Multiblock ADMM for nonsmooth nonconvex optimization with nonlinear coupling
constraints

Le Thi Khanh Hien Department of Mathematics and Operational Research,
University of Mons, Belgium<EMAIL_ADDRESS>Dimitri Papadimitriou
3NLab, Huawei Belgium Research Center (BeRC)
<EMAIL_ADDRESS>
This paper proposes a multiblock alternating direction method of multipliers
for solving a class of multiblock nonsmooth nonconvex optimization problem
with nonlinear coupling constraints. We employ a majorization minimization
procedure in the update of each block of the primal variables. Subsequential
and global convergence of the generated sequence to a critical point of the
augmented Lagrangian are proved. We also establish iteration complexity and
provide preliminary numerical results for the proposed algorithm.

ADMM, nonlinear coupling constraints, majorization minimization, composite
optimization

##  1 Introduction

We consider the following multiblock optimization problem with nonlinear
coupling constraints

$\operatorname*{minimize}_{x,y}\;F(x_{1},\ldots,x_{m})+h(y)\quad\text{
s.t}\quad\phi(x)+\mathcal{B}y=0,$ (1)

where $x$ can be decomposed into $m$ blocks $x=(x_{1},\ldots,x_{m})$ with
$x_{i}\in\mathbb{R}^{n_{i}}$, $n=\sum_{i=1}^{m}n_{i}$, $y\in\mathbb{R}^{q}$,
$\phi$ is a nonlinear mapping from $\mathbb{R}^{n}$ to $\mathbb{R}^{s}$
defined by $\phi(x)=(\phi_{1}(x),\ldots,\phi_{s}(x))$, $\mathcal{B}$ is a
linear map, $F(x)=f(x)+g(x)$, $g(x)=\sum_{i=1}^{m}g_{i}(x_{i})$, where
$f:\mathbb{R}^{n}\to\mathbb{R}$ is a continuous (but possibly nonsmooth)
function, $g_{i}:\mathbb{R}^{n_{i}}\to\mathbb{R}\cup\\{+\infty\\}$ are proper
lower semi-continuous (lsc) functions for $i=1,2,\dots,m$, and
$h:\mathbb{R}^{q}\to\mathbb{R}$ is a differentiable function. Note that $F$
and $h$ can be nonconvex. Throughout the paper, we assume the following
assumptions. (A1) $F$ satisfies $\partial
F(x)=\partial_{x_{1}}F(x)\times\ldots\times\partial_{x_{m}}F(x)$, where
$\partial F$ denote the limiting subdifferential of $F$ (see [RockWets98,
Definition 8.3] for its definition).

(A2) $\nabla h$ is $L_{h}$-Lipschitz continuous.

(A3) $\sigma_{\mathcal{B}}:=\lambda_{\min}(\mathcal{B}\mathcal{B}^{*})>0$
($\sigma_{\mathcal{B}}$ is the smallest eigenvalue of
$\mathcal{B}\mathcal{B}^{*}$).

(A4) $F(x)+h(y)$ is lower bounded. (Note that Assumption (A1) is satisfied
when $f$ is a sum of a continuously differentiable function and a block
separable function; see [Attouch2010, Proposition 2.1].) While the Alternating
Direction Method of Multipliers (ADMM) has gained significant attention and
proven highly effective in addressing multiblock composite optimization
problems with linear constraints, see e.g., [Hien2021_iADMM, Wang2019], the
exploration of ADMM’s applicability to nonconvex nonsmooth problems with
_nonlinear coupling constraints_ remains relatively limited. The authors in
[Bolte2018_ADMM] considers Problem (1) with $m=1$ and a general nonlinear
coupling constraint. They propose a universal framework to study global
convergence analysis of Lagrangian sequences, introduce the notion of
information zone, and propose an adaptive regime to detect this zone in
finitely many steps and force the iterates to remain in the zone. Note that
the essential assumption for ensuring the information zone’s role and the
global convergence of the Lagrangian sequence in the framework of
[Bolte2018_ADMM] is the boundedness of the multiplier sequence, as depicted in
[Bolte2018_ADMM, Lemma 1]. Moreover, the choice of parameters within the
context of [Bolte2018_ADMM] is closely tied to the upper bound constant of the
multiplier sequence, as discussed in [Bolte2018_ADMM, Remark 7]. In practice,
determining this upper bound for the multiplier can be an exceedingly
challenging, if not impossible, task. The authors in [Cohen2021] consider
Problem (1) with $m=1$. They design a proximal linearized ADMM in which the
proximal parameter is generated dynamically by a backtracking procedure.
Although the boundedness assumption of the multiplier sequence is not required
in [Cohen2021], the backtracking procedure used in [Cohen2021], on one hand,
requires to repeatedly evaluate the values of $f$ and $\phi$ to guarantee a
descent property in updating $x$, and on the other hand, relies on the
boundedness assumption of the generated sequence to guarantee the boundedness
of the proximal parameters produced by the backtracking procedure. However,
[Cohen2021] does not provide a sufficient condition to guarantee the important
boundedness assumption.

In this paper, we propose a multiblock alternating direction method of
multipliers (mADMM) for solving Problem (1). Different from the general
framework of [Bolte2018_ADMM], we specify the update of the primal _block_
variables by embedding majorization minimization (MM) procedure. MM step can
recover proximal point, proximal gradient and mirror descent step by using
suitable surrogate functions depending on the structure of the objective
function, see [Titan2020, Razaviyayn2013]. Hence, employing MM and conducting
the convergence analysis based on the surrogate functions allow us to not only
better explore the structure of the problem but also unify the convergence
analysis of special cases. We prove the subsequential convergence standard
assumptions, and prove the global convergence with some additional
assumptions. We also establish the iteration complexity and provide numerical
results for mADMM (we note that iteration complexity was not established and
numerical results were not presented in [Bolte2018_ADMM] and [Cohen2021]).

The paper is organized as follows. In the next section, we provide some
preliminary knowledge to support the forthcoming analysis. In Section 3, we
describe mADMM and conduct the convergence analysis for it. We also provide a
sufficient condition such that the generated sequence by mADMM is bounded. We
report numerical results in Section LABEL:sec:numerical and conclude the paper
in Section LABEL:sec:conclusion.

Notation. We denote $[m]=\\{1,\ldots,m\\}$. For a mapping
$\phi:\mathbb{R}^{n}\to\mathbb{R}^{s}$, we denote
$\nabla\phi(x)=[\nabla\phi_{1}(x)\ldots\nabla\phi_{s}(x)]$
($\nabla\phi(x)^{\top}$ would be the Jacobian), and
$\nabla_{x_{i}}\phi(x)=[\nabla_{x_{i}}\phi_{1}(x)\ldots\nabla_{x_{i}}\phi_{s}(x)]$.

##  2 Preliminaries

###  2.1 Augmented Lagrangian and $\varepsilon$-stationary point.

The augmented Lagrangian function of Problem (1) is

$\begin{array}[]{ll}&\mathcal{L}_{\beta}(x,y,\omega)=\varphi_{\beta}(x,y,\omega)+\sum_{i=1}^{m}g_{i}(x_{i}),\\\
\text{where}&\varphi_{\beta}(x,y,\omega)=f(x)+h(y)+\langle\omega,\phi(x)+\mathcal{B}y\rangle+\frac{\beta}{2}\|\phi(x)+\mathcal{B}y\|^{2}.\end{array}$
(2)

and $\beta>0$ is a penalty parameter. Let $(x,y,\omega)$ be a critical point
of $\mathcal{L}_{\beta}$, that is

$\begin{array}[]{ll}&0\in\partial_{x_{i}}F+\nabla_{x_{i}}\phi(x)\big{(}\omega+\beta(\phi(x)+\mathcal{B}y)\big{)},\\\
&\nabla
h(y)+\nabla\mathcal{B}y\big{(}\omega+\beta(\phi(x)+\mathcal{B}y)\big{)}=0,\quad\phi(x)+\mathcal{B}y=0.\end{array}$
(3)

The conditions in (3) imply that
$0\in\partial_{x_{i}}F+\nabla_{x_{i}}\phi(x)\omega$, $\nabla
h(y)+\mathcal{B}^{*}\omega=0$, and $\phi(x)+\mathcal{B}y=0$. Hence $(x,y)$ is
a stationary point of Problem (1). Conversely, if $(x,y)$ is a stationary
point of Problem (1) then there exists $\omega$ such that $(x,y,\omega)$
satisfies (3), which means that $(x,y,\omega)$ is a critical point of
$\mathcal{L}_{\beta}$. Therefore, finding a stationary point of (1) is
equivalent to finding a critical point of the augmented Lagrangian
$\mathcal{L}_{\beta}$.

