f_{\mathcal{N}^{T}}(\epsilon;\mu,\sigma,a,b)$, satisfies
$\epsilon_{N^{T}}$-DP, where

$\displaystyle\epsilon_{N^{T}}=\ln\left[\cfrac{\mu+\cfrac{\sigma\cdot(\phi(\alpha)-\phi(\beta))}{(\Phi(\beta)-\Phi(\alpha))}}{\diff{M_{N^{T}}(t)}{t}|_{t}=-\Delta
q}\right]$ (14)

in which $\phi(\cdot)$ is the probability density function of the standard
normal distribution, $\phi(\cdot)$ is its cumulative distribution function and
$\alpha=\frac{a-\mu}{\sigma}$ and $\beta=\frac{b-\mu}{\sigma}$.

######  Lemma 4.14.

Randomized DP using truncated Gaussian distribution can satisfy the necessary
condition in Equation 13.

Therefore, truncated Gaussian distribution may improve over the baseline, and
this can be computed by optimizing the privacy-utility trade-off using the
Lagrange multiplier function in Equation 12. In particular, our simulation
shows that, this distribution can also be effective for smaller $\epsilon$
(stronger privacy guarantees).

###  4.6 Expanding the Search Space with Combined PDFs

Theorem 4.2 can be directly applied to design a utility-maximizing Randomized
DP Laplace mechanism with a sufficiently large search space (infinite number
of different random variables). Since the Laplace mechanism has already been
studied under $\ell_{1},\ell_{2}$ and entropy, we will focus on the usefulness
metric.

######  Corollary 4.15 (Optimal Utility for Combined RVs).

If $x_{1},x_{2},\cdots,x_{n}$ are $n$ independent random variables with
respective MGFs $M_{x_{i}}(t)=\mathbb{E}(e^{tx_{i}})$ for $i=1,2,\cdots,n$,
then for the linear combination $Y=\sum\limits_{i=1}^{n}a_{i}x_{i}$, the
optimal usefulness (similar relation holds for other metrics) under
$\epsilon$-differential privacy constraint is given as

$\displaystyle U_{Y}(\epsilon,\Delta
q,\gamma)=1-\min\limits_{\mathcal{A,U}}\bigg{\\{}\prod\limits_{i=1}^{n}M_{x_{i}}(-a_{i}\gamma)\bigg{\\}}$
(15) subject to
$\displaystyle\epsilon=ln\left[\cfrac{\sum\limits_{j=1}^{n}a_{j}\cdot
E_{x_{j}}(\frac{1}{b})}{\sum\limits_{j=1}^{n}a_{j}\cdot
M^{\prime}_{x_{j}}(a_{j}\cdot-\Delta
q)\cdot\prod\limits_{\begin{subarray}{c}i=1\\\ i\neq
j\end{subarray}}^{n}M_{x_{i}}(-a_{i}\cdot\Delta q)}\right]$

where $\mathcal{A}=\\{a_{1},a_{2},\cdots,a_{n}\\}$ is the set of the
coefficients and $\mathcal{U}=\\{u_{1},u_{2},\cdots,u_{n}\\}$ is the set of
parameters of the probability distributions of RVs $x_{i},\ \forall i\leq n$.

Similar to the case of single RVs, we can compute the optimal solution for
this optimization problem using the Lagrange multiplier function in Equation
12.

We will focus on all RVs that are produced using linear combinations of the
Gamma, uniform and truncated Gaussian distributions (which include both weak
and strong privacy-preserving PDFs). Therefore, the corresponding Lagrange
multiplier function is

$\displaystyle\mathcal{L}(a_{1},a_{2},a_{3},k,\theta,a_{u},b_{u},\mu,\sigma,a_{\mathcal{N}^{T}},b_{\mathcal{N}^{T}},\Lambda)$
(16) $\displaystyle=M_{\Gamma(k,\theta)}(-a_{1}\gamma)\cdot
M_{U(a_{u},b_{u})}(-a_{2}\gamma)$ $\displaystyle\cdot
M_{\mathcal{N}^{T}(\mu,\sigma,a_{\mathcal{N}^{T}},b_{\mathcal{N}^{T}})}(-a_{3}\gamma)+\Lambda\cdot(\ln\Bigg{[}\cfrac{\mathsf{N}}{\mathsf{D}}\Bigg{]}-\epsilon)$

