#  Reconfigurable Intelligent Surface Aided Mobile Edge Computing over
Intermittent mmWave Links††thanks: The work of Airod, Di Lorenzo and Calvanese
Strinati has been partially funded by the H2020 project RISE-6G no. 101017011.

Fatima Ezzahra Airod1, Mattia Merluzzi1, Paolo Di Lorenzo2,3, Emilio Calvanese
Strinati1
1CEA-Leti, Université Grenoble Alpes, F-38000 Grenoble, France
2DIET department, Sapienza University of Rome, Italy, 3CNIT, Parma, Italy
email:{fatima-ezzahra.airod, mattia.merluzzi, emilio.calvanese-
strinati}<EMAIL_ADDRESS>
###### Abstract

The advent of Reconfigurable Intelligent Surfaces (RISs) in wireless
communication networks unlocks the way to support high frequency radio access
(e.g. in millimeter wave) while overcoming their sensitivity to the presence
of deep fading and blockages. In support of this vision, this work exhibits
the forward-looking perception of using RIS to enhance the connectivity of the
communication links in edge computing scenarios, to support computation
offloading services. We consider a multi-user MIMO system, and we formulate a
long-term optimization problem aiming to ensure a bounded end-to-end delay
with the minimum users’ average transmit power, by jointly selecting uplink
user precoding, RIS reflectivity parameters, and computation resources at a
mobile edge host. Thanks to the marriage of Lyapunov stochastic optimization,
projected gradient techniques and convex optimization, the problem is
efficiently solved in a per-slot basis, requiring only the observation of
instantaneous realizations of time-varying radio channels and task arrivals,
and that of communication and computing buffers. Numerical simulations show
the effectiveness of our method and the benefits of the RIS, in striking the
best trade-off between power consumption and delay for different blocking
conditions, also when different levels of channel knowledge are assumed.

##  I Introduction

The advent of the sixth generation of mobile communication systems (6G)
unveils several ambitions that span from the support of new services, to
completely new key performance indicators. Indeed, we are facing an
unprecedented revolution of applications, such as immersive virtual reality,
connected autonomous systems, and the industrial Internet of Things, all
verticals that require real time data transmission and processing [1]. Of
course, to be effective, these services require to be enabled with new levels
of dependability, reliability and sustainability. From a radio access
perspective, the adoption of higher frequency such as millimeter wave (mmWave)
and TeraHertz (THz) bands certainly enhances radio access network capacity,
although at the price of a higher sensitivity to the presence of spatial
blockages and, in general, to deep fading events that may hinder the
aforementioned vision on performance [2, 3]. To this end, Reconfigurable
Intelligent Surfaces (RISs) have recently emerged as a promising candidate to
counteract the above mentioned issue, thanks to their ability to
opportunistically shape the wireless propagation environment. More precisely,
RISs are composed of scattering elements that can be adaptively configured to
shape the incident wave through adjustable phase shifts, with the aim of
improving system performance in specific locations in space and time [4, 5].
Therefore, owing to their abilities of customizing time-varying wireless
propagation environments, RISs mark undeniably the dawn of the 6G era. Another
key technological enabler of the 6G vision, already introduced in 5G, is
Multi-access Edge Computing (MEC), which brings storage and computing
resources close to end users, aiming to enable a new class of connect-compute
services [6]. As such, the interplay between RISs and MEC plays a key role in
improving network performance, thanks to the double benefit of computation and
communication aspects, to be tackled and optimized jointly. In this paper, we
focus on computation offloading services, whose goal is to move the execution
of computation demanding applications from resource-poor end devices to nearby
Mobile Edge Hosts (MEHs), to enable energy efficient, low latency, and
reliable processing [7]. In particular, we investigate on the promising
convergence of RISs and MEC, mainly focusing on a joint optimization of radio
and computing resources, down to the wireless propagation environment
properties.
Related works. Most of investigated works in the literature have been focused
on addressing computation offloading upon appropriate wireless environments,
tackling the joint optimization of communication and computation resources in
MEC-enabled wireless networks [6, 9]. However, inevitably, moving towards
higher frequency bands to cope with large data volumes is no more suitable for
MEC systems due to the unpredictable and intermittent nature of wireless
links. Indeed, blocking events may deteriorate the overall network
performance. In line with this, the recent literature has involved the
prominence of RISs to boost the performance of MEC systems in terms
reliability [10, 7, 11]. Nevertheless, to the best of our knowledge, a dynamic
joint optimization of computing resources and RIS-aided MIMO radio parameters
is lacking.
Contribution. In this work, we propose an algorithm aiming to dynamically
configure RIS parameters, users’ uplink precoding, and computation resources,
with the goal of minimizing users’ transmit power, with guaranteed finite E2E
delay of the offloading service. Thanks to the theory of Lyapunov stochastic
optimization, we are able to split a long-term problem into consecutive
deterministic optimization problems, based on instantaneous observations of
context parameters. The solution of the latter, from a radio perspective,
builds on an alternating optimization strategy that couples a projected
gradient step for the RIS parameters [13], and a water-filling solution for
the users’ precoding [14]. The computation resource allocation problem is also
solved with low complexity.

##  II System Model

In this work, we consider a dynamic system, in which a set $\mathcal{U}$ of
$N$ users continuously generate data locally and offload them to an MEH
through the wireless connection with an Access Point (AP). In such a scenario,
context parameters such as wireless channels and data arrivals vary over time,
thus calling for a dynamic cross-layer optimization involving users’ precoding
for uplink transmission, RIS reflectivity matrix, and computation resources at
the MEH. Then, we consider time as organized in time slots $t=1,2,\ldots$ of
equal duration $\tau$. Given a random variable $X$, we denote by
$\overline{X}$ its long-term average:

$\overline{X}=\lim_{T\to\infty}\frac{1}{T}\sum\nolimits_{t=1}^{T}\mathbb{E}\\{X(t)\\}$
(1)

###  II-A Communication model

####  II-A1 RIS reflectivity model

An RIS can be modeled as an array of $M$ nearly passive elements, whose phases
can be opportunistically tuned. Therefore, the RIS can be characterized by its
reflectivity matrix that is represented, at time $t$, as
$\mathbf{\Theta}(t)=\textrm{diag}\\{r_{1}(t),...,r_{M}(t)\\}$, where $r_{i}$,
$i=1...,M$, are the complex reflection coefficients of each element,
characterized by a fixed amplitude, which we assume to be unitary, and an
adjustable phase shift $\theta_{i}$, therefore we have
$r_{i}=\alpha_{i}e^{j\theta_{i}}$. In the sequel, we will denote by
$\mathbf{r}(t)$ the vector $\mathbf{r}(t)=\\{r_{i}(t)\\}_{i=1}^{M}$.