###### Definition 2.1

We call $(x,y)$ an $\varepsilon$-stationary point of (1) if there exists
$\omega$ and $\chi_{i}\in\partial_{x_{i}}F(x)$ such that
$R_{i}=\|\chi_{i}+\nabla_{x_{i}}\phi(x)\omega\|\leq\varepsilon$,
$R_{y}=\|\nabla h(y)+\mathcal{B}^{*}\omega\|\leq\varepsilon$, and
$R_{c}=\|\phi(x)+\mathcal{B}y\|\leq\varepsilon.$

###  2.2 Block surrogate function.

The notations used in this subsection are independent of the notations of
other sections. Let us first remind the proximal gradient method (PG), see
e.g., [Boyd2014], for solving the following composite optimization problem

$\min_{x\in\mathbb{R}^{n}}f(x)+g(x),$ (4)

where $f$ is a continuous differentiable function, $\nabla f(x)$ is assumed to
be $L$-Lipschitz continuous and $g$ is a lower semicontinuous function. The PG
step is

$x^{k+1}\in\operatorname{Prox}_{\frac{1}{L}g}(x^{k}-\frac{1}{L}\nabla
f(x^{k})),$ (5)

where the proximal mapping is defined as

$\operatorname{Prox}_{g}(x):=\arg\min_{y}g(y)+\frac{1}{2}\|y-x\|^{2}.$

As $\nabla f$ is $L$-Lipschitz continuous, the descent lemma (see
[Nesterov2004]) gives us

$f(x)\leq f(x^{k})+\langle\nabla
f(x^{k}),x-x^{k})+\frac{L}{2}\|x-x^{k}\|^{2}.$ (6)

Denote the right hand side of (6) by $u(x,x^{k})$. We notice that

$u(x,x)=f(x)\quad\mbox{and}\quad u(x,y)\geq f(x),\,\forall\,x,y.$ (7)

Each function $u(\cdot,\cdot)$ that satisfies (7) is called a surrogate
function of $f$. The PG step in (5) can be rewritten in an equivalent form

$x^{k+1}=\arg\min_{x}u(x,x^{k})+g(x),$ (8)

where $u$ is the Lipschitz gradient surrogate defined by the right hand side
of (6). By choosing suitable surrogate functions, many first order methods
such as proximal methods, mirror descent methods, etc., can also be rewritten
in the majorization minimization (MM) form. Convergence analysis of MM
algorithm (which can be roughly described in (8)) would unify the convergence
analysis of the algorithms that correspond to different choices of the
surrogates.

When $x$ has multiple blocks $x=(x_{1},\ldots,x_{m})$,
$x_{i}\in\mathbb{R}^{n_{i}}$, and we assume that $g(x)=\sum g_{i}(x_{i})$,
block coordinate descent (BCD) method is a well-known approach to solve (4).
BCD updates one block at a time while fixing the values of the other blocks.
The condition in (7) is then extended to deal with functions that have
multiblock variables as follows. We adopt the definition of block surrogate
functions from [Razaviyayn2013].

######  Definition 2.2 (Block surrogate function)

Let $\mathcal{X}_{i}\subseteq\mathbb{R}^{n_{i}}$,
$\mathcal{X}=\mathcal{X}_{1}\times\ldots\times\mathcal{X}_{m}\subseteq\mathbb{R}^{n}$.
A continuous function $u_{i}:\mathcal{X}_{i}\times\mathcal{X}\to\mathbb{R}$ is
called a block surrogate function of $f$ on $\mathcal{X}$ with respect to
block $x_{i}$ if

$u_{i}(z_{i},z)=f(z)\quad\mbox{and}\quad u_{i}(x_{i},z)\geq f(x_{i},z_{\neq
i})\,\,\mbox{for all}\,\,x_{i}\in\mathcal{X}_{i},z\in\mathcal{X},$

where $(x_{i},z_{\neq i})$ denotes
$(z_{1},\ldots,z_{i-1},x_{i},z_{i+1},\ldots,z_{m}).$ The block approximation
error is defined as $e_{i}(x_{i},z):=u_{i}(x_{i},z)-f(x_{i},z_{\neq i}).$

When writing the MM step for the update of block $i$, cf. (10), we can
perceive that the current point is $z$ and the value $z_{i}$ of block $i$ will
be updated by $\arg\min_{x_{i}}u_{i}(x_{i},z_{\neq i})+g_{i}(x_{i})$.

###### Example 2.3

  1. (i)

A proximal surrogate (see e.g., [Attouch2009, Attouch2013]) is

$u_{i}(x_{i},z)=f(x_{i},z_{\neq
i})+\kappa_{i}D_{\mathfrak{h}_{i}}(x_{i},z_{i}),$

where $\kappa_{i}$ is a scalar that can depend on $z$, and
$D_{\mathfrak{h}_{i}}(x_{i},z_{i})$ is a Bregman divergence defined by

$D_{\mathfrak{h}_{i}}(x_{i},z_{i})=\mathfrak{h}_{i}(x_{i})-\mathfrak{h}_{i}(z_{i})-\langle\nabla\mathfrak{h}_{i}(z_{i}),x_{i}-z_{i}\rangle,$
(9)

where $\mathfrak{h}_{i}:\mathbb{R}^{n_{i}}\to\mathbb{R}$ is a strongly convex
function and can be adaptively chosen in the course of the update of $x_{i}$.
If $\mathfrak{h}_{i}(x_{i})=\frac{1}{2}\langle x_{i},Q_{i}x_{i}\rangle$, where
$Q_{i}$ is a positive definite matrix that could also depend on $z$, then the
proximal surrogate becomes the typical extended proximal surrogate

$u_{i}(x_{i},z)=f(x_{i},z_{\neq
i})+\frac{1}{2}\kappa_{i}\|x_{i}-z_{i}\|_{Q_{i}}^{2}.$

The approximation error is
$e_{i}(x_{i},z)=\kappa_{i}D_{\mathfrak{h}_{i}}(x_{i},z_{i}).$

  2. (ii)

A quadratic surrogate (see e.g., [Emilie2016, Ochs2019]) is defined as

$u_{i}(x_{i},z)=f(z)+\langle\nabla_{x_{i}}f(z),x_{i}-z_{i}\rangle+\frac{\kappa_{i}}{2}\|x_{i}-z_{i}\|_{Q_{i}}^{2},$

where $\kappa_{i}\geq 1$, $f$ is assumed to be twice differentiable, and
$Q_{i}$ (which can depend on $z$) is a positive definite matrix such that
$Q_{i}-\nabla_{x_{i}}^{2}f(x_{i},z_{\neq i})$ is also a positive definite
matrix. The approximation error function is

$e_{i}(x_{i},z)=f(z)-f(x_{i},z_{\neq
i})+\langle\nabla_{x_{i}}f(z),x_{i}-z_{i}\rangle+\frac{\kappa_{i}}{2}\|x_{i}-z_{i}\|_{Q_{i}}^{2}.$

If $Q_{i}=L_{i}I$, where $I$ is a identity matrix and $L_{i}$ is a positive
number, then the quadratic surrogate reduces to a Lipschitz gradient surrogate
in the following.

  3. (iii)

A Lipschitz gradient surrogate (see e.g., [Xu2013]) is defined as

$u_{i}(x_{i},z)=f(z)+\langle\nabla_{x_{i}}f(z),x_{i}-z_{i}\rangle+\frac{\kappa_{i}L_{i}}{2}\|x_{i}-z_{i}\|^{2},$

where $\kappa_{i}\geq 1$ and we assume
$x_{i}\mapsto\nabla_{x_{i}}f(x_{i},z_{\neq i})$ is $L_{i}$-Lipschitz
continuous. Note that $L_{i}$ can depend on $z$.

  4. (iv)

A Bregman surrogate (see e.g., [HienNicolas_KLNMF, Hien2022, melo2017]) is
defined as

$u_{i}(x_{i},z)=f(z)+\langle\nabla_{x_{i}}f(z),x_{i}-z_{i}\rangle+\kappa_{i}L_{i}D_{\mathfrak{h}_{i}}(x_{i},z_{i}),$

where $\kappa_{i}\geq 1$, $\mathfrak{h}_{i}:\mathbb{R}^{n_{i}}\to\mathbb{R}$
is a strongly convex function, $D_{\mathfrak{h}_{i}}(x_{i},z_{i})$ is a
Bregman divergence defined in (9), the function $x_{i}\mapsto f(x_{i},z_{\neq
i})$ is assumed to be $L_{i}$-relative smooth ($L_{i}$ can depend on $z$) to
$\mathfrak{h}_{i}$, that is, the function $x_{i}\mapsto
L_{i}\mathfrak{h}_{i}(x_{i})-f(x_{i},z_{\neq i})$ is convex (see
[lu2018relatively]). If $\mathfrak{h}_{i}(x_{i})=\frac{1}{2}\|x_{i}\|^{2}$
then the block Bregman surrogate reduces to the Lipschitz gradient surrogate.

##  3 Multiblock ADMM for solving Problem (1)

We describe mADMMfor solving Problem (1) in Algorithm 1.

Algorithm 1 mADMM for solving Problem (1)

Notations. For a given $k$ (the outer iteration index) and $i$ (the cyclic
inner iteration index) we denote
$x^{k,i}=(x^{k+1}_{1},\ldots,x^{k+1}_{i},x^{k}_{i+1},\ldots,x^{k}_{s})$ and
let $x^{k+1}=x^{k,m}$.