where the numerator and the denominator $\mathsf{N,\ D}$ are

$\displaystyle\mathsf{N}=$ $\displaystyle(a_{1}\cdot
k\cdot\theta)+(a_{2}\cdot\frac{a+b}{2})+(a_{3}\cdot(\mu+(\cfrac{\sigma\cdot\phi(\alpha)-\phi(\beta))}{(\Phi(\beta)-\Phi(\alpha))}))$
$\displaystyle\mathsf{D}=a_{1}\cdot
M^{\prime}_{\Gamma(k,\theta)}(-a_{1}\cdot\Delta q)\cdot
M_{U(a_{u},b_{u})}(-a_{2}\cdot\Delta q)$ $\displaystyle\cdot
M_{\mathcal{N}^{T}(\mu,\sigma,a_{\mathcal{N}^{T}},b_{\mathcal{N}^{T}})}(-a_{3}\cdot\Delta
q)$ $\displaystyle+a_{2}\cdot M_{\Gamma(k,\theta)}(-a_{1}\cdot\Delta q)\cdot
M^{\prime}_{U(a_{u},b_{u})}(-a_{2}\cdot\Delta q)$ $\displaystyle\cdot
M_{\mathcal{N}^{T}(\mu,\sigma,a_{\mathcal{N}^{T}},b_{\mathcal{N}^{T}})}(-a_{3}\cdot\Delta
q)$ $\displaystyle+a_{3}\cdot M_{\Gamma(k,\theta)}(-a_{1}\cdot\Delta q)\cdot
M_{U(a_{u},b_{u})}(-a_{2}\cdot\Delta q)$ $\displaystyle\cdot
M^{\prime}_{\mathcal{N}^{T}(\mu,\sigma,a_{\mathcal{N}^{T}},b_{\mathcal{N}^{T}})}(-a_{3}\cdot\Delta
q)$

Finally, Algorithm 1 details our Randomized DP Laplace mechanism using linear
combinations of these three PDFs. In Section LABEL:exp:sec, using experiments
and simulation results, we show that Algorithm 1 can indeed outputs near-
optimal results.

Input : Dataset $D$, Privacy budget $\epsilon$, Query $q(\cdot)$, Metric and
its parameters (from data recipient)

Output : Query result $q(D)+Lap(b_{r})$ which is $\epsilon$-DP and is near-
optimal w.r.t. the utility requirement

$\Delta q\leftarrow$ Sensitivity ($q(\cdot)$)

Find optimal parameters from Lagrange Multiplier $\mathcal{L}(\epsilon,\Delta
q,\text{metric})=$
$\\{a_{1}^{opt},a_{2}^{opt},a_{3}^{opt},k^{opt},\theta^{opt},a^{opt}_{u},b^{opt}_{u},\mu^{opt},\sigma^{opt},a_{\mathcal{N}^{T}}^{opt},b_{\mathcal{N}^{T}}^{opt}\\}$

$X_{1}\sim\Gamma(k^{opt},\theta^{opt})$

$X_{2}\sim U(a^{opt}_{u},b^{opt}_{u})$

$X_{3}\sim\mathcal{N}^{T}(\mu^{opt},\sigma^{opt},a_{\mathcal{N}^{T}}^{opt},b_{\mathcal{N}^{T}}^{opt})$

$\frac{1}{b_{r}}=a_{1}^{opt}\cdot X_{1}+a_{2}^{opt}\cdot
X_{2}+a_{3}^{opt}\cdot X_{3}$

return $q(D)+Lap(b_{r})$

Algorithm 1 Randomized DP with 3 PDFs

## References

  * [1] J. Le Ny and M. Mohammady, ”Differentially Private MIMO Filtering for Event Streams,” in IEEE Transactions on Automatic Control, vol. 63, no. 1, pp. 145-157, Jan. 2018, doi: 10.1109/TAC.2017.2713643.
  * [2] Meisam Mohammady, Lingyu Wang, Yuan Hong, Habib Louafi, Makan Pourzandi, and Mourad Debbabi. 2018. Preserving Both Privacy and Utility in Network Trace Anonymization. In Proceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security (CCS ’18). Association for Computing Machinery, New York, NY, USA, 459–474. https://doi.org/10.1145/3243734.3243809.
  * [3] Meisam Mohammady, Shangyu Xie, Yuan Hong, Mengyuan Zhang, Lingyu Wang, Makan Pourzandi, and Mourad Debbabi. 2020. R2DP: A Universal and Automated Approach to Optimizing the Randomization Mechanisms of Differential Privacy for Utility Metrics with No Known Optimal Distributions. In Proceedings of the 2020 ACM SIGSAC Conference on Computer and Communications Security (CCS ’20). Association for Computing Machinery, New York, NY, USA, 677–696. https://doi.org/10.1145/3372297.3417259