####  II-A2 Channel model

We consider a MIMO system, in which the AP is equipped with $N_{a}$ antennas,
while each user is equipped with $K$ antennas. For each user $k$, the E2E
channel matrix $\mathbf{H}_{k}(t)$ at time $t$ is composed of: i) a direct
channel $\mathbf{H}_{k,d}(t)\in\mathbb{C}^{N_{a}\times K}$ between the user
and the AP; ii) an indirect link, comprising the channel
$\mathbf{H}_{{k},r}(t)\in\mathbb{C}^{{M}\times K}$ between the user and the
RIS, and the channel $\mathbf{H}_{{r,a}}(t)\in\mathbb{C}^{N_{a}\times M}$
between the RIS and the AP. Also, since we consider blocking, we define
$\beta_{k,a}(t)\in\\{0,1\\}$ ($\beta_{k,r}(t)\in\\{0,1\\}$), which equals $1$
if the direct (indirect) link experiences a blockage event. Then, the overall
channel matrix of user $k$ can be written as follows [15] (we omit the index
$t$ for ease of notation):

$\normalsize\mathbf{H}_{k}=(1-\beta_{k,a})\mathbf{H}_{k,d}+(1-\beta_{k,r})\mathbf{H}_{{r,a}}\Theta\mathbf{H}_{k,r}.$
(2)

Note that, in the sequel, we will denote by $p_{k,a}$ ($p_{k,r}$) the blocking
probability of the direct (indirect) link, i.e. the probability that
$\beta_{k,a}$ ($\beta_{k,r}$) equals $1$, which can be computed as the
expectation of $\beta_{k,a}$ ($\beta_{k,r}$). Uplink transmission is a
fundamental phase of computation offloading services, foreseen to also
increase future uplink traffic [16]. Thus, let us now formalize the uplink
parameters, whose optimization will be presented in the sequel. Letting
$\mathbf{Q}_{k}(t)\in\mathbb{C}^{K\times K}$ be the transmit covariance matrix
of user $k$ at time $t$, the experienced data rate reads as follows

$R_{k}(t)=W_{k}\log_{2}\left|\mathbf{I}+\sigma_{k}^{-2}\mathbf{H}_{k}(t)\mathbf{Q}_{k}(t)\mathbf{H}_{k}^{H}(t)\right|$
(3)

where $W_{k}$ represents the bandwidth assigned to user $k$, and the noise
power is $\sigma_{k}^{2}=N_{0}W_{k}$, with $N_{0}$ the noise power spectral
density. Here, we assume that users are served thorough orthogonal channels
with a frequency division multiplexing. Obviously, besides the current channel
state conditions, the data rate depends on the user transmit covariance and
the RIS parameters, which we will jointly optimize in the sequel.

###  II-B Queuing Model and delay

Computation offloading services generally entail three phases, along with
their respective delays: i) uplink communication buffering and transmission of
input data; ii) computation buffering and computation; iii) downlink
communication buffering and transmission of results. In this work, we consider
the first two delays, although considering the last one would not
substantially change the system model, as presented in [7]. Therefore, we
model the E2E delay thorough a queueing system, comprising an uplink
communication buffer $B_{l,k}(t)$, and a computation buffer $B_{r,k}(t)$ for
each user $k$.
Communication buffer: The uplink buffer of each user $k$ is fed by new
arrivals $A_{k}(t)$ at time $t$, and drained by transmitting bits over the
wireless interface at rate $R_{k}(t)$ (cf. (3)). Given a slot of duration
$\tau$, the queue evolves as:

$\normalsize B_{l,k}(t+1)=\max\left(0,B_{l,k}(t)-\tau
R_{k}(t)\right)+A_{k}(t)$ (4)

Computation buffer: Assuming that all computation tasks are offloaded to the
MEH, we consider a computation queue for each UE, which is fed by the its
arriving data in uplink, and drained by the computation performed at the MEH.
We assume a linear relation between the number of transmitted bits and the CPU
cycles. Then, denoting by $J_{k}$ the number of CPU cycles per bit, the remote
computation queue evolves as:

$\displaystyle\normalsize B_{r,k}(t+1)=$
$\displaystyle\max\left(0,B_{r,k}(t)-\tau f_{k}(t)/J_{k}\right)$
$\displaystyle+\min(B_{l,k}(t),\tau R_{k}(t)),$ (5)

where $f_{k}(t)$ represents the amount of resources (in CPU cycles/s)
allocated to user $k$ during time slot $t$. Due to Little’s law, the average
E2E delay experienced by each device is proportional to the sum queue length
[6]:
$\overline{D_{k}}=\tau\frac{\overline{B_{l,k}}+\overline{B_{r,k}}}{\overline{A_{k}}}$.

##  III Problem formulation

In this paper, we jointly optimize users’ uplink covariance matrix, RIS
parameters, and computation resources at the MEH, to minimize the users’
transmit power under queue stability constraints. The problem can be
formulated as follows:

$\displaystyle\min_{\\{\mathbf{Q}_{k}(t)\\}_{k},{\mathbf{r}(t)},\\{f_{k}(t)\\}_{k}}\quad\sum\nolimits_{k\in\mathcal{U}}\overline{\textrm{Tr}(\mathbf{Q}_{k})}$
(6) subject to $\displaystyle(a)\;\overline{B_{l,k}}<\infty,\quad\forall
k\;\;(b)\;\overline{B_{r,k}}<\infty,\quad\forall k$
$\displaystyle(c)\;\mathbf{Q}_{k}(t)\succcurlyeq 0,\quad\forall
k\;\,(d)\;\textrm{Tr}(\mathbf{Q}_{k}(t))\leq P_{k}^{\max},\quad\forall k$
$\displaystyle(e)\;|r_{i}(t)|=1,\;\forall i\quad\,(f)\;f_{k}(t)\geq
0,\quad\forall k$
$\displaystyle(g)\;\sum\nolimits_{k\in\mathcal{U}}f_{k}(t)\leq f_{\max}.$