Choose $x^{0}$, $y^{0}$, $\omega^{0}$ and $\beta$ satisfies
(LABEL:para_condition_2). Set $k=0$.

repeat

for $i=1,...,m$ do

Update
$x^{k+1}_{i}\in\arg\min_{x_{i}}\\{u_{i}(x_{i},x^{k,i-1},y^{k},\omega^{k})+g_{i}(x_{i})\\}.$
(10)

end for

Update $y$ as in (14).

Update $\omega$ as follows
$\omega^{k+1}=\omega^{k}+\beta\big{(}\mathcal{\phi}(x^{k+1})+\mathcal{B}y^{k+1}\big{)}.$
(11)

Set $k\leftarrow k+1$.

until some stopping criteria is satisfied.

We note that $x_{i}$, $i\in[m]$, $y$ and $\omega$ are the blocks of variables
of $\varphi_{\beta}$ defined in (2).

Update of block $x_{i}$. We choose block surrogate functions of
$\varphi_{\beta}$ with respect to $x_{i}$, $i\in[m]$, such that they satisfy
Condition 1 and one of the two conditions - Condition 2 or Condition 3.

###### Condition 1

For $i\in[m]$, there exists an “upper bound” error $\bar{e}_{i}(x_{i},z)$ such
that the approximation error
$e_{i}(x_{i},z)=u_{i}(x_{i},z)-\varphi_{\beta}(x_{i},z_{\neq i})$ satisfies

  * $\bullet$

$e_{i}(x_{i},z)\leq\bar{e}_{i}(x_{i},z)$ for all $(x_{i},z)$, i.e., $e_{i}$ is
upper bounded by $\bar{e}_{i}$, and

  * $\bullet$

we have $\bar{e}_{i}(z_{i},z)=0$ and $\nabla_{x_{i}}\bar{e}_{i}(z_{i},z)=0$
for all $z$.

###### Condition 2

The error $e_{i}(x_{i},z)$ satisfies
$e_{i}(x_{i},z)\geq\eta_{i}(z,\beta)D_{i}(x_{i},z_{i})$ for all $x_{i}$, $z$,
where $\eta_{i}(z,\beta)$ is a scalar that can depend on the values of $z$ and
$\beta$. Here $D_{i}(\cdot,\cdot)$ satisfies

$D_{i}(x_{i}^{k+1},x_{i}^{k})\geq 0,\,\mbox{and
if}\,D_{i}(x_{i}^{k+1},x_{i}^{k})\to
0\,\mbox{when}\,k\to\infty\,\mbox{then}\,\|x_{i}^{k+1}-x_{i}^{k}\|\to 0.$ (12)

A simple example of $D_{i}(\cdot,\cdot)$ is
$D_{i}(x_{i}^{k+1},x_{i}^{k})=\frac{1}{2}\|x_{i}^{k+1}-x_{i}^{k}\|^{2}$.

###### Condition 3

The function $t(x_{i},z)=u_{i}(x_{i},z)+g_{i}(x_{i})$ satisfies

$t(x_{i},z)\geq
t(x_{i}^{\prime},z)+\langle\xi,x_{i}-x_{i}^{\prime}\rangle+\eta_{i}(z,\beta)D_{i}(x_{i}^{\prime},x_{i})$

for all $x_{i},x_{i}^{\prime},z$ and any
$\xi\in\partial_{x_{i}}t(x_{i}^{\prime},z)$. Here $D_{i}(\cdot,\cdot)$
satisfies the conditions in (12). For the case
$D_{i}(x_{i}^{\prime},x_{i})=1/2\|x_{i}^{\prime}-x_{i}\|^{2}$, Condition 3
reduces to the strong convexity of $x_{i}\mapsto t(x_{i},z)$ with constant
$\eta_{i}(z,\beta)$.

###### Remark 3.1

Note that all surrogates in Example 2.3 satisfy Condition 1 with
$\bar{e}_{i}(x_{i},z)=e_{i}(x_{i},z)$.

The surrogate in Example 2.3(i) satisfies Condition 2 with
$\eta_{i}=\kappa_{i}$ and $D_{i}=D_{\mathfrak{h}_{i}}$. Consider the Bregman
surrogate in Example 2.3(iv). Since $x_{i}\mapsto
L_{i}\mathfrak{h}_{i}(x_{i})-f(x_{i},z_{\neq i})$ is convex, we have

$L_{i}\mathfrak{h}_{i}(x_{i})-f(x_{i},z_{\neq
i})-(L_{i}\mathfrak{h}_{i}(z_{i})-f(z_{i},z_{\neq i}))-\langle
L_{i}\nabla\mathfrak{h}_{i}(z_{i})-\nabla_{x_{i}}f(z_{i},z_{\neq
i}),x_{i}-z_{i}\rangle\geq 0,$

which implies
$L_{i}D_{\mathfrak{h}_{i}}(x_{i},z_{i})+f(z)+\langle\nabla_{x_{i}}f(z),x_{i}-z_{i}\rangle-f(x_{i},z_{\neq
i})\geq 0.$ Therefore, if we take $\kappa_{i}>1$ then the approximation error
of the Bregman surrogate satisfies Condition 2 with
$\eta_{i}=(\kappa_{i}-1)L_{i}$ and $D_{i}=D_{\mathfrak{h}_{i}}$. Specifically,

$e_{i}(x_{i},z)=f(z)+\langle\nabla_{x_{i}}f(z),x_{i}-z_{i}\rangle+\kappa_{i}L_{i}D_{\mathfrak{h}_{i}}(x_{i},z_{i})-f(x_{i},z_{\neq
i})\geq(\kappa_{i}-1)L_{i}D_{\mathfrak{h}_{i}}(x_{i},z_{i}).$

Similarly, we can show that the quadratic surrogate and the Lipschitz gradient
surrogate in Example 2.3 also satisfy Condition 2.

Consider the Lipschitz gradient surrogate in Example 2.3 (iii). If $g_{i}$ is
convex then we see that $x_{i}\mapsto u_{i}(x_{i},z)+g_{i}(x_{i})$ is
$\kappa_{i}L_{i}$-strongly convex. Hence, if $g_{i}$ is convex then Condition
3 is satisfied with $\eta_{i}(z,\beta)=L_{i}(z,\beta)$ and
$D_{i}(x_{i}^{\prime},x_{i})=\frac{1}{2}\|x_{i}^{\prime}-x_{i}\|^{2}$.
Similarly, we can prove that if $g_{i}$ is convex, then the quadratic
surrogate and the Bregman surrogate in Example 2.3 also satisfy Condition 3.

Update of block $y$. As $h$ is assumed to be $L_{h}$-smooth, we use the
following surrogate

$u_{m+1}(y,\tilde{x},\tilde{y},\omega)=h(\tilde{y})+\langle\nabla
h(\tilde{y}),y-\tilde{y}\rangle+\frac{L_{h}}{2}\|y-\tilde{y}\|^{2}+\bar{\varphi}_{\beta}(\tilde{x},y,\omega),$
(13)

where
$\bar{\varphi}_{\beta}(x,y,\omega):=f(x)+\langle\omega,\phi(x)+\mathcal{B}y\rangle+\frac{\beta}{2}\|\phi(x)+\mathcal{B}y\|^{2}$.
The update of $y$ is

$\begin{array}[]{ll}&y^{k+1}\in\arg\min_{y}u_{m+1}(y,x^{k+1},y^{k},\omega^{k})\\\
&=\arg\min\limits_{y}\big{\\{}\langle\nabla
h(y^{k})+\mathcal{B}^{*}\omega^{k},y\rangle+\frac{\beta}{2}\|\phi(x^{k+1})+\mathcal{B}y\|^{2}+\frac{L_{h}}{2}\|y-y^{k}\|^{2}\big{\\}}\\\
&=(\beta\mathcal{B}^{*}\mathcal{B}+L_{h}\mathbf{I})^{-1}\big{(}L_{h}y^{k}-\nabla
h(y^{k})-\mathcal{B}^{*}(\omega^{k}+\beta(\phi(x^{k+1})))\big{)}.\end{array}$
(14)

###  3.1 Subsequential convergence.

We establish subsequential convergence for mADMM in this section. Let us first
prove some sufficient decreasing properties for the update of $x_{i}$ and $y$.

###### Proposition 3.2

(i) The update in (10) guarantees a sufficient decreasing:

$\mathcal{L}_{\beta}(x^{k,i},y^{k},\omega^{k})+\eta_{i}^{k}D_{i}(x^{k+1}_{i},x_{i}^{k})\leq\mathcal{L}_{\beta}(x^{k,i-1},y^{k},\omega^{k}),$
(15)

where $\eta_{i}^{k}=\eta_{i}(x^{k,i},y^{k},\omega^{k},\beta)$, which is the
scalar in Condition 2 or Condition 3.