# Quasiparticle spectroscopy in technologically-relevant niobium using London
penetration depth measurements

Sunil Ghimire  Ames National Laboratory, Ames, IA 50011, U.S.A.  Department of
Physics & Astronomy, Iowa State University, Ames, IA 50011, U.S.A.  Kamal R.
Joshi  Ames National Laboratory, Ames, IA 50011, U.S.A.  Amlan Datta  Ames
National Laboratory, Ames, IA 50011, U.S.A.  Department of Physics &
Astronomy, Iowa State University, Ames, IA 50011, U.S.A.  Aidan Goerdt  Ames
National Laboratory, Ames, IA 50011, U.S.A.  Makariy A. Tanatar  Ames National
Laboratory, Ames, IA 50011, U.S.A.  Department of Physics & Astronomy, Iowa
State University, Ames, IA 50011, U.S.A.  Deborah Schlagel  Ames National
Laboratory, Ames, IA 50011, U.S.A.  Matthew J. Kramer  Ames National
Laboratory, Ames, IA 50011, U.S.A.  Jayss Marshall  Rigetti Computing, 775
Heinz Ave., Berkeley, CA 94710, U.S.A.  Cameron J. Kopas  Rigetti Computing,
775 Heinz Ave., Berkeley, CA 94710, U.S.A.  Joshua Y. Mutus  Rigetti
Computing, 775 Heinz Ave., Berkeley, CA 94710, U.S.A.  Alexander Romanenko
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, U.S.A.  Anna
Grassellino  Fermi National Accelerator Laboratory, Batavia, Illinois 60510,
U.S.A.  Ruslan Prozorov<EMAIL_ADDRESS>Ames National Laboratory, Ames,
IA 50011, U.S.A.  Department of Physics & Astronomy, Iowa State University,
Ames, IA 50011, U.S.A.

###### Abstract

London penetration depth was measured in niobium foils, thin films, single
crystals, and superconducting radio-frequency (SRF) cavity pieces cut out from
different places. The low-temperature ($T<T_{c}/3$) variation, sensitive to
the low-energy quasiparticles with states inside the superconducting gap,
differs dramatically between different types of samples. With the help of
phenomenological modeling, we correlate these different behaviors with known
pair-breaking mechanisms and show that such measurements may help distinguish
between different pair-breaking mechanisms, such as niobium hydrides and two-
level systems (TLS). The conclusions also apply to SRF cavities when tracking
the temperature-dependent quality factor and the resonant frequency.

##  I Introduction

Superconductors are characterized by unique properties that make them
particularly attractive for quantum computing [1, 2, 3, 4, 5, 6, 7] and
accelerator technologies [8, 9]. Niobium is often used as at least some part
of these technologies. This is due in large part to its low resistivity at low
temperatures, high thermal conductivity, and highest among elements
superconducting transition temperature, $T_{c}\approx 9.3$ K [10, 11, 12, 13,
14, 15, 16, 17]. These attributes make niobium a popular choice for
fabricating qubits based on Josephson junctions [18]. Niobium is a vital
material in accelerator technology due to its capacity to carry microwaves
without significant losses. The high quality factor (Q-factor), which
signifies the efficiency of energy storage in a resonator relative to energy
loss, along with low surface resistivity at relatively high magnetic fields,
makes niobium an excellent choice for superconducting radio frequency (SRF)
cavities used in particle accelerators [19, 8, 20, 21, 22, 23, 24] .

While a significant effort has been devoted to study properties of niobium
over years with first significant results appearing in the late 1930s [25],
there are still fundamental aspects that require further research. Measurement
capabilities as well as theoretical understanding of superconductors evolved
immensely and new studies bring novel results to this day. For example, based
on the first-principles microscopic theory of anisotropic superconducting and
normal state in Nb [26], it was recently proposed that, intrinsically, niobium
is a type-I superconductor [27]. (Perhaps, all elemental superconductors are!)
However, in real samples and devices, disorder always tips the balance over to
the type-II side, but not too far from the boundary separating these two
regimes. In this situation, the electromagnetic response is close to non-local
since the coherence length and the London penetration depth are comparable, of
the order of 30-50 nm [27].

As all refractory metals, niobium has some physical-chemical issues that
complicate and sometimes impede its use in applications. One of the most
pressing issues in Nb SRF cavities is the so-called “hydrogen Q - disease”, a
severe degradation of the quality factor, Q [28, 29]. Niobium has significant
affinity for hydrogen and can intake it even from water and ambient moisture.
At room temperature, small hydrogen moves through niobium lattice as a free
molecular gas. However, niobium hydrides form upon cooling below 150-180 K
and, depending on hydrogen concentration, steric effects (volume mismatch) may
irreversibly damage initially perfect crystalline structure [28, 29]. This
damage is practically impossible to remove. Furthermore, in the applications
relying on the superconducting properties, the most important part of any Nb
structure is the surface layer where electromagnetic field penetrates or
supercurrent flows. When the Nb part is inevitably exposed to air, a few
nanometers thick layers of different niobium oxides and sub-oxides form and,
depending on their nature, may drastically degrade device properties [30].
Moreover, oxygen diffusing deeper into the bulk may bind hydrogen forming two-
level systems (TLS) that create bound states at low energies, deep inside the
superconducting gap, which is extremely detrimental for quantum coherence. The
TLS-related losses represent a significant portion of the contemporary
research in applied superconductivity [31, 32, 33].

Therefore, continuing studies of niobium’s intrinsic properties, in particular
its electromagnetic response, are still needed to improve qubits,
accelerators, and other technologies that leverage the unique traits of this
metal. The ongoing significant effort in development these technologies
emphasizes the importance of such research, which directly influences
advancements in many sectors, including cryptography, optimization, and high-
energy physics [34].