The constraints of (6) have the following meaning: $(a)$-$(b)$ the local and
remote queues of each user are stable; $(c)$ the transmit covariance matrix of
each user is semidefinite positive; $(d)$ the uplink transmit power of each
user is lower than a maximum value $P_{k}^{\max}$; $(e)$ the RIS reflectivity
entries are complex exponential; $f)$ the CPU cycle frequency allocated to
each user by the MEH is non-negative; $(g)$ The sum all CPU cycle frequencies
assigned to each user is at most equal to the MEH CPU maximum frequency
$f_{\max}$. Problem (6) is a priori very complex to solve, as it involves time
averages performed on variables whose statistics are supposed to be unknown.
To solve it in an efficient way, we leverage on Lyapunov stochastic
optimization [17], which allows us to define a sequence of deterministic
problems, based on instantaneous observations of context parameters. In
particular, following [17], and defining the vector
$\mathbf{b}(t)=[\\{B_{l,k}(t)\\}_{k},\\{B_{r,k}(t)\\}_{k}]$, we can write the
Lyapunov function as
$L(\mathbf{b}(t))=\frac{1}{2}\sum\nolimits_{k\in\mathcal{U}}[B_{l,k}^{2}(t)+B_{r,k}^{2}(t)]$
[17], which is a measure of the overall congestion state of the system. Our
aim is to drive the network towards stability, with the minimum transmit
power. To this end, as in [17], let us define first the drift-plus-penalty
(DPP) function
$\Delta_{p}(t)\\!=\\!\mathbb{E}\\{\\!L(\mathbf{b}(t+1))-L(\mathbf{b}(t))\\!+\\!V\sum_{k\in\mathcal{U}}\textrm{tr}(\mathbf{Q}_{k}(t))|\mathbf{b}(t)\\!\\},$
which is the one slot conditional expected change of the Lyapunov function,
with a penalty factor, weighted by a parameter $V$, used to trade-off users’
transmit powers and queue backlogs, thus shaping the desired trade-off between
power consumption and E2E delay. Interestingly, queues’ stability ($(a)$-$(b)$
in (6)) is guaranteed if the DPP is bounded by a finite constant for all $t$
[17]. As in [17], we now proceed by minimizing a suitable upper bound of the
DPP. The upper bound, whose derivations are omitted due to the lack of space
(see, e.g., [17]) reads as

$\displaystyle\Delta_{p}(t)\leq
C+\mathbb{E}\big{\\{}\sum\nolimits_{k\in\mathcal{U}}[\left(B_{r,k}(t)-B_{l,k}(t)\right)\tau
R_{k}(t)$ $\displaystyle\\!+A_{k}(t)B_{l,k}(t)-\tau
B_{r,k}(t)f_{k}(t)/J_{k}+V\textrm{tr}(\mathbf{Q}_{k}(t))]|\mathbf{b}(t)\big{\\}},$

where $C$ is a positive constant, omitted due to the lack of space. By
greedily minimizing this upper bound in each time slot (i.e. removing the
expectation), queues’ stability is guaranteed, as well as the asymptotic
optimality of the solution as $V$ increases, with the cost of increased queue
backlogs (i.e. higher E2E delay) [17]. It is easy to show that the resulting
problem can be split, in each time slot $t$, into a radio resource allocation
sub-problem, including the optimization of user covariance matrices and RIS
parameters, and a computation resource allocation sub-problem, to optimize the
MEH CPU scheduling. The overall proposed dynamic resource allocation procedure
is described in Algorithm 1, whose steps are described in the following. In
particular, Sec. III-A describes the implementation of step $1$, to optimize
radio resources. The implementation of step $2$ follows in Sec. III-B.

###  III-A Radio resource allocation sub-problem

The radio resource allocation sub-problem (step $1$ of Algorithm 1) involves
$\\{\mathbf{Q}_{k}(t)\\}_{k}$ and $\mathbf{r}(t)$, and is formulated as

$\displaystyle\small\min_{\\{\mathbf{Q}_{k}(t)\\}_{k},{\mathbf{r}(t)}}\;$
$\displaystyle\sum\limits_{k\in\mathcal{U}}\left(V\textrm{Tr}(\mathbf{Q}_{k}(t))-\tau(B_{l,k}(t)-B_{r,k}(t))R_{k}(t)\right)$
$\displaystyle\textrm{subject
to}\quad(c)\textrm{-}(e)\;\textrm{of}\;\eqref{lg_eq}$ (7)

Problem (III-A) is non convex, due to the non linear equality constraint
$(e)$. However, given the RIS parameters, the problem is convex and enjoys a
simple water-filling solution [14]. Then, the solution of (III-A) is built on
an iterative optimization algorithm that alternatively optimizes (III-A) with
respect to the RIS phase shift using the projected gradient descent method
(PGM), as in [13], and optimally updates the uplink covariance matrices of all
users through the water-filling method [14]. Steps $1.1$ and $1.2$ are
implemented as follows.

####  III-A1 RIS optimization step (Step $1.1$ of Algorithm 1)

The projected gradient step (step $1.1$) with respect to $\mathbf{r}(t)$ comes
with low complexity, as both the gradient and the projection can be written in
closed form [13, Eq. ($17$a)]. However, differently from [13], we deal with a
multi-user case. Nevertheless, thanks to the decoupling obtained through the
Lyapunov optimization framework, in this case, the gradient is a weighted sum
of different terms (corresponding to different users), where the weights
include both communication and computation queues. This naturally introduces a
scheduling of the RIS, which is therefore optimized to prioritize users with
worse queueing states. The gradient with respect to $\mathbf{r}$ reads as
follows [13]:

$\displaystyle\nabla_{\mathbf{r}}f(\textbf{r},\left\\{\textbf{Q}_{k}\right\\}_{k})=-\tau\sum\nolimits_{k\in\mathcal{U}}W_{k}(B_{l,k}-B_{r,k})$
$\displaystyle\times\textrm{diag}\left(\mathbf{H}_{r,a}^{H}\left(I+\mathbf{Z}_{k}\mathbf{Q}_{k}\mathbf{Z}_{k}^{H}\right)^{-1}\mathbf{Z}_{k}\mathbf{Q}_{k}\overline{\mathbf{H}}_{k,R}^{H}\right),$
(8)

where $\mathbf{Z}_{k}=\mathbf{H}_{k}/\sigma$, and
$\overline{\mathbf{H}}_{k,R}=\mathbf{H}_{k,r}/\sigma$, and the operator
$\textrm{diag}(L)$ saves the diagonal elements of an $N_{L}\times N_{L}$
matrix $L$ into a vector. Finally, since $|r_{i}|=1$ must hold for all
$i=1,\ldots M$, the projection onto the unit circle reads as [13]:

$P_{\Theta}(r_{i})=r_{i}/|r_{i}|,\qquad\forall i=1,\ldots,M.$ (9)

Step $1.1$ of Algorithm 1 is implemented through (III-A1) and (9).