(ii) The update in (14) guarantees a sufficient decreasing:

$\mathcal{L}_{\beta}(x^{k+1},y^{k+1},\omega^{k})+\frac{\delta}{2}\|y^{k+1}-y^{k}\|^{2}\leq\mathcal{L}_{\beta}(x^{k+1},y^{k},\omega^{k}),$
(16)

where $\delta=L_{h}+\beta\lambda_{\min}(\mathcal{B}^{*}\mathcal{B})$.

###### Proof 3.3

Proof. We remark that, if we denote $x_{m+1}=y$, $g_{m+1}(y)=0$ and
$D_{m+1}(y,y^{\prime})=\frac{1}{2}\|y-y^{\prime}\|^{2}$, then we can write the
update of block $x_{i}$, $i\in[m]$, and the update of block $y$ in (14) in the
following unified form, supposed the current iterate is
$z=(\tilde{x}_{1},\ldots,\tilde{x}_{m+1},\omega)$,

$x_{i}^{+}=\arg\min_{x_{i}}\\{u_{i}(x_{i},z)+g_{i}(x_{i})\\},$ (17)

where $u_{i}$ is a surrogate function of $\varphi_{\beta}$ with respect to
block $x_{i}$, $i\in[m+1]$.

(i) In the following, we use $\eta_{i}$ for $\eta_{i}(z,\beta)$. Suppose
Condition 2 holds. Then we have

$u_{i}(x_{i}^{+},z)-\varphi_{\beta}(x_{i}^{+},z_{\neq
i})\geq\eta_{i}D_{i}(x_{i}^{+},z_{i}).$ (18)

On the other hand, from (17) we have $u_{i}(x_{i},z)+g_{i}(x_{i})\geq
u_{i}(x_{i}^{+},z)+g_{i}(x_{i}^{+})$ for all $x_{i}.$ Choose $x_{i}=z_{i}$,
then combine with (18). We obtain

$u_{i}(z_{i},z)+g_{i}(z_{i})\geq
u_{i}(x_{i}^{+},z)+g_{i}(x_{i}^{+})\geq\varphi_{\beta}(x_{i}^{+},z_{\neq
i})+\eta_{i}D_{i}(x^{+}_{i},z_{i})+g_{i}(x_{i}^{+}).$ (19)

Furthermore, $u_{i}(z_{i},z)=\varphi_{\beta}(z)$. The result follows from
(19).

Suppose Condition 3 holds. Choosing $x_{i}^{\prime}=x_{i}^{+}$ and
$x_{i}=z_{i}$ in Condition 3, we have

$u_{i}(z_{i},z)+g_{i}(z_{i})\geq
t(x_{i}^{+},z)+\langle\xi,z_{i}-x_{i}^{+}\rangle+\eta_{i}D_{i}(x_{i}^{+},z_{i}),\quad\forall\xi\in\partial_{x_{i}}t(x_{i}^{+},z).$
(20)

From (17) we have $0\in\partial_{x_{i}}t(x_{i}^{+},z)$. Hence, it follows from
(20) that

$\varphi_{\beta}(z)+g_{i}(z_{i})\geq
u_{i}(x_{i}^{+},z)+g_{i}(x_{i}^{+})+\eta_{i}D_{i}(x_{i}^{+},z_{i}).$

Moreover, we have $u_{i}(x_{i}^{+},z)\geq\varphi_{\beta}(x_{i}^{+},z_{\neq
i})$. Hence the result follows.

(ii) Since $y\mapsto u_{m+1}(y,\tilde{x},\tilde{y},\omega)$ defined in (13) is
$(L_{h}+\beta\lambda_{\min}(\mathcal{B}^{*}\mathcal{B}))$-strongly convex,
similarly to the proof of Part (i) for the case that Condition 3 is satisfied,
we can prove that (16) is satisfied. $\square$

Denote $\Delta x^{k}_{i}=x^{k}_{i}-x^{k-1}_{i}$, $\Delta y^{k}=y^{k}-y^{k-1}$,
$\Delta\omega^{k}=\omega^{k}-\omega^{k-1}$.

###### Proposition 3.4

The values of $\eta_{i}^{k}$ and $\delta$ are given in Proposition 3.2. Let
$\hat{\delta}=2L_{h}$. We have

$\begin{array}[]{ll}&\mathcal{L}_{\beta}(x^{k+1},y^{k+1},\omega^{k+1})+\sum\limits_{i=1}^{m}\eta^{k}_{i}D_{i}(x^{k+1}_{i},x^{k}_{i})+\frac{\delta}{2}\|\Delta
y^{k+1}\|^{2}\\\
&\leq\mathcal{L}_{\beta}(x^{k},y^{k},\omega^{k})+\frac{1}{\beta\sigma_{\mathcal{B}}}\|\mathcal{B}^{*}\Delta\omega^{k+1}\|^{2},\end{array}$
(21)

and

$\|\mathcal{B}^{*}\Delta\omega^{k+1}\|^{2}\leq
3\left((L_{h}^{2}+\hat{\delta}^{2})\|\Delta
y^{k+1}\|^{2}+\hat{\delta}^{2}\|\Delta y^{k}\|^{2}\right).$ (22)


# Relevance of sample geometry on the in-plane anisotropy of Sr$x$Bi2Se3
superconductor

Xinrun Mi  Low Temperature Physics Lab, College of Physics & Center of Quantum
Materials and Devices, Chongqing University, Chongqing 401331, China  Yecheng
Jing  Low Temperature Physics Lab, College of Physics & Center of Quantum
Materials and Devices, Chongqing University, Chongqing 401331, China  Kunya
Yang  Low Temperature Physics Lab, College of Physics & Center of Quantum
Materials and Devices, Chongqing University, Chongqing 401331, China  Yuhan
Gan  Low Temperature Physics Lab, College of Physics & Center of Quantum
Materials and Devices, Chongqing University, Chongqing 401331, China  Aifeng
Wang  Low Temperature Physics Lab, College of Physics & Center of Quantum
Materials and Devices, Chongqing University, Chongqing 401331, China  Yisheng
Chai  Low Temperature Physics Lab, College of Physics & Center of Quantum
Materials and Devices, Chongqing University, Chongqing 401331, China  Mingquan
He<EMAIL_ADDRESS>Low Temperature Physics Lab, College of Physics &
Center of Quantum Materials and Devices, Chongqing University, Chongqing
401331, China

(today)

###### Abstract

Possible emergence of nematic superconductivity that breaks its underlying
lattice symmetry in doped topological insulator Bi2Se3,establishes this system
as a unique candidate of topological superconductors. Exclusion of possible
extrinsic causes for the two-fold superconductivity is essential to clarify
its topological nature. Here, we present electrical transport on Sr$x$Bi2Se3
superconductors with rectangular and circular geometries. The occurrence of
the two-fold symmetric in-plane upper critical field $H_{c2}$ is found to be
weakly geometry dependent. However, the anisotropic ratio between the maximum
and minimum in-plane upper critical fields varies significantly among samples
with different shapes. Compared with the rectangular sample, the anisotropic
ratio is largely suppressed in the circular sample which has higher geometric
rotational symmetry. Our results imply that sample geometry plays a
subdominant role, but circular shape is more ideal to reveal the two-fold
superconductivity of SrxBi2Se3 in the vortex state.

##  I Introduction

The emergence of spontaneous symmetry breaking in addition to $U(1)$ gauge
symmetry in superconductors often leads to unconventional superconductivity,
such as $d$-wave, chiral $p$-wave superconductivity. One more prominent
example can be found in a nematic superconductor with odd-parity pairing
symmetry, which breaks the lattice rotational symmetry in the superconducting
state [1, 2]. Even more intriguing of such a nematic superconductor is its
topological nature, which could realize the time-reversal-invariant
topological superconducting state [1, 2]. Expectations of finding Majorana
fermion excitation in the nematic topological superconductors have attracted
much research efforts in the community of condense matter physics [3].

Material realization of the nematic superconductor is made possible by the
discoveries of superconductivity in Cu, Sr or Nb doped topological insulator
Bi2Se3. Soon after the report on superconductivity of Cu intercalated Bi2Se3
[4], F. Liang et al. [1] suggested the possibility of realizing an odd-parity
superconducting state in this system. Inspired by the observation of an
unusual two-fold symmetric spin susceptibility that breaks the trigonal
lattice symmetry of CuxBi2Se3 in the superconducting state [5], the ’nematic
superconductor’ hosting a two-dimensional $E_{u}$ odd-pairing state was
proposed by F. Liang to explain the unconventional symmetry breaking effects
[2]. Since then, numerous studies have reported signatures of nematic
superconductivity in $A_{x}$Bi2Se3 (A= Cu, Sr or Nb) using various techniques
including electrical transport [6, 7, 8, 9, 10], specific heat [11, 12, 13],
magnetic torque [14], thermal-expansion [15], scanning tunneling microscope
(STM) [16]. However, to which extent the observed two-fold symmetric
superconductivity is originated from an intrinsic topological nature still
remains elusive. Other factors, such as built-in strain or distortion [17,
18], structural inhomogeneity and impurity phases [19, 20, 21, 22],
inhomogeneously distributed superconducting regions [4, 23, 24, 25, 16],
cannot be completely ruled out at this stage. Exclusion of possible extrinsic
origins is of great importance to clarify the putative topological nature of
the nematic superconductivity found in doped Bi2Se3 systems.