With so many factors that may affect the properties, hence the ultimate
performance of superconducting devices, the question is how to distinguish
between different mechanisms that create problems? For example, there is no
universal suppression of properties by generic hydrides. Their influence
depends on concentration, morphology, and size distribution. Likewise, the
influence of surface layers of oxides and hydrides depend on their thickness,
conductivity, magnetism and, of course, chemical makeup [35, 31, 22, 36, 37,
38].

To (partially) address these questions, we performed precision measurements of
the London penetration depth using sensitive tunnel-diode resonator technique.
Seven representative samples were studied: different parts of niobium SRF
cavity; a thin film used in a superconducting transmon qubit; commercial foil;
and single crystals for comparison. The samples were not treated in any
special way and most contained the hydrides from previous handling, but not
all as we could observe directly in low-temperature polarized microscope.

At the low temperatures, below roughly $T_{c}/3$, the superconducting gap is
constant and the temperature - dependent superfluid density is determined by
the quasiparticles that were created by different non-thermal pair-breaking
mechanisms, such as TLS or spin-flip scattering. In order to understand the
results, we modeled penetration depth using Dynes model of superconductivity,
first by itself with only two parameters, pair-breaking and pair-conserving,
$\Gamma$ and $\Gamma_{s}$, respectively, and then extended to incorporate TLS
into the total density of states. As a result, we found a unique fingerprint
of TLS in our measurements, thus providing protocol on how to identify TLS and
distinguish them from other sources of pair-breaking scattering.

Figure 1: Polarized-light images of a [110] single crystal full of hydrogen.
(a) room-temperature image of the crystal that was never cooled down; (b) the
same crystal at 5 K showing profound hydrides outgrowths; (c) room temperature
image of the crystal warmed after cooling showing scars - damage from the
hydrides. These scars cannot be removed even by heating the sample to sub-
melting temperatures indicating significant plastic deformation induced by the
hydrides.

##  II Experimental

###  II.1 Samples

Seven niobium samples (four different types) were used in this work. Indeed,
many samples of each type were measured during this study. The sample types
are: (1,2) Two single crystals, both cut from the same large ingot and
polished using water. They absorbed large amounts of hydrogen. (3) Commercial
Nb foil from Alfa Aesar, 250 $\mu$m thick, 99.98% purity. It has been kept in
a desiccator and showed no presence of hydrides. (4) Sputtered 160 nm thick
niobium film, also kept dry and showing no obvious hydride formation.
Identical films are used to make transmon qubits [39, 38]. (5,6,7) Samples cut
out from different places of a real SRF cavity. A surface thermal map was
constructed where surface temperature variations were measured in different
spots during the resonance. In some places, called “hot spots” temperature
rose by up to 1 K, whereas in other, called “cold spots”, changed only a
little by 40 mK. Details of this mapping and measurements are found elsewhere
[40]. All samples were cut and dry-polished down to sub-mm size to fit in our
measurement setup.

###  II.2 London penetration depth

The London penetration depth was measured by using a sensitive tunnel diode
resonator (TDR) technique [41, 42, 43, 44, 45, 46, 47]. Essentially, TDR is a
tank circuit where the sample is inserted into a single-layer-of-turns
inductor that produces a small, $H_{ac}<2~{}\text{$\mu$T}$, AC magnetic field
at around $f_{0}\approx 14$ MHz. Connected in series tunnel diode, biased to
the regime of negative differential resistance, compensates for losses in the
circuit and for a certain impedance matching conditions, the circuit starts
resonating spontaneously, usually below 70 K or so. If the diode and the
circuit are well stabilized and isolated, the resolution of the device is
about 1 part per billion resolving 0.01 Hz changes on top of 10 MHz main
frequency. When a sample responds magnetically, the total inductance changes
leading to a frequency shift, $\Delta f/f_{0}$, proportional to the samples
magnetic susceptibility, $\chi\left(T\right)$, with a sample-dependent
calibration constant, $\Delta f\left(T\right)/f_{0}=G\chi\left(T\right)$. The
susceptibility may then be converted to the London penetration depth, using,
$\left(1-N\right)\chi=\lambda/R\tanh\left(R/\lambda\right)-1$, knowing
demagnetizing factor, $N$ [48], and the effective sample dimension, $R$ [47].
Since at low temperatures (practically below $0.8T_{c}$) the $\tanh$ term can
be dropped and temperature dependence of the susceptibility analyzed since
$\chi\left(T\right)\sim\lambda\left(T\right)$. Detailed description of the
measurement procedure [43], calibration [42, 48, 47] and applications [44, 45,
46] provide a complete description of this unique technique capable of
resolving sub-angstrom changes in the London penetration depth,
$\lambda\left(T\right)$, in sub-mm sized crystals. Here we use it to study
niobium samples from different sources.

Figure 2: Magneto-optical Faraday images of three different cutouts from the
SRF cavity. (a) cold spot from cavity’s inner surface showing small tubercles-
like hydride structure; (b) a cutout 1 mm deep from cold spot showing no
hydrides; (c) hot spot from the surface with characteristic boomerang-shaped
large hydrides.