####  III-A2 Uplink covariances optimization (Algorithm 2)

From (III-A), it can be easily observed that, once the RIS configuration is
fixed, the problem with respect to $\\{\mathbf{Q}_{k}(t)\\}_{k}$ admits a low
complexity solution. First of all, the problem is separable among the $N$
users. Moreover, for a generic user $k$, if $B_{l,k}(t)\leq B_{r,k}(t)$, both
terms in (III-A) are monotone non-decreasing functions of the user transmit
power. Therefore, in this case, the optimal solution is
$\mathbf{Q}_{k}^{*}(t)=\mathbf{0}_{K}$, i.e. user $k$ does not transmit (step
$1.2.a$ of Algorithm 1). This holds true also in the case in which all links
are blocked. Instead, for a generic user $k$ for which $B_{l,k}(t)>B_{r,k}(t)$
holds, the problem is convex and is similar to the one presented in [14], thus
admitting the water-filling procedure in Algorithm 2.

###  III-B Computation resource allocation sub-problem

The second sub-problem, necessary to implement step $2$ of Algorithm 1, is
formulated as follows:

$\displaystyle\normalsize\max_{f_{k}(t)}\quad$
$\displaystyle\sum\nolimits_{k\in\mathcal{U}}B_{r,k}(t)f_{k}(t)/J_{k}$ (10)
subject to $\displaystyle a)\;0\leq
f_{k}\leq\min\left(f_{\max},B_{r,k}(t)J_{k}/\tau\right),\quad\forall k\;$
$\displaystyle b)\;\sum\nolimits_{k\in\mathcal{U}}f_{k}(t)\leq f_{\max},$

where, for efficiency purposes, we added the constraint in $(a)$ that prevents
each user to be allocated more frequency than the one needed to empty the
remote queue. Problem (10) is linear, and the optimal frequencies can be
iteratively found by assigning the whole available frequency to the user with
the highest ratio $B_{r,k}(t)/J_{k}$. If this leaves available frequency, the
left part is assigned to the subsequent users until draining the whole CPU
power of the server $f_{\max}$, or serving all users [7].

Algorithm 1 Dynamic optimization of RIS-assisted MEC

$V$, $\mathcal{U}=\\{1,\ldots,N\\}$ $N_{slots}$, $\tau$, $P_{k}^{max}$,
$B_{l,k}(0)$, $B_{r,k}(0)$, $J_{k}$, $\forall k\in\mathcal{U}$, $f_{\max}$,

for $t=1:N_{slots}$ do

step 1: Optimize $\\{\mathbf{Q}_{k}(t)\\}_{k}$ and $\mathbf{r}(t)$.

for $n=1:I_{\max}$ do

step 1.1:
$\mathbf{r}^{n+1}=P_{\Theta}(r_{n}-\rho\nabla_{\mathbf{r}}f(\mathbf{r}^{n},\\{\mathbf{Q}_{k}^{(n)}\\}_{k}))$

step 1.2:

for $k=1,\ldots,N$ do

a: If $B_{l,k}\leq B_{r,k}$, $\mathbf{Q}_{k}^{(n+1)}=\mathbf{0}_{K}$, else

b: Update optimal $\\{\mathbf{Q}_{k}^{(n+1)}\\}_{k}$ with Algorithm 2

end for

end for

step 2: Optimize $\\{f_{k}(t)\\}_{k}$ as in Section III-B

step 3: Compute $R_{k}(t),\forall k\in\mathcal{U}$ as in (3);

step 4: Update $B_{l,k}$ and $B_{r,k}$ as in (4) and (II-B), respectively.

end for

Algorithm 2 Uplink covariance optimization for user $k$ [14]

step 1: Compute
$\mathbf{H}_{k}^{H}\mathbf{H}_{k}=\mathbf{U}^{H}\mathbf{\Sigma}\mathbf{U}$,
with $\mathbf{\Sigma}$ a diagonal matrix with non-negative elements
$\sigma_{i},i=1,...,K$

step 2: Check if $\sum_{i=1}^{K}\max\left(0,\frac{\tau
W_{k}(B_{l,k}-B_{r,k})}{V}-\frac{1}{\sigma_{i}}\right)\leq P_{k}^{\max}$
holds. If yes, then let $\mu^{*}=0$

and $\lambda^{*}_{i}=\max\left(0,\frac{\tau
W_{k}(B_{l,k}-B_{r,k})}{V}-\frac{1}{\sigma_{i}}\right)$, else,

step 3: Take all $\sigma_{i},i=1,\ldots,K$ in a decreasing order, i.e. as,
$\sigma_{d(1)}>\sigma_{d(2)}>\ldots>\sigma_{d(K)}$.

step 4:

Start with $S_{0}=0$.

for $i=1:K$ do

Let $S_{i}=S_{i-1}+\frac{1}{\sigma_{d(i)}}$ and
$\mu^{*}=\frac{i}{S_{i}+P}-\frac{V}{\tau W_{k}(B_{l,k}-B_{r,k})}$. If
$\mu^{*}\geq 0$, $\frac{1}{\mu^{*}+\frac{V}{\tau
W_{k}(B_{l,k}-B_{r,k})}}-\frac{1}{\sigma_{d(i)}}\geq 0$, and
$\frac{1}{\mu^{*}+\frac{V}{\tau
W_{k}(B_{l,k}-B_{r,k})}}-\frac{1}{\sigma_{d(i+1)}}\leq 0$, then stop the loop,
otherwise, move to the next iteration.

end for

step 5: Let
$\displaystyle\lambda^{*}_{i}=\max\left[0,\frac{1}{\mu^{*}+\frac{V}{\tau
W_{k}(B_{l,k}-B_{r,k})}}-\frac{1}{\sigma_{i}}\right]$.

step 6: $\mathbf{Q}_{k}^{*}=\mathbf{U}^{H}\mathbf{\Lambda}^{*}\mathbf{U}$,
with $\mathbf{\Lambda}^{*}$ diagonal matrix with entries $\lambda^{*}_{i}$