Many experimental identifications of nematic superconductivity in
$A_{x}$Bi2Se3 were performed in the vortex state by applying magnetic fields
that rotate in-plane [3]. Various factors could affect the in-plane anisotropy
in the vortex state especially for crystals with irregular shapes. Firstly,
the demagnetization effect exists inevitably in any samples with finite sizes,
which could vary substantially along different in-plane magnetic field
directions except for circular or spherical shapes. The geometric effects can
have further impact on the flux penetration, in a way that the penetration of
flux lines at sharp corners is more difficult compared to smooth areas [26].
One more possibility could be the influence of surface superconductivity,
considering the weakly van-der Waals coupled layer structure of $A_{x}$Bi2Se3.
As already pointed out by Saint-James and de Gennes in 1963 [27], for the in-
plane field configuration, nucleation of superconductivity on the surfaces
enables a higher surface critical field $H_{c3}=1.69H_{c2}$ than that of the
bulk $H_{c2}$. Importantly, the $H_{c3}$ depends on sample geometry and
substantial enhancement of $H_{c3}$ to values higher than $1.69H_{c2}$ is
expected at edges or vertices [28, 29, 30]. Thus, different in-plane magnetic
field configurations with respect to irregular shapes containing various sharp
edges may significantly affect the in-plane anisotropic superconducting
properties. Detail analysis of the impacts induced by sample geometry on the
nematic superconductivity is, however, rarely explored.

Fig. 1: (a) Normalized electric resistance $R/R_{n}$(5 K) of the rectangular
(S-R) and circular samples (S-C). (b)(c) The angular variation of resistance
$R(\theta)$ of S-R and S-C samples measured near $T_{c}$ in a fixed in-plane
magnetic field $\mu_{0}H$=0.5 T. The photographs in (b) and (c) are optical
images of S-R and S-C. The current is along the in-plane direction for S-R,
which is perpendicular to the magnetic field $I\perp H$ at $\theta=0^{\circ}$.
The current was applied out-of-plane and the Corbino-like electrodes were used
for S-C. (d) Comparison of $R(\theta)$ recorded in $\mu_{0}H$=0.5 T for S-R
and S-C at 2 K.

In this article, we demonstrate the impacts of sample geometry on the in-plane
anisotropic superconductivity of SrxBi2Se3. We managed to engineer one single
piece of crystal into different shapes including rectangular bar and circular
disk, which enables direct investigation of geometric effects. It is found
that the in-plane anisotropic magneto-transport properties in the vortex state
depend strongly on crystal shapes, but the two-fold symmetry remains generic.
The anisotropic ratio between the maximum and minimum in-plane $H_{c2}$ varies
with sample shapes, which appears to be much smaller in the circular sample
compared with that in rectangular sample. Our findings suggest that sample
geometry does affect the in-plane anisotropy of SrxBi2Se3, and that it is
better to use circular geometry when studying anisotropic superconducting
properties in the vortex state.

##  II Methods

Single crystals of SrxBi2Se3 with nominal strontium $x$=0.2 were grown by the
melt-growth method [31]. Stoichiometric mixture of high purity Sr
piece$(99.5\%)$, Bi powder$(99.999\%)$ and Se powder$(99.999\%)$ was sealed in
a evacuated quartz tube. Then, the quartz tube was heated to 850 ∘C and kept
at this temperature for 48 h. After that, the quartz tube was cooled to 610 ∘C
slowly with 3 ∘C /h followed by fast quenching in water. The in-plane magnetic
field angle dependent electrical transport measurements were performed in a
Physical Property Measurement System (PPMS, Quantum Design Dynacool 9 T),
using a home-made insert equipped with a rotator. The electric resistance of
samples with rectangular and circular shapes was measured using standard four-
probe and Corbino-like methods, respectively.

##  III Results And Discussions

Figure 1(a) presents the zero field electric resistance of a rectangular bar
shaped crystal ( S-R) and a circular disk sample (S-C), in the vicinity of the
superconducting transition. The resistance has been normalized to its normal
state value $R_{n}$($T$=5 K) for better comparison. Both samples belong to a
single piece of crystal. The circular sample was cut from the rectangular
sample, and sharp edges were polished away subsequently. The current $I$ was
applied in-plane (out-of-plane) for the rectangular (circular) sample [see
Figs. 1(b)(c)]. Nearly identical transitions are found in both samples, which
show zero resistance below $T_{c}^{zero}$=2.5 K. The sample superconducting
$T_{c}$ allows us to compare the geometrical effects directly in the coming
discussions.

Fig. 2: (a) Magnetoresistance of the S-C sample measured at 1.8 K with
magnetic field applied along various in-plane directions. The upper critical
fields $H_{c2}$ are identified as the position where the magnetoresistance
reaches 50% of the normal state resistance, as marked out by gray dots. (b)
Angular variation of the upper critical field $H_{c2}(\theta)$. The red solid
line is a theoretical description according to Eq. 1 for an anisotropic
superconductor.

In Figs. 1(b)(c), we present the angular dependence of the resistance
$R(\theta)$, recorded in a fixed field $\mu_{0}H$=0.5 T with magnetic field
rotating within the basal plane ($H\perp c$), for the rectangular and circular
samples. Clear two-fold symmetry is observed in $R(\theta)$ below $T_{c}$ for
all samples, irrespective of the different geometries, suggesting a
subdominant role played by the crystal shape. Despite the generic two-fold
symmetry, the geometric effects still leave some fingerprints on $R(\theta)$.
One can see that the anisotropy of $R(\theta)$ is more evident in the
rectangular sample than that in the circular sample. This difference is seen
more clearly in Fig. 1(d), which directly compares $R(\theta)$ of the two
samples at 2 K. The anisotropic ratio between the maximum and minimum
resistance is $\alpha=R_{max}/R_{min}$= 35.7 for the rectangular sample, which
becomes much smaller in the circular ($\alpha=2$) sample. In addition, the
two-fold symmetry disappears at 2.5 K in the circular sample, which is still
visible in the rectangular sample at the same temperature. These results imply
that different geometries do have impacts on the in-plane anisotropy, although
the two-fold symmetry remains generic. We will come back to the discussion of
the geometric effects in more detail later.

The emergence of two-fold anisotropic $R(\theta)$ in the superconducting state
points to a two-fold symmetric upper critical field $H_{c2}(\theta)$. Fig.
2(a) shows the magnetoresistance of the circular sample measured at 2 K with
magnetic field pointing to various fixed angles. Clear anisotropy is seen, in
which the superconductivity is suppressed in lower (higher) magnetic fields at
$\theta=0^{\circ}$ ( $\theta=90^{\circ}$), compared with other angles. The
angular variation of $H_{c2}(\theta)$ can be better visualized in in Fig.
2(b). The values of $H_{c2}(\theta)$ are determined according to the criteria
when the magnetoresistance reaches 50% of the normal state resistance [see the
dots in Fig. 2(a)]. The two-fold symmetry of $H_{c2}(\theta)$ is obvious,
which suggests a similar profile of the superconducting gap $\Delta(\theta)$,
considering the direct link between $H_{c2}$ and $\Delta$. Indeed, the angular
variation of $H_{c2}(\theta)$ can be traced nicely by [the solid line in Fig.
2(b)]:

$H_{c2}(\theta)=\frac{H_{c2}(0^{\circ})}{\left(\cos^{2}\theta+\varGamma^{-2}\sin^{2}\theta\right)^{1/2}},$
(1)

suggesting the anisotropic nature of the superconducting order parameter [7,
9]. Here, the anisotropic ratio
$\varGamma=H_{c2}(90^{\circ})/H_{c2}(0^{\circ})$=2.5 and
$\mu_{0}H_{c2}(0^{\circ})$=0.31 T give the best fit at 1.8 K. Signature of
three-fold symmetry of the underlying lattice is invisible, which might be the
consequence of nematic superconductivity although the exact origin is still
elusive [3].

In Figure 3, we explore the anisotropic in-plane upper critical field in more
detail. Now the magnetic field is directing along fixed angles where $H_{c2}$
reaches minimum [$H_{c2}^{min}(\theta=0^{\circ}$), Figs. 3(a)(d)] and maximum
[$H_{c2}^{max}(\theta=90^{\circ}$), Figs. 3(b)(e)] values. From these results,
the $H-T$ phase diagrams can be obtained, as shown in Figs. 3 (c)(f). Again,
the criteria where the resistance reaches 50% of the normal state value is
chosen for $H_{c2}(T)$. Clearly, significant anisotropy is found and the
superconductivity is suppressed faster along the $H_{c2}^{min}(T)$ line than
that of the $H_{c2}^{max}(T)$ boundary in both samples. The separation between
$H_{c2}^{min}(T)$ and $H_{c2}^{max}(T)$ becomes largely reduced, when
switching from the rectangular sample to the circular sample. At 2 K, the
anisotropic ratio of the in-plane upper critical field,
$\varGamma=H_{c2}^{max}/H_{c2}^{min}$, reads $\varGamma=$ 3.1 and 1.7 for the
rectangular and circular samples, respectively. Compared with the rectangular
sample, $\varGamma$ is reduced by 45$\%$ in the circular sample, suggesting
sizable contributions from the shape anisotropy.