###  II.3 Low-temperature optical and magneto-optical imaging

The two-dimensional distribution of the magnetic induction was mapped in real-
time employing magneto-optical imaging using Faraday effect in transparent
ferrimagnetic indicators (bismuth-doped iron garnets) placed on top of the
samples. Details of the technique can be found in our previous studies of Nb
[49, 15]. The closed-cycle flow-type optical 4He cryostat exposed the cooled
sample to an Olympus polarized-light microscope. The magnetic induction on the
sample surface polarizes in-plane magnetic moments in the indicator, and the
distribution of this polarization component along the light propagation is
visualized through double Faraday rotation. In the images, only the magnetic
field is visible due to a mirror sputtered at the bottom of the indicator.

The same microscope with the cryostat were used for direct observation in
linearly polarized light. Due to polarization, all surface features become of
higher contrast, because they often cause some rotation of the polarization
plane upon reflection and since we work in (almost) crossed polarizer/analyzer
configuration.

##  III Results and Discussion

###  III.1 Optical measurements

We start with optical characterization of the samples. Figure 1 shows
polarized light imaging one of the [110] oriented (out of the page direction)
single crystals cut from a big piece in presence of water, so it absorbed a
significant amount of hydrogen. Panel (a) shows this crystal at the room
temperature, never cooled down. No signature of surface features are seen.
Panel (b) shows the same crystal at 5 K revealing large hydride formations. In
our studies, we found that the hydrides tend to grow along principal
directions and, therefore elongated shape here is not surprising. The
direction, however, would depend on the strain, which is unknown. Figure 1(c)
again shows a room-temperature image, but now after the cool-down. Scars left
by the hydrides are clearly visible. These significant deformations and
rapture of the previously perfect crystal would remain even if the crystal is
heated up only few degrees below melting point of niobium. Therefore, the
penetration depth data shown below were collected on a crystal full of large
hydrides.

Next we examine three different cutouts from the SRF cavity. Two are from the
inner surface and one from the depth of 1 mm beneath. Figure 2 shows magneto-
optical images of each part in the remanent state of a trapped magnetic flux.
(The sample is cooled in a magnetic field to low temperature and the field is
turned off.) Importantly, sample surface is not visible, only trapped magnetic
flux distribution. Figure 2 (a) shows the cold spot (where temperature
variation did not exceed 40 mK). There are hydrides in shape of small
tubercles. Panel (b) shows cutout from 1 mm depth under the cold spot
revealing clean surface without the hydrides indicating that hydrogen
distribution is highly non-uniform depth-wise. The third panel (c), shows a
hot spot (temperature rose more than 1 K in a resonating cavity) revealing
large hydrides similar to those observed in a single crystal, Fig.1.

We will now examine temperature-dependent superfluid density in these samples.
This measurement brings information about the superconducting gap structure
and possible pair-breaking mechanisms. In addition to crystals and cavity
cutouts, we also measured a thin film used in the fabrication of transmons as
well as commercial foil - to cover all possible states of niobium samples.
Thin films from the same batches were characterized in great detail elsewhere
[50].

###  III.2 London penetration depth

Figure 3: Magnetic susceptibility of two single crystals, Alfa Aesar foil and
160 nm Rigetti thin film. (a) main panel - full range, inset - zooming into
the superconducting transition. (b) Low-temperature variation of London
penetration depth fitted to the power-law,
$\Delta\lambda\sim\left(T/T_{c}\right)^{n}$, from the base temperature,
$0.044T_{c}$ to $0.3T_{c}$ yielding the values of the exponent $n$ indicated.
Magenta bar show the corresponding scale of $\lambda\left(T\right)$ variation
in nm.

The first set of measurements on two single crystals, Alfa Aesar foil and
Rigetti thin film is shown in Fig.3. For comparison between different samples
the curves were normalized to represent ideal magnetic susceptibility that
starts at $\chi=0$ above $T_{c}$ and reaches $\chi=-1$ at the low temperature.
The detailed shape of $\chi\left(T\right)$ is unaffected by this scaling. The
inset in panel (a) of Fig.3 zooms at the superconducting transition showing
the sharpest transition in Nb film and very similar $T_{c}$ in two crystals.
All these values are somewhat lower than often quoted 9.3 K, likely due to
disorder-induced pair-breaking in this quite anisotropic material [26, 51].
Figure 3(b) shows the low-temperature variation revealing quite different
behaviors of $\lambda\left(T\right)$. The power-law fitting,
$\Delta\lambda\sim\left(T/T_{c}\right)^{n}$ from the base temperature,
$0.044T_{c}$ to $0.3T_{c}$ produced the indicated values of the exponent, $n$,
which range from exponential attenuation, $n>4$, to a convex downturn with
$n=0.7$, which is extremely unusual for any superconductor. In both single
crystals we observe a downturn below roughly 1.3 K, whereas this behavior is
absent in a thin film and a foil.