##  IV Numerical results

In this section, numerical results are provided to assess the performance of
our strategy. We consider a scenario with $N=6$ users aiming to offload their
tasks to a MEH collocated at the AP serving the users. Each user is assigned
an equal portion of the total bandwidth $B=1$ MHz, while the noise power
spectral density is set to $N_{0}=-174$ dBm/Hz. The slot duration is set to
$\tau=10$ ms. The arrival rate is $1$ Mbps with Poisson distribution, for all
users. The maximum available CPU cycle frequency is $f_{\max}=4.5$ GHz and
$J_{k}=500$ $\forall k$ (cf. (II-B)). All channels (cf. (2)) are generated for
a typical mmWave operating frequency, $f=28$ GHz, as in [15], with: $K=4$,
$N_{a}=4$, and $M=64$. The maximum transmit power for a single user $k$ is set
to $P_{k}^{\max}=100$ mW. In the sequel, for the sake of comparison, we
consider both scenarios with and without the RIS. Also, we assume two
different degrees of channel knowledge: i) Alg. 1: instantaneous knowledge of
$\beta_{k,r}(t)$ and $\beta_{k,a}(t)$ (cf. 2) is assumed, and Alg. 1 is used
for radio resource allocation; ii) Alg. 1, statistical: only a statistical
knowledge of the blockage, i.e. the blocking probabilities $p_{k,a}$ and
$p_{k,r}$ is assumed. In this case, Algorithm 1 is used, but $\beta_{k,r}(t)$
and $\beta_{k,a}(t)$ are replaced by $p_{k,a}$ and $p_{k,r}$ in (2), for the
optimization. Obviously, the data rate experienced by each user is computed
with the true channel in (2). Furthermore, for all cases, we consider also the
case in which the RIS phase shifts are randomly selected; whereas, for Alg. 1,
we also consider the case in which, after step 1.1 of Algorithm 1, the RIS
phase shifts are quantized with $2$ bits, which is a practical constraint of
RIS implementation [7].

Figure 1: Delay-power trade-off Figure 2: Avg. transmit power vs. AP blocking
probability

As a first result, in Fig. 1, we show the trade-off between average E2E delay
and transmit power, for different blocking probabilities $p_{k,a}$ on the
direct link. We specifically plot the results for $p_{k,a}=0$ and
$p_{k,a}=0.5$111For $p_{k,a}=0$, $\beta_{k,a}(t)=1,\forall k,t$, so that only
Alg. 1 is shown, obtained by increasing the Lyapunov trade-off parameter $V$
from right to left. For all curves we can notice how, by increasing $V$, the
system average transmit power decreases while the average service delay
increases. Also, all scenarios with the optimized RIS outperform the scenario
without the RIS, although with negligible gain for the case without blocking.
This suggests that the benefits of the RIS are more significant in case of
high blocking probability of the direct link. Also, the Alg. 1, statistical
strategy performs better than the non RIS case, except for the random phase
shifts case. Finally, for Alg. 1, it can be noticed that a 2-bit quantization
of RIS phase shifts yields parallel performance as the ideal case (i.e.,
continuous) with a very negligible gap, thus suggesting that our method can be
exploited also for practical RIS optimization.
To further highlight the previously mentioned remarks, we illustrate, through
Fig. 2, the gain in terms of average transmit power of each strategy with
respect to the non RIS case, as a function of the direct link
blocking222Problem (6) is not feasible for $p_{k,a}>0.7$ without the RIS, for
a fixed E2E delay bound of $150$ ms, obtained by tuning the trade-off
parameter $V$. As expected, as the blocking probability increases, the gain
notably increases with the Alg. 1 strategy, (up to 10 dB for $p_{k,a}=0.7$),
also with quantized phases. Conversely, the gain of Alg. 1, statistical is
visible only for higher blocking probabilities, due to the fact that, in this
case, the channel knowledge is well-matched to the real channel states.
Eventually, this implies that unreliable blocking knowledge is critical for
the performance. Optimizing the RIS through step $1.1$ of Algorithm 1 leads to
a better exploitation of the indirect path. However, for the random phase
case, it can be noticed that no gain is achieved (less than 2 dB in the best
case), which is obvious since we have no control on the RIS. More
specifically, it can be noticed that for $p_{k,a}$ around $0.5$, the channel
knowledge is completely mismatched. Instead, at high blocking probabilities
$p_{k,a}\geq 0.6$, the mismatch is reduced and higher gain is achieved.
Overall, we can conclude that the use of an RIS is prominent to satisfy a
reliable MEC-based task offloading in case of bad conditions of the direct
link, i.e., higher $p_{k,a}$, for all strategies. More specifically, the
benefit of the RIS starts to be noticed with a lower $p_{k,a}$ for the best
strategy, while it becomes more visible with higher $p_{k,a}$ for the worst
strategy.

##  V Conclusion

In this work, we have explored the effectiveness of using an RIS to counteract
the blocking (e.g. unreliability) when increasing communication frequency, in
the case of computation offloading services. To this end, we considered a
blocking aware framework through which we investigated the dynamic joint
optimization of computing resources and RIS-aided multi-user MIMO
communication parameters. Then, for dynamic configuration, we applied Lyapunov
optimization tools to transform a complex long-term optimization problem into
a per-slot deterministic problem that requires only instantaneous observations
of the context parameters and properly defined state variables. Numerical
results show the inherent gain of empowering MEC with RISs, while assuming
different degrees of channel knowledge along with different scenarios and
blocking conditions.