Fig. 3:  Temperature dependence of the resistance measured at fixed fields
applying along (a)(d) the minimum [$H_{c2}^{min}(\theta=0^{\circ}$)], and
(b)(e) the maximum [$H_{c2}^{max}(\theta=90^{\circ}$)] upper critical fields
directions for S-R (a)(b) and S-C (d)(e) samples. (c)(f) The anisotropic upper
critical field determined from the data shown in (a)(b) and (d)(e).

In Figure 4, we compare the geometric effects directly using the data
presented in Fig. 1 and Fig. 3. As shown in Figs. 4(a), the angular variation
of $R(\theta)$ shows strong sample dependence when other conditions are kept
the same ($T$=2 K, $\mu_{0}H$=0.5 T). Only a single periodicity of 180∘ is
found in both samples with the same maximum and minimum positions, indicating
negligible contributions from multi-domain structures. This suggests that the
observed variations among different samples are mainly coming from geometry
effects. The anisotropy is suppressed significantly in the circular sample, as
also seen in the anisotropic ratio $\varGamma$ displayed in Fig. 4(b). The
anisotropic superconducting properties are likely amplified by the shape
anisotropy in the rectangular sample.

For a finite sized sample with an ellipsoidal shape, the internal field
$H_{in}$ experienced by the sample differs from the external applied field
$H_{0}$ by:

$H_{in}=H_{0}-NM,$ (2)

where $N$ is the demagnetization factor, $M$ is the magnetization.
Superconductors expel magnetic flux, and $M<$0 in the superconducting state,
which gives a larger internal field than the field applied $H_{in}>H_{0}$. The
demagnetization effects are rather complicated for other shapes. Following R.
Prozorov et al. [32], the effective demagnetization factors for a rectangular
cuboid and a circular disk can be approximated as:

$N^{-1}=1+\frac{3}{4}\left(\frac{l_{\parallel}}{l_{\perp}}+\frac{l_{\parallel}}{t}\right),$
(3)

and

$N^{-1}=2+\frac{1}{\sqrt{2}}\frac{d}{t},$ (4)

respectively. For the rectangular cuboid, the symbol $l_{\parallel}$ is the
length along the field direction, $l_{\perp}$ is the size perpendicular to the
magnetic field. The parameter $d$ is the diameter of the circular disk, and
$t$ represents the thickness in both shapes [see also Fig. 4(c)]. Rough
estimations of the demagnetization factors of our samples in certain field
directions are shown in Fig. 4(c). Simplifications have been made here
assuming ideal rectangular and circular geometries. One can see that, for the
rectangular shape, the demagnetization factors differ by a factor of 2.5
between $\theta=0^{\circ}$ and $\theta=90^{\circ}$ directions. Such an
geometrical anisotropy is absent in the circular shape at any in-plane
directions. Along $\theta=0^{\circ}$, the demagnetization factor $N$ is
similar in both shapes. For $\theta=90^{\circ}$, on the other hand, $N$ is
much smaller in the rectangular shape than that in the circular geometry.
Consequently, the rectangular sample experiences a smaller internal magnetic
field $H_{in}(\theta=90^{\circ})$, and superconductivity can survive up to a
larger upper critical field identified by the externally applied field. The
anisotropic ratio in the rectangular sample ($\varGamma_{S-R}/2.5=1.24$)
actually gets close to that in the circular sample ($\varGamma_{S-C}=1.7$), if
the anisotropic demagnetization effects are taken into account despite our
oversimplified model. Note that the effect of demagnetization is overestimated
here since $-M<H_{0}$ in the vortex state. In addition to the anisotropic
demagnetization factor, the sharp edges or vertices in the rectangular shape
can delay the entrance of vortices [26], or even accumulate higher surface
upper critical fields [28, 29, 30]. These effects are hard to quantify, but
certainly complicates the anisotropy further. One more possibility is the
influence of Lorenz force, which is absent only for $H\parallel I$ at
$\theta=90^{\circ}$ in the rectangular sample. The anisotropy brought by
Lorenz force is minimized in the circular sample with current flowing along
the $c$-axis. All these factors contribute to the observed different
anisotropic ratios in different geometries. The circular shape appears to be
more suitable to probe the intrinsic anisotropy.

Fig. 4: Comparison of the (a) angular dependence of resistance $R_{\theta}$
and (b) anisotropic ratio $\varGamma$(2 K) of the upper critical field between
the S-R and S-C samples. (c) Estimation of the demagnetization factors $N$ of
the S-R and S-C samples when the magnetic field is applied along
$\theta=0^{\circ}$ and $\theta=90^{\circ}$ directions. For S-R, the length
parallel (perpendicular) to the field direction is defined as $l_{\parallel}$
($l_{\perp}$). The diameter of the S-C sample is $d$. The thicknesses of both
samples are labeled as $t$.

##  IV Conclusions

In summary, we have investigated the geometric effects on the in-plane
anisotropy of the nematic superconductor SrxBi2Se3. The two-fold symmetry of
the superconducting state appears to be generic, irrespective to samples with
different shapes. The anisotropy, on the other hand, varies significantly with
sample geometries. Compared with the rectangular sample, the anisotropic ratio
of the in-plane upper critical field is largely reduced in the circular
sample. Both anisotropic demagnetization factors, flux and/or surface
superconductivity accumulation at sharp vertices, may affect the in-plane
anisotropy in samples with anisotropic shapes. Our findings suggest that the
sample geometry plays a subdominant role in the anisotropic superconductivity,
but can not be neglected completely in the study of nematic superconductivity
in the vortex state.

## Acknowledgments

We thank Guiwen Wang and Yan Liu at the Analytical and Testing Center of
Chongqing University for technical support. This work has been supported by
National Natural Science Foundation of China (Grant Nos.11904040, 12047564),
Chongqing Research Program of Basic Research and Frontier Technology, China
(Grant No. cstc2020jcyj-msxmX0263), Fundamental Research Funds for the Central
Universities, China(2020CDJQY-A056, 2020CDJ-LHZZ-010, 2020CDJQY-Z006),
Projects of President Foundation of Chongqing University,
China(2019CDXZWL002). Y. Chai acknowledges the support by National Natural
Science Foundation of China (Grant No. 11674384, 11974065). A. Wang
acknowledges the support by National Natural Science Foundation of China
(Grant No. 12004056).

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# Pulsed CW laser for long-term spectroscopic measurements at high power in
deep-UV

Nikita Zhadnov  1,* Artem Golovizin  1,2 Irene Cortinovis  1 Ben Ohayon  1,3
Lucas de Sousa Borges  1 Gianluca Janka1,4 and Paolo Crivelli1  1Institute for
Particle Physics and Astrophysics, ETH, Zurich, 8093, Switzerland 2P.N.
Lebedev Physical Institute, Moscow, 119991, Russia 3Technion - Israel
Institute of Technology, Haifa, 3200003, Israel 4Laboratory for Muon Spin
Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland
<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

We present a novel technique for in-vacuum cavity-enhanced UV spectroscopy
that allows nearly continuous measurements over several days, minimizing
mirror degradation caused by high-power UV radiation. Our method relies on
pulsing of the cavity’s internal power, which increases the UV intensity to
maximum only for short periods when the studied atom is within the cavity mode
volume while keeping the average power low to prevent mirror degradation.
Additionally, this method significantly decreases laser-induced background on
charged particle detectors. The described 244 nm laser system is designed for
1S-2S two-photon CW spectroscopy of muonium in the Mu-MASS project. It was
tested to provide intracavity powers above 20 W, requiring maintenance only a
few times a day. The pulsing technique demonstrates minimal impact on the
radiation frequency, with no observed shifts exceeding 15 kHz. Our approach
represents a promising new technique for high-precision spectroscopy of atoms
in harsh UV environments and demonstrates the feasibility of CW spectroscopy
of muonium.

##  1 Introduction

High intensity deep-ultraviolet (deep-UV) continuous wave (CW) lasers open up
new opportunities for light-matter interaction for testing fundamental
physical theories and developments in applied science [1, 2]. Outperforming
pulsed lasers in frequency stability, they are an essential tool for precision
spectroscopy of transitions from the ground state in hydrogen [3, 4, 5] and
its isotopes [6], antihydrogen [7] and muonium [8, 9, 10]. Deep-UV CW lasers
allow laser cooling of hydrogen [11], mercury ions and atoms [12, 13, 14],
cadmium [15], AlF [16], AlCl [17] and HgF [18]. Progress of UV laser
technologies is beneficial for development of new optical atomic clocks [15,
13, 19, 20] and single photon excitation of Rydberg states [21, 22]. Collinear
resonance ionization spectroscopy using CW lasers [23, 24] is at the forefront
of measuring nuclear properties, such as shape, spin, and moments, through
hyperfine structure analysis. Studying certain atomic species with this
technique requires the use of deep-UV laser light for their excitation [25,
26].