Figure 4: The information similar to Fig.3, but for the SRF cavity cutouts.
Two from the surface, cold and hot spot, and the third from the depth of 1 mm
under cold spot. While the transitions are higher, there is significant spread
with widest transition width of about 0.4 K in the hot spot sample and the
narrowest in the inner cold spot. (b) Low-temperature variation shows a
distinct convex downturn, which is very unusual for $\lambda\left(T\right)$ in
any superconductor. The inner cold spot shows saturation behavior consistent
with a relatively clean superconducting gap.

The second set of data, shown in Fig.4, presents the results obtained in the
SRF cavity cutouts in a way formally similar to the first set, Fig.3. Full
temperature variation is presented in the main panel of Fig.4(a) and the inset
zooms into the transition region. While the transitions are higher than in the
first set, there is a more significant spread with the widest transition width
of about 0.4 K in the hot spot sample and the narrowest in the inner cold
spot. Panel (b) of Fig.4 presents low-temperature variation, showing even more
pronounced downward curvature, but only for one sample that came from the cold
spot. This fact will be the key for the interpretation of these results. Such
distinct convex downturn is very unusual for $\lambda\left(T\right)$ in any
(good and uniform) superconductor. The inner cold spot shows saturation
behavior consistent with a relatively clean superconducting gap and the
surface cold spot shows an intermediate behavior.

In principle, such behavior of the single crystals in Fig.3 and the hot spot
sample in Fig.4 could be due to the proximity effect when superconductivity is
induced in the surface metallic layer and the overall diamagnetic screening
increases thus leading to the downturn in susceptibility. This was shown
directly using TDR technique in MgB2 wires that has excess magnesium on the
surface [52]. Pambianchi _et al._ measured effective penetration depth in
proximity-coupled Nb/Al bilayer films where they observed power law with
exponent n$\leq$ 1 distinctly different from the exponential behavior of Nb
[53]. However, in both works, the effect required a quite thick layer of
another normal metal on niobium surface. It was suggested that niobium hydride
phase, Nb4H3, which is also a candidate own superconductivity with a critical
temperature of 1.2 K [54]. However, our niobium samples are exposed to air and
poorly or non-conducting niobium oxides are formed on the surface thus
eliminating the possibility of the formation of a good metallic layer. The
oxides are very robust and withstand heating almost to melting temperature of
Nb [35, 55]. Careful atom probe and transmission electron microscopy (TEM)
studies showed no evidence of the metallic hydride phase on sample surfaces.
In case of cavity cutouts, why would these layers only form over hot spots?
Finally, the most important point is that proximity effect leads to the
enhanced diamagnetism and the downturn is the departure from the original
$\lambda(T)$ curve downward upon cooling. What we observe is opposite - the
whole curve is shifted up indicating significant contamination of the
superconducting gap with quasiparticle states rather than additional
screening. Still, keeping in mind this possibility, we now turn to a more
general analysis of the obtained results.

##  IV Theoretical analysis

###  IV.1 General remarks

Here we present a simple phenomenological analysis where we use a
semiclassical connection between London penetration depth and the density of
states. The results were verified by Matsubara summation formalism. Being
phenomenological does not mean being qualitative. We use self-consistency
equation to calculate the order parameter and then the superfluid density, and
obtain quantitative prediction for temperature-dependent penetration depth. We
follow the semiclassical approach introduced by Chandrasekhar et al. [56].
Some examples of using this theory for different superconductors can be found
in Ref.[44]. The model gives direct connection between the superfluid density,
$n_{s}\left(T\right)$, and the density of states, $N\left(E\right)/N_{n}$
normalized by its value at the Fermi level in the normal state, $N_{n}$. Here
the energy of Bogolubov quasiparticles is, $E=\sqrt{\epsilon^{2}+\Delta^{2}}$
and the normal metal band energy, $\epsilon$, is measured from the Fermi
level. The response of supercurrent to a vector potential,
$\mathbf{J=\mathbb{-R}A}$, is determined by the so-called response tensor,
$\mathbb{R}$, which consists of two parts, diamagnetic and paramagnetic. The
full expression involves the average over the Fermi surface of a possibly
anisotropic gap function. In case of niobium we can safely use spherical Fermi
surface and constant gap (although the gap is somewhat anisotropic [26]. Then
the superfluid density, normalized on its value in the clean case (which is
the total electron density of normal metal), reads:

$n_{s}=1+2\intop_{0}^{\infty}\frac{\partial f\left(E\right)}{\partial
E}\frac{N\left(E\right)}{N_{n}}dE$ (1)

where the order parameter enters the density of states as,

$\frac{N\left(E\right)}{N_{n}}=\frac{E}{\sqrt{E^{2}-\Delta^{2}}}$ (2)

The derivative of the Fermi function is,

$\frac{\partial f\left(E\right)}{\partial
E}=-\frac{1}{4t}\text{sech}^{2}\left(\frac{E}{2t}\right)$ (3)

where $t=T/T_{c}$ (note that this is actual superconducting transition
temperature that may be lower than the “clean” value that we denote $T_{c0}$)
and quasiparticle energy is dimensionless in units of $k_{B}T_{c}$ and we set
$k_{B}=1$ everywhere. The quantity of interest, the London penetration depth
is then obtained from Eq.1 as,

$\frac{\lambda\left(T\right)}{\lambda_{0}}=\frac{1}{\sqrt{n_{s}}}$ (4)

where $\lambda_{0}$ is a clean-limit London penetration depth. Therefore, with
scattering, this ratio will be greater than 1 even at $T=0$.