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# Sagnac-type neutron displacement-noise-free interferometeric gravitational-
wave detector

Yuki Kawasaki  Department of Physics, Nagoya University, Nagoya, Aichi
464-8602, Japan;  Shoki Iwaguchi  Department of Physics, Nagoya University,
Nagoya, Aichi 464-8602, Japan;  Tomohiro Ishikawa  Department of Physics,
Nagoya University, Nagoya, Aichi 464-8602, Japan;  Atsushi Nishizawa  Research
Center for the Early Universe (RESCEU), Graduate School of Science, The
University of Tokyo, Tokyo 113-0033, Japan;  Masaaki Kitaguchi  The Kobayashi-
Maskawa Institute for the Origin of Particles and the Universe, Nagoya
University, Nagoya, Aichi 464-8602, Japan;  Department of Physics, Nagoya
University, Nagoya, Aichi 464-8602, Japan;  Yutaka Yamagata  RIKEN Center for
Advance Photonics, RIKEN, Hirosawa 2-1, Wako, Saitama 351-0198, Japan;  Yanbei
Chen  Theoretical Astrophysics 350-17, California Institute of Technology,
Pasadena, California 91125, USA;  Bin Wu  Department of Physics, Nagoya
University, Nagoya, Aichi 464-8602, Japan;  Ryuma Shimizu  Department of
Physics, Nagoya University, Nagoya, Aichi 464-8602, Japan;  Kurumi Umemura
Department of Physics, Nagoya University, Nagoya, Aichi 464-8602, Japan;
Kenji Tsuji  Department of Physics, Nagoya University, Nagoya, Aichi 464-8602,
Japan;  Hirohiko Shimizu  Department of Physics, Nagoya University, Nagoya,
Aichi 464-8602, Japan;  The Kobayashi-Maskawa Institute for the Origin of
Particles and the Universe, Nagoya University, Nagoya, Aichi 464-8602, Japan;
Yuta Michimura  LIGO Laboratory, California Institute of Technology, Pasadena,
California 91125, USA;  Research Center for the Early Universe (RESCEU),
Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan;
Kazuhiro Kobayashi  Equipment Development Support Section, Technical Center of
Nagoya University, Nagoya, Aichi 464-8602, Japan;  Takafumi Onishi  Equipment
Development Support Section, Technical Center of Nagoya University, Nagoya,
Aichi 464-8602, Japan;  Seiji Kawamura  Department of Physics, Nagoya
University, Nagoya, Aichi 464-8602, Japan;  The Kobayashi-Maskawa Institute
for the Origin of Particles and the Universe, Nagoya University, Nagoya, Aichi
464-8602, Japan;

###### Abstract

The detection of low-frequency gravitational waves on Earth requires the
reduction of displacement noise, which dominates the low-frequency band. One
method to cancel test mass displacement noise is a neutron displacement-noise-
free interferometer (DFI). This paper proposes a new neutron DFI
configuration, a Sagnac-type neutron DFI, which uses a Sagnac interferometer
in place of the Mach-Zehnder interferometer. We demonstrate that a sensitivity
of the Sagnac-type neutron DFI is higher than that of a conventional neutron
DFI with the same interferometer scale. This configuration is particularly
significant for neutron DFIs with limited space for construction and limited
flux from available neutron sources.
Keywords: Gravitational wave; Neutron interferometer; Displacement-noise free
interferometer; Sagnac interferometer

##  1 Introduction

Since 2015, when LIGO first detected gravitational waves (GWs), numerous GW
events have been observed [1, 2, 3]. The frequencies of GWs depend on their
sources. For example, GWs in the low-frequency band around 1 Hz include
important physical phenomena such as “primordial gravitational waves (PGWs)”
[4]. However, the sensitivity of GW detectors in the low frequency band is
limited by displacement noise. This noise arises from the fluctuating
displacement of the detector’s test masses, due to thermal noise, noise from
the suspension system, and seismic noise. One way to minimize this
displacement noise is to send GW detectors up into space, as for LISA [5, 6]
and DECIGO [7, 8]. Space-based GW detectors can reduce some displacement
noise, such as suspension system noise, and ground vibration noise by floating
the test mass in space. Cooling mirrors and suspension systems in space can
also reduce thermal noise. However, space-based detectors require a
significant amount of time, effort, and cost. The sensitivity in the low-
frequency band also is limited by radiation pressure noise, which is one of
the sources of displacement noise. Thus, reducing displacement noise is also
essential.
Displacement-noise-free interferometery (DFI) is one of the methods for
canceling all displacement noise [9]. In a coordinate system in the
transverse-traceless gauge, GW signals and displacement noise can be
distinguished [10, 11]. Therefore, multiple interferometer signals can be used
to cancel displacement noise while preserving GW signals [12, 13]. This is the
principle behind a DFI.
DFI is most sensitive in a limited frequency band that corresponds to the
inverse of the time required for the beam to propagate between test masses.
For example, a DFI with several kilometers of arm length is mostly sensitive
near $10^{5}\,\mathrm{Hz}$, which is much higher than the 1 Hz range where
displacement noise is dominant. This is because of the high propagation speed
of the laser light. Therefore, the neutron DFI was devised, in which a neutron
beam is injected into the DFI. Neutrons have a finite mass, and their speed is
much slower than that of photons. Therefore, even a DFI with several
kilometers of arm length has good sensitivity around 1 Hz. Furthermore, the
ability to select the speed of the incident neutrons is an advantage for
optimizing the sensitivity of a DFI.
Various configurations of neutron DFI have already been developed [14, 15,
16]. In this paper, we present a neutron DFI configuration that uses a Sagnac-
type interferometer. In the conventional configuration (single, two-velocity),
shown in Figure 1, a single Mach-Zehnder interferometer is injected with
neutrons of two velocities from two relative directions [15]. After entering
the Mach-Zehnder interferometer, the neutrons are divided into right and left
paths by a beamsplitter (BS) and propagate along the two sides of the
interferometer under the influence of GWs. The photodetector (PD) observes the
interference state of the neutrons in the right and left paths. A Sagnac-type
neutron DFI is injected with neutrons of four velocities from one direction.
The PD observes the interference state of clockwise- and counterclockwise-
propagating neutrons. This configuration allows a Sagnac-type neutron DFI to
achieve higher sensitivity than a conventional neutron DFI of the same size
interferometer and with the same number of neutrons.

In this paper, we discuss the Sagnac-type neutron DFI in terms of
configuration and sensitivity. We discuss the configuration in Section 2, the
method of canceling displacement noise in Section 3, the GW signals from
neutron DFIs in Section 4, noise and sensitivity comparisons with the single,
two-velocity neutron DFI in Section 5, and consider characteristics of the
sensitivity curve in Section 6. Finally we present conclusions in Section 7.

Figure 1: Configuration of single two-velocity neutron DFI as viewed from the
z-axis direction. The black bars represent the test masses, mirrors and
beamsplitters. The red and blue arrows show the trajectories of the neutrons
with incident velocity $v_{1}$ and $v_{2}$ projected into horizomtal plane,
respectively. The solid arrows are the trajectories of neutrons incident
through beamsplitter (BS) A, while the dashed arrows represent the
trajectories of neutrons incident through BS B. Mirrors C and D are located on
the x-axis and BSs A and B are located on the y-axis. The origin in the xy-
plane represents the center of the square formed by the neutron orbits. Figure
2: Configuration of a Sagnac-type neutron DFI viewed along the z-axis. The
black bars represent test masses, mirrors and beamsplitter. The blue lines
show the trajectories of the neutrons. Mirrors C and D are placed on the
x-axis, while BS A and mirror B are placed on the y-axis. The origin in the
xy-plane is the center of the square drawn by the neutron orbit. The origin on
the z-axis is the height at which a neutron trajectory intersects the plane of
BS A.