Currently, the highest power deep-UV laser systems are capable of generating
up to 2 W of CW light [17, 27]. Moreover, a number of experiments such as the
laser cooling of hydrogen [11] and muonium spectroscopy [10] require tens of
watts of such radiation, which can be obtained by amplification in an optical
cavity. A big challenge that appears at such intensities of UV light,
especially in vacuum, is the degradation of optical components: mirrors,
crystals, lenses, and windows [28, 29, 30].

Cavity mirrors exposed to maximum power suffer the most from degradation. The
speed of degradation is known to increase with the UV light power which
optical surfaces experience. Previous research [10] demonstrated that an
ultra-high vacuum (UHV, $10^{-8}$ mbar) Fabry-Pérot cavity, with oxide multi-
layer coatings mirrors ($\mathrm{HfO_{2}/Al_{2}O_{3}}$) on a
$\mathrm{SiO_{2}}$ substrate, experiences more than a two-fold decrease of
finesse in one hour at $5\,$W of intracavity power. Fluoride-coated mirrors
($\mathrm{MgF_{2}/LaF_{3}}$) on $\mathrm{CaF_{2}}$ substrates demonstrate a
much more stable behavior: having slightly worse optical performance, they can
maintain up to $10\,$W of laser power in UHV on one-hour timescales.
Nevertheless, more than these record-breaking characteristics are required in
some applications. One possible way to maintain the optical quality of the
mirrors is to operate them in an oxygen atmosphere. This approach demonstrated
continuous operation for more than 4 hours at the power of $16\,$W at
$10^{-3}\,$mbar of oxygen [10]. However, most precision spectroscopy
applications demand UHV conditions, which are extremely hard to combine with
this technique. A compromise solution may be periodic conditioning of the
mirrors with oxygen in the presence of UV light. This requires breaks in the
experiment but allow for restoring optical characteristics between measurement
periods. The most likely cause of the degradation of fluoride mirrors is
hydrocarbon contamination. During recovery, UV light generates ozone and
atomic oxygen and decomposes hydrocarbon contaminants into components. The
latter reacts with oxygen atoms and form simpler volatile molecules desorbed
from the surface. Unfortunately, working with more than $10\,$W of intracavity
power escalates degradation and requires spending more time on mirror recovery
than on actual measurements.

Another important issue relates to resonance ionisation spectroscopy [8]. In
such experiments, charged particles (ions, muons) are most often detected
using microchannel plates (MCP). These devices are sensitive to deep-UV
photons. Since it is very problematic to provide complete protection of an MCP
from scattered light without reduction of its sensitive area, this effect
inevitably leads to a decrease in signal-to-noise ratio.

The laser system presented in this article was developed for the Mu-MASS
project aiming at high-precision CW laser spectroscopy of 1S-2S transition in
muonium. In this experiment, muons, coming from the low energy muons (LEM)
beamline at PSI [31], hit a mesoporous thin film silica target. Around 60 $\%$
are converted into muonium atoms and 40 $\%$ diffuse back into vacuum before
decaying[32]. Some of those atoms pass through a standing wave of $244\,$nm
light created with an enhancement cavity and can be excited to the 2S state.
The 2S atoms are then photoionised with a 355 nm pulsed laser and the
resulting muon is detected by an MCP (for details see [33]). Due to this
transition’s low two-photon excitation cross-section value and the small
number of available muonium atoms, having an excitation rate of 1 event per
hour requires $25\,$W of laser power on resonance. Therefore, to collect a
statistically significant dataset, the high-power laser has to be stable for
several days. The possibility of continuous conditioning of the mirrors for
the Mu-MASS project is limited by the fact that oxygen contamination induces
degradation of the LEM moderator [34]. Even though differential pumping was
implemented between the cavity mirrors and the moderator zone, a reduction of
the muon flux by a factor of two was measured in a timescale of a few minutes.
Overcoming this obstacle would require impractical and complicated additions
to the Mu-MASS vacuum system, involving multiple stages of differential
pumping. This article introduces a new approach to enable long-term, high-
power laser spectroscopic measurements in the deep-UV region for Mu-MASS and
similar experiments requiring clean UHV conditions.

##  2 Methods

The experiment was conducted using the Mu-MASS laser setup, as shown on Fig.
1. The setup consists of several key components, including a high-power
infrared Yb fiber amplifier, a CLBO crystal-based UV second harmonic
generator, and a vacuum enhancement cavity. These components have been
previously described in [35, 11, 27, 10]. For this experiment, fluoride
mirrors were chosen for both the input and output couplers of the enhancement
cavity due to their durability in UV. To the best of our knowledge, this laser
system represents the most powerful $244\,$nm CW laser source currently
available.

Figure 1: Experimental setup for muonium spectroscopy. ECDL - external cavity
diode laser, TA - tapered amplifier, SHG - second harmonic generator, LBO -
lithium triborate, PBS - polarising beam splitter, AOM - acousto-optic
modulator, BD - beam dump, EOM - electro-optic modulator, L1-L5 - mode
matching lenses, PD - photodetector, PM - power monitor, CLBO - caesium
lithium borate, IC - input coupler, OC - output coupler, HR - high reflector,
PZT - piezoelectric transducer stack. Toptica laser assembly generates 4.5 W
of $488\,$nm light, which goes through AOM and then converted to 244 nm using
a SHG cavity with a CLBO crystal. The resulting UV light, reaching up to 1.2
W, is coupled to an enhancement cavity inside the ultrahigh vacuum chamber.

Spectroscopic measurements often work in repetitive mode: the atoms or
molecules of interest are prepared, interact with laser radiation and one
measures their excitation efficiency. Usually, it requires the laser light to
be available on demand or periodically [3, 8, 24]. Therefore, to slow down the
effect of UV-induced degradation without sacrificing the ultimate laser power
one can deliver the light only for the necessary measurement period, avoiding
constant irradiation of optical components. A similar technique was used for
visible laser for the CRIS (collinear resonance ionization spectroscopy)
project [24], where the laser power was pulsed with a Pockels cell. As CRIS
works with a bunched beam, the pulsing of CW light is used to obtain higher
spectroscopic resolution while suppressing the background. The use of cavities
for amplification and second harmonic generation creates challenges for the
pulsing technique: it does not allow decreasing the laser power below the
level necessary to maintain the cavities’ length in resonance with the laser.

A laser power control system was implemented using a quartz crystal-based AOM
(Fig. 1) that is designed for high light intensities. Pulsing the AOM RF
amplitude lets us quickly adjust the zero-order beam intensity at 488 nm, with
a base-to-peak power level ratio of more than 10. By aligning the AOM to
maximize the first diffraction order (with a diffraction efficiency of over
$>90\%$), we direct the zero order to the UV SHG and stop the first order with
a beam dump. Bragg diffraction of light on a sound wave leads to spoilage of
the zero-order mode shape by “eating away" the intensity in the central part
of the beam more than at the edges. This effect is advantageous in reducing
light coupling to the SHG not only by direct power reduction but also by worse
mode matching. Using this technique and taking into account the nonlinear
efficiency of second harmonic generation, we are able to maintain a baseline
power of UV of just a few mW, which can peak up to $1.2\,$W when the AOM is
not active. The optimal pulse duration was determined by computer simulation
of muonium formation and its escape from the target to maximize the
probability of 1S-2S excitation. The resulting value is around 1 microsecond.
To keep the SHG and the enhancement cavity in resonance with the laser
wavelength, Pound-Drever-Hall locking scheme [36] is used. The bandwidths of
the feedback loops are constrained by the first resonance frequencies of the
piezoelectric transducers utilized to regulate the positions of the mirrors.
The enhancement cavity uses a center-of-mass piezo transducer mount [37],
which provides the widest feedback bandwidth, however still less than
$100\,$kHz. As these cavity-locking systems are slow to react to
$1\,\mu$s-scale pulsing of laser power, they act as low-pass filters for such
fast disturbances, allowing the cavities to remain locked throughout the
measurement. In the described laser system, even few-tens-of-microseconds-long
pulses do not cause the cavities to unlock.

Figure 2: The laser power pulse is measured at three positions in the optical
scheme: (a) in the zero-order right after the AOM, (b) incoming to the
enhancement cavity, and (c) after transmission through the enhancement cavity.
The RF pulse, which triggers the AOM, occurs at $t=0$ s, and the delay of the
laser pulse is caused by the travel time of the sound wave in the AOM crystal
and the exponential behavior of power increase after passing through the SHG
cavity and the enhancement cavity, with finesses of 150 and 200 (non-
degraded), respectively. Eventually, the intracavity UV power response is
characterized by a delay of 460 ns, followed by an exponential rise with a
time constant of 190 ns.