Importantly, this approach does not specify how the order parameter is
obtained. In the original papers of the semiclassical approach where very
clean superconductors were considered [57], it was sufficient to use the so-
called Einzel ansatz to analytically represent the gap with high accuracy
[58]. This is insufficient for our purposes as we would like to explore
“contaminated” gaps, thus we will use the self-consistent solutions for
$\Delta\left(T,\Gamma\right)$, where $\Gamma$ is pair-breaking scattering rate
[21].

###  IV.2 Dynes superconductors

The simplest way to introduce pair-breaking into the density of states is to
consider Dynes superconductors. There are several comprehensive works with a
detailed analysis of the Dynes model and its applicability to realistic and
quite complicated cases. A full set of all thermodynamic and transport
parameters was calculated and is available for reference. We follow to
extensive coverage of the Dynes model by Gurevich and Kubo [59, 60, 61, 62]
and Herman and Hlubina [63, 64, 65]. In this model, instead of Eq.2, the
density of states is given by,

$\frac{N\left(E\right)}{N_{n}}=\Re\left[\frac{E+i\Gamma}{\sqrt{\left(E+i\Gamma\right)^{2}-\Delta^{2}}}\right]$
(5)

Dynes superconductors are gapless. At $E=0$, Eq.5 gives
$N\left(E\right)=N_{n}\Gamma/\sqrt{\Gamma^{2}+\Delta^{2}}$. However, for small
rates (usually found in tunneling experiments [66, 55]), $\Gamma\ll\Delta$,
one obtains quite useful estimates. In general the meaning of this
phenomenological parameter is that $\hbar/\Gamma$ is the quasiparticle
lifetime. It also has meaning of pair-breaking scattering time, because Dynes
$\Gamma$ is analogous to Abrikosov-Gor’kov (AG) scattering parameter [67],

$\rho=\frac{\hbar}{2\pi T_{c}\tau}=\frac{\Gamma}{2\pi}$ (6)

(actually, in the original paper, AG used $\rho$ without “2” in the
denominator), and the $T_{c}$ for Dynes superconductor is obtained from a
classical AG formula [60],

$\psi\left(\frac{\Gamma}{2\pi
t_{c}}+\frac{1}{2}\right)-\psi\left(\frac{1}{2}\right)+\ln\left(t_{c}\right)=0$
(7)

where $t_{c}=T_{c}/T_{c0}$ and $\psi\left(x\right)$ is digamma function. The
transition temperature becomes zero when $\Gamma_{crit}=0.8819$.

Figure 5: The change of the London penetration depth, calculated using Eqs.
8,5,1 and 4. (Here we put $\lambda_{0}=1$). At larger scattering rates it
approaches quadratic temperature dependence as predicted by Gurevich and Kubo
[68].

To calculate the superfluid density, we need to solve the self-consistency
equation for the order parameter written in terms of Matsubara sum [60],

$\log(t)-\sum_{n=0}^{n_{max}}\left(\frac{2\pi
t}{\sqrt{\Delta^{2}+\left(\Gamma+2\pi
t\left(n+\frac{1}{2}\right)\right)^{2}}}-\frac{2}{2n+1}\right)=0$ (8)

For each value of $\Gamma$ and for each temperature, $t$, Eq.8 yields
$\Delta\left(t,\Gamma\right)$. The number of terms, $n_{max}$ depends on the
required accuracy and the lower the temperature, the more terms are needed.
Here we used $n_{max}=10000$.

Now we are set to calculate the London penetration depth as function of
temperature for different values of $\Gamma$ and compare with the experiment.
One difficulty is that the formulation in form of Eq.1 does not include the
effect of pair-conserving scattering, which nevertheless affects the
superfluid density. For Dynes superconductor this is solved by calculating
superfluid density using Matsubara sum as introduced by Herman and Hlubina
[64],

$n_{s}=2\pi
t\sum_{n=0}^{n_{max}}\frac{\Delta^{2}}{\Omega_{n}^{2}(\Omega_{n}+\Gamma_{s})}$
(9)

where $\Omega_{n}=\sqrt{\Delta^{2}+(\Gamma+\pi t(2n+1))^{2}}$. Here two
scattering parameters are present, the pair-breaking $\Gamma$ that also enters
the original Dynes equation, Eq.5 and pair-conserving scattering rate,
$\Gamma_{s}$.