##  2 Configuration of a Sagnac-type neutron DFI

The configuration of a Sagnac-type interferometer is shown in Figure 2 from a
bird’s-eye view (xy plane) and in Figure 3 from the side view (yz plane).
Gravity acts in the z direction.The propagation of the neutron trajectory in
the xy plane is a square with a side length of $L$. BS A and mirrors B, C, and
D are placed at the vertices of the square. Neutrons incident on A are divided
into two directions and propagate in the interferometer in clockwise and
counterclockwise orbits. Neutrons with four different velocities $v_{i}$
($i=1,2,3,4$) are injected into the Sagnac-type neutron DFI. The horizontal
and vertical velocities of neutrons with incident velocity $v_{i}$ ($n_{i}$)
are denoted by $v_{i}^{\mathrm{h}}$ and $v_{i}^{\mathrm{v}}$, respectively. As
shown in Figure 3, $n_{i}$ hit mirrors $\mathrm{C}_{i}$ and $\mathrm{D}_{i}$
after a time $T_{i}(=L/v_{i}^{\mathrm{h}})$ from the incident, and at mirror
$\mathrm{B}_{i}$ after another $T_{i}$. Mirror $\mathrm{B}_{i}$ is angled in
the xz plane so that it is perpendicular to the trajectory of $n_{i}$ in the
yz plane. Then, only the x-component of the neutron velocity is preserved, and
the neutron is reflected back. After reflection at mirror $\mathrm{B}_{i}$,
both clockwise and counterclockwise orbits are symmetrical about the y-axis.
Then, $2T_{i}$ after the reflection at mirror $\mathrm{B}_{i}$ the clockwise
neutrons pass through BS A, and the counterclockwise neutrons are reflected at
BS A. Their interference states are observed by the PD. In this paper, we set
$t=0$ to be the time when the clockwise and counterclockwise neutrons are
simultaneously reflected by mirror $\mathrm{B}_{i}$.

Figure 3: Pathway of neutrons in the yz plane. Rectangles and squares indicate
mirrors and beamsplitters, respectively. Mirrors $\mathrm{B}_{i}$ (i = 1 to 4)
are angled in the yz-plane to be perpendicular to the trajectories of $n_{i}$.
The height at which all neutrons hit the BS A is $0\,\mathrm{m}$

###  2.1 Neutron trajectory

Let us consider the trajectory of $n_{i}$ taking into account the effect of
falling due to gravity. Here, we ignore the mirror displacement temporarily
and focus only on the neutrons trajectories. The coordinate at which the
neutrons collide with A is denoted by

$\bm{x}_{i\mathrm{A}}=(0,\frac{L}{\sqrt{2}},0).$ (1)

The velocity of the neutrons incident on A is denoted by

$\bm{v}_{i}=(\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},-\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},v_{i}^{\mathrm{v}}).$
(2)

After a time interval $T_{i}$ elapses after the neutrons’ incidence on A, the
neutrons collide with mirrors C and D. The velocities of the neutrons as they
propagate from A to C and from A to D are given by

$\displaystyle\bm{v}_{i\mathrm{AC}}(t)$
$\displaystyle=(-\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},-\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},v_{i}^{\mathrm{v}}-g(t+2T_{i}))~{}(-2T_{i}\leq
t\leq-T_{i}),$ (3) $\displaystyle\bm{v}_{i\mathrm{AD}}(t)$
$\displaystyle=(\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},-\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},v_{i}^{\mathrm{v}}-g(t+2T_{i}))~{}(-2T_{i}\leq
t\leq-T_{i}).$ (4)

The coordinates at where the neutrons impact mirrors C and D are

$\displaystyle\bm{x}_{i\mathrm{C}}$
$\displaystyle=(-\frac{L}{\sqrt{2}},0,v_{i}^{\mathrm{v}}T_{i}-\frac{1}{2}gT^{2}_{i}),$
(5) $\displaystyle\bm{x}_{i\mathrm{D}}$
$\displaystyle=(\frac{L}{\sqrt{2}},0,v_{i}^{\mathrm{v}}T_{i}-\frac{1}{2}gT^{2}_{i}).$
(6)

The velocities of the neutrons as they propagate from C to B and from D to B
and the coordinate where they impact mirror B are given by

$\displaystyle\bm{v}_{i\mathrm{CB}}(t)$
$\displaystyle=(\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},-\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},v_{i}^{\mathrm{v}}-g(t+2T_{i}))~{}(-T_{i}<t\leq
0),$ (7) $\displaystyle\bm{v}_{i\mathrm{DB}}(t)$
$\displaystyle=(-\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},-\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},v_{i}^{\mathrm{v}}-g(t+2T_{i}))~{}(-T_{i}<t\leq
0),$ (8) $\displaystyle\bm{x}_{i\mathrm{B}}$
$\displaystyle=(0,-\frac{L}{\sqrt{2}},2v_{i}^{\mathrm{v}}T_{i}-\frac{1}{2}g(2T_{i})^{2}).$
(9)

After reflection at mirror B, oriented in the yz plane to be perpendicular to
the trajectories of the neutrons, the clockwise-propagating neutrons propagate
from B to C and from C to A. The counterclockwise-propagating neutrons
propagate from B to D and from D to A. The velocities of the neutrons are
given by

$\displaystyle\bm{v}_{i\mathrm{BD}}(t)$
$\displaystyle=(\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},-v_{i}^{\mathrm{v}}-g(t-2T_{i}))~{}(0<t\leq
T_{i}),$ (10) $\displaystyle\bm{v}_{i\mathrm{DA}}(t)$
$\displaystyle=(-\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},-v_{i}^{\mathrm{v}}-g(t-2T_{i}))~{}(T_{i}<t\leq
2T_{i}),$ (11) $\displaystyle\bm{v}_{i\mathrm{BC}}(t)$
$\displaystyle=(-\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},-v_{i}^{\mathrm{v}}-g(t-2T_{i}))~{}(0<t\leq
T_{i}),$ (12) $\displaystyle\bm{v}_{i\mathrm{CA}}(t)$
$\displaystyle=(\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},-v_{i}^{\mathrm{v}}-g(t-2T_{i}))~{}(T_{i}<t\leq
2T_{i}).$ (13)