The accurate control of delays in the optical and radio wave lines is crucial
for the successful implementation of the described method. Figure 2
illustrates a single laser power pulse at three different points in the
optical setup: at the zero-order of the AOM, after the UV SHG, and inside the
enhancement cavity (see Figure 1). Intracavity power was determined from the
transmitted light power measurement, taking into account the transmission
coefficient of the output coupler. To minimize the AOM delay, we positioned
the laser beam as close as possible to the ultrasonic emitter on the edge of
the AOM crystal, right up to the point where the beam starts to cut into the
crystal. Due to the delay and the time constants of the cavities, it takes
$1\,\mu$s for the UV light to reach 95% of its peak power level after the AOM
RF amplitude pulse. Much shorter delays, rise times and higher extinction
ratios can be achieved with the use of Pockels cell instead of AOM [24]. In
that case, the power rise time would be limited by the cavities.

A triggering rate of 5 kHz (typical for the LEM muon beam at PSI), results in
an average duty cycle of $5\times 10^{-3}$ for $1\,\mu$s long laser pulses.
With a base UV power of $5\,$mW and a peak power of $1.2\,$W, this enables the
average power to be reduced by more than 100 times compared to the peak power.
In UHV conditions, even this huge reduction in the UV load on the mirrors does
not allow overcoming the degradation effect completely, but should
dramatically reduce its rate. A second interesting feature, which should be
provided by fast AOM power control, is the opportunity to suppress photon-
induced background signals on MCP detectors.

##  3 Results and discussion

The performance of the laser system was monitored during the first attempt of
muonium CW spectroscopy. The lasers and the vacuum chamber were transported
from ETH Zurich to PSI. The task for the laser system during the measurements
was to work stably at high peak power for several days.

To maintain a high enough enhancement factor in the course of UV laser light
operation, the cavity mirrors need recovery from hydrocarbon contamination
[10]. During the recovery process, the vacuum chamber of LEM was shut off to
protect the moderator, and oxygen was pumped in through the needle valves near
the mirrors (Fig. 1) up to the pressure of $10^{-2}\,$mbar. At an intracavity
power of about $1\,$W the enhancement recovery process has a time constant of
2.5 minutes (Fig. 3). The entire procedure including oxygen evacuation down to
$10^{-6}\,$mbar was taking almost half an hour, which is fast enough to not
significantly affect the duration of the measurements.

Figure 3: The cavity’s enhancement factor during oxygen conditioning was
determined by measuring the cavity transmission. The conditioning was
performed with $1\,$W of intracavity UV light and an oxygen pressure of
$10^{-2}\,$mbar. The recovery process had a time constant of $2.5\,$minutes.
The fluctuations on the graph are due to the influence of the oxygen flow from
the needle valves.

The UV SHG output radiation was operated at the base power of $5\,$mW,
responding to each trigger signal with a pulsed increase in power up to more
than $1\,$W. The average trigger rate during the experiment was $2.5\,$kHz.
The incoming and transmitted laser peak power were logged during the whole
measurement to monitor the real-time cavity enhancement. A fragment of these
measurements is presented in Fig. 4. The experiment showed that with an
initial intracavity pulse power of 25 watts, the resonator degrades to 20
watts in approximately 5 hours and requires recovery (Fig. 4). A notable
observation is that there were no signs of degradation on the output coupler
of the UV SHG during the pulsing regime, which had been a concern in the past.

We should mention that the enhancement factor and UV SHG output power in [10]
is larger than in this work. During the measurement, the enhancement was
worsened by vibrations that disturb the coupling of laser light into the
cavity mode. The piezo’s ability to suppress vibrations will be improved by
shortening the cavity and placing it vertically. The second harmonic generator
suffered from dust contamination and misalignment due to temperature changes,
showing slightly worse behavior than optimal. Dust protection and periodic
alignment help to maintain its stable operation.

Figure 4: Fragment of intracavity and UV SHG output peak power measurements.
At $t=2\,$h the quality of the enhancement cavity lock was improved, and at
$t=8$ h mirror recovery started. In the first graph, the percentage of uptime
is shown. Fluctuations of the peak power are caused by imperfect cavity lock
quality and vibrations. The descending intracavity power is due to mirror
degradation and a decrease of UV generation efficiency.

The laser system was operating for about 80 hours during five days of
measurements. Breaks in work were mainly due to interruptions in the operation
of the muon beam, background noise tests, and other measurements in which the
laser was not involved. It is worth noting that the installation of the laser
setup in the LEM beamline took around 40 hours, and even during the days of
measurement, misalignments due to laser transportation were improved over the
course of the measurement.

During propagation through a transparent medium, a laser pulse can acquire a
chirp due to the effects of chromatic dispersion, nonlinearities, or rapid,
laser-induced changes of the refractive index. To analyze this effect, we
assembled an additional setup for UV light frequency measurement. This
includes an alternative beam path where the laser light does not experience
pulsing. One watt of blue light produced by the Toptica box is split off with
a polarizing beamsplitter (see Fig. 1 after LBO SHG) and directed to another
AOM, which provides detuning, and then to a single-pass BBO crystal to
generate 244 nm light. The two beams are combined and directed to a
photodiode, which produces a beat signal at a doubled detuning frequency. In
our experiment, this was $124.445\pm 0.002\,$MHz. Since we are interested in
frequency changes smaller than the muonium 1S-2S transition linewidth (144
kHz), a single laser pulse waveform does not provide enough data due to the
spectral limit set by its short duration. Furthermore, the silicon
photodetector used in the deep-UV region has a very moderate sensitivity, and
the second harmonic produced by the BBO crystal has low intensity, which
reduced the signal-to-noise ratio. To collect a statistically significant
sample, we analyzed 2000 waveforms. We took the first $750\,$ns of each pulse,
filtered them with a band-pass Fourier filter, and calculated the average
frequency by measuring the difference between neighboring zero-crossings of
the sine signal at the beat frequency. This gave us a normally-distributed
sample of two thousand instantaneous frequencies, which were used to obtain an
average value of $124.445\pm 0.006\,$MHz. This allows us to conclude with a
95% confidence level that no frequency shift above $15\,$kHz is introduced by
the pulsing scheme.

Figure 5: (a) Power pulse inside the enhancement cavity followed by a dip. (b)
The temporal distribution of the number of counts detected by the MCP
detector. The region where the radiation power is minimal is highlighted. For
the description of the Mu-MASS experiment see [33].

The control of AOM power not only enables pulsing of UV intensity but also
allows for an almost complete turn-off of the UV light by maximizing AOM
diffraction efficiency (Fig. 5(a)). This means that the UV light intensity
inside the vacuum chamber can be decreased by at least a factor of 10 in
comparison to the base level for a few microseconds after the pulse, creating
a reduced background time window for less noisy particle detection. To reach
the MCP detector, which is removed at a distance of half a meter from the
enhancement cavity, photons need to pass through a 90-degree bend in the
vacuum tube [33]. For extra protection of the detectors from scattered laser
radiation, we installed four light-absorbing screens (Acktar MaxiBlack foil)
with axial holes in the vacuum chamber. These screens isolate the mirror area
from the central chamber (Fig. 1). The signal of the MCP was measured for 12
hours with $1\,\mu$s intracavity UV peak power of $800\,$mW and a following
$3\,\mu$s long minimum-power time window. Turning the radiation off made it
possible to reduce the background rate by a factor of two, as shown in Figure
5(b): from $7.5\pm 0.3\,$Hz to $3.1\pm 0.4\,$Hz.

##  4 Conclusion

In summary, this study introduces a new and innovative method for precision
continuous wave laser ultraviolet spectroscopy that addresses the challenge of
optical surface degradation commonly encountered in such experiments. On-
demand CW UV laser light pulsing enables the use of cavity-enhanced radiation
in ultrahigh vacuum conditions, with a power output range of $20-25\,$W.
Spectroscopic measurements with this technique can last at least a week. To
maintain a high enhancement coefficient, mirror recovery with oxygen
conditioning is only required 4-5 times a day, which has minimal impact on the
measurement process. Precise control over UV radiation power is beneficial for
resonance ionization spectroscopy, as it allows for rapid turn-offs of the
laser radiation to minimize UV pickup by MCP detectors during the registration
time window.

The presented laser system was designed for 1S-2S spectroscopy of muonium in
the Mu-MASS project. With minor improvements, the pulsed CW laser system
demonstrates the feasibility of muonium continuous-wave spectroscopy and paves
the way for further studies of this unique atomic system.

Funding This work is supported by the ERC consolidator grant 818053-Mu-MASS
(P.C.) and the Swiss National Science Foundation under grant 197346 (PC). B.O.
acknowledges support from the European Union’s Horizon 2020 research and
innovation program under the Marie Skłodowska-Curie grant agreement No.
101019414. Acknowledgments We would like to acknowledge Zak Burkley for his
essential contribution of the laser system. We would like to thank Dylan Yost
for the very useful discussions.

Disclosures The authors declare no conflicts of interest.

Data Availability Statement Data underlying the results presented in this
paper are not publicly available at this time but may be obtained from the
authors upon reasonable request.

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