Figure 5 shows the variation of the London penetration depth, calculated using
Eqs. 8,5,1 and 4. (Here $\lambda_{0}=1$). Identical results are obtained from
Eq.9 with $\Gamma_{s}=0$. At fairly small rates, the behavior is sub-
quadratic, but clearly non-exponential indicating significant deviation from
the clean (exponential) case. At larger scattering rates penetration depth
approaches quadratic temperature dependence as predicted by Gurevich and Kubo
[68]. Compared with our experimental results, such behavior would be
consistent with the Alfa Aesar foil, Fig.3 and, perhaps to some extend with
the cold spot shown in Fig.4.

Figure 6: Temperature dependent London penetration depth, $\Delta\lambda(T)$,
calculated for a fixed Dyson pair-breaking $\Gamma=0.5$ and
$\Gamma_{s}=0,5,50$. While, expectedly, the slope becomes steeper, the
behavior remains close to quadratic (note that, unlike Fig.5, this is linear
scale.)

Still, we are not even close to reproducing the downturn. Perhaps, pair-
conserving $\Gamma_{s}$ will help? Figure 6 shows penetration depth for a
fixed Dyson pair-breaking, $\Gamma=0.5$, and pair-conserving,
$\Gamma_{s}=0,5,50$ (unlike $\Gamma$, the pair-conserving $\Gamma_{s}$ can
assume any values, but dirty behavior starts with the values above 1, so 50 is
an exaggeration to see the difference. While the temperature variation became
much steeper, the functional form remains sub-quadratic. To explore other
possibilities we turn to the two-level systems.

###  IV.3 Two-level systems and penetration depth

Figure 7: Temperature dependent London penetration depth, calculated using
Eq.1 with (red line) and without (black line) TLS contribution to the density
of states. The total $N\left(E\right)/N_{n}$ is shown in the inset. A
relatively small peak, with the parameters indicated, causes such a dramatic
change in the penetration depth. Importantly, it naturally produces the
downturn in the penetration depth.

There is a significant interest in TLS, which are intensely studied
theoretically and experimentally. Here we are only interested in their
possible effect on the London penetration depth. For that, it is sufficient to
model TLS as small peaks deep inside the superconducting energy gap in the
density of states. Since the superfluid density is the integral over all
energies, we do not think the precise details of the TLS peak may drastically
change the outcome. Here we model the TLS density of states as a Lorentzian,

$TLS(E)=\frac{h}{\left(\frac{E-E_{p}}{w}\right)^{2}+1}$ (10)

so that this peak of height $h$ and width $w$ is located at energy $E_{p}$
inside the Dynes gap. An example is shown in the inset in Fig.7. The total DOS
is obtained by adding Eq.5 and Eq.10. Then Eq.1 is used to calculate the
superfluid density shown in Fig.8. The black curves on both figures are the
results for Dynes superconductor with small $\Gamma=0.01$ showing close to
isotropic s-wave BCS classical exponential attenuation at low temperatures.
The introduction of a small TLS peak changes situation drastically. This is
because its location deep inside the gap. According to Eq.1, the reduction of
the superfluid density is proportional to the product of DOS and derivative of
the Fermi function. The latter is steep at low temperatures and when it
reaches even a small peak, the result is significant.

Clearly, this mechanism naturally explains our observations of the downturn in
the penetration depth. While the hydrides are clearly not good for
applications because they increase overall surface impedance and lead to
significant dissipation, the downturn is not their signature. Further research
is needed to clarify the nature of TLS which is much more complicated than
used here and involved dynamic processes, we believe our measurements and
analysis presented a complimentary way to identify TLS in niobium based
applications.

Figure 8: Temperature dependent superfluid density with and without TLS. The
Dynes curve was computed for small $\Gamma=0.01$ and the TLS parameters are
shown in the inset in Fig.7. A pronounced depression at low temperatures may
explain various issues appearing upon deep cooling. Inset shows self-
consistent order parameter as function of temperature.

##  V Conclusions

Dyson pair-breaking $\Gamma$ changes temperature variation from exponential to
the power law, but not faster than quadratic behavior. The pair-conserving
$\Gamma_{s}$ only increases the rate of change, but does not significantly
alter the functional form of $\lambda(T)$. The two-level systems residing
inside the gap have profound effect changing exponential attenuation to sub-
linear downward concave curvature of $\lambda(T)$. We emphasize that this was
a limited study designed to probe a variety of different types of samples.
Systematic measurements of each type are needed to identify the actual cause
behind the TLS-like behavior.

###### Acknowledgements.

We thank James Sauls, John Zasadzinski, Alex Gurevich and Maria Iavarone for
useful discussions. This work was supported by the U.S. Department of Energy,
Office of Science, National Quantum Information Science Research Centers,
Superconducting Quantum Materials and Systems Center (SQMS) under contract
number DE-AC02-07CH11359. The research was performed at the Ames National
Laboratory, operated for the U.S. DOE by Iowa State University under contract
# DE-AC02-07CH11358.

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