The $n_{i}$ propagate through this trajectory. The velocities of clockwise-
and counterclockwise-propagating neutrons $\bm{v}_{i}^{l}$ $(l=\mathrm{c,cc})$
can be defined as

$\bm{v}_{i}^{\mathrm{c}}(t)=\left\\{\begin{array}[]{l}\bm{v}_{i\mathrm{AD}}(t)~{}(-2T_{i}\leq
t\leq-T_{i})\\\ \bm{v}_{i\mathrm{DB}}(t)~{}(-T_{i}<t\leq 0)\\\
\bm{v}_{i\mathrm{BC}}(t)~{}(0<t\leq T_{i})\\\
\bm{v}_{i\mathrm{CA}}(t),~{}(T_{i}<t\leq 2T_{i})\\\ \end{array}\right.$ (14)
$\bm{v}_{i}^{\mathrm{cc}}(t)=\left\\{\begin{array}[]{l}\bm{v}_{i\mathrm{AC}}(t)~{}(-2T_{i}\leq
t\leq-T_{i})\\\ \bm{v}_{i\mathrm{CB}}(t)~{}(-T_{i}<t\leq 0)\\\
\bm{v}_{i\mathrm{BD}}(t)~{}(0<t\leq T_{i})\\\
\bm{v}_{i\mathrm{DA}}(t),~{}(T_{i}<t\leq 2T_{i})\\\ \end{array}\right.$ (15)

###  2.2 Neutron pathway for DFI realization

The realization DFI necessitates combining pairs of neutron signals that
contain the displacement noise from the same test mass. This is because
displacement noise from different points, even on the same test mass, are not
correlated. To cancel the displacement noise at a point on C(D), two neutron
signals that hit the same point on C(D) are required. Similarly, to cancel the
displacement noise at a point on A, we need two signals that reflect at the
same point on A, with the displacement noise at a point on C(D) already
canceled. To summarize, $n_{1}$ and $n_{2}$ and $n_{3}$ and $n_{4}$ must
impact the same point on C(D), and all four neutrons must impact the same
point on A. We determine $v_{i}^{\mathrm{h}}$ and $v_{i}^{\mathrm{v}}$ for
each neutron group, $n_{i}$, to satisfy these conditions using the following
procedure:
1\. Set four horizontal incident neutron velocities $v_{i}^{\mathrm{h}}$
($i=1~{}\text{to}~{}4$).
2\. Set $v_{2}^{\mathrm{h}}$ as the slowest horizontal velocity and
$v_{4}^{\mathrm{h}}$ as the fastest horizontal velocity.
3\. Determine $v_{2}^{\mathrm{v}},v_{4}^{\mathrm{v}}$ such that $v_{2}$ and
$v_{4}$ are within the range of neutron velocities that can be injected. In
this paper, we set the slowest neutron velocity as 75 m/s and the fastest
neutron velocity as 100 m/s. (This is the range of neutron velocities for
which comparable fluxes can be expected.)
4\. Calculate $z_{\mathrm{C}i}$, the height at which $n_{i}$ impact mirror
$\mathrm{C}_{i}$ ($\mathrm{D}_{i}$) as

$z_{\mathrm{C}i}=v_{i}^{\mathrm{v}}T_{i}-\frac{1}{2}gT_{i}^{2},$ (16)

and determine $v_{1}^{\mathrm{v}}$ and $v_{3}^{\mathrm{v}}$ to satisfy
$z_{\mathrm{C}1}=z_{\mathrm{C}2}$ and $z_{\mathrm{C}3}=z_{\mathrm{C}4}$. Note
that the height at which all neutrons impact mirror A is $0\,\mathrm{m}$.
Figure 4 shows the neutron pathways that satisfy the conditions for the
neutron velocities.

Figure 4: Neutron pathways for complete displacement noise cancelation. Each
colored line corresponds to the trajectory of one group of neutrons, $n_{i}$.
All neutrons enter the interferometer through the same point on A. The
neutrons represented by the red and blue lines impact the same point,
$\mathrm{C}_{1,2}$, while the neutrons represented by the yellow and green
lines impact the same points, $\mathrm{C}_{3,4}$. All neutrons are then
reflected by the mirrors $\mathrm{B}_{i}$ and follow trajectories that are
symmetric about the y-axis.

##  3 Method of canceling displacement noise

###  3.1 Displacement noise in the time domain

First, let us consider only the displacement noise experienced by neutrons,
$n_{i}$. When considering the displacement noise that the neutrons receive
from mirrors $\mathrm{B}_{i}$, $\mathrm{C}_{i}$, and $\mathrm{D}_{i}$, we omit
the subscripts for the mirrors B, C, and D. We set $\bm{d}_{j}$ as the
displacement of test mass $j$ ($j=$A, B, C, D) in the direction normal to its
surface. The magnitude of the phase change of the neutrons,
$\phi_{ij}^{\mathrm{dis}}(t)$ caused by $\bm{d}_{j}$ is expressed in terms of
the wavenumber of the neutrons, $\bm{k}_{i}$ as

$\phi_{ij}^{\mathrm{dis}}(t)=2\bm{k}_{i}^{l}(t)\cdot\bm{d}_{j}(t),~{}~{}\bm{k}_{i}^{l}=\frac{m}{\hbar}\bm{v}_{i}^{l}(t).$
(17)

Here, $m$ is the mass of the neutrons, and $\hbar$ is Dirac’s constant. Table
1 shows the displacement noise that the clockwise and counterclockwise-
propagating neutrons experience. Note that $t=0$ corresponds to the time when
the neutrons are simultaneously reflected by mirror B. Also, since clockwise-
propagating neutrons traverse BS A twice, they do not receive displacement
noise from BS A.

Table 1: Phase changes that the neutrons groups, $n_{i}$, receive. Test mass | A | C | B | D | A
---|---|---|---|---|---
clockwise $\phi_{i}^{\mathrm{c},\mathrm{dis}}$ | 0 | $\phi_{i\mathrm{C}}^{\mathrm{dis}}(T_{i})$ | $\phi_{i\mathrm{B}}^{\mathrm{dis}}(0)$ | $\phi_{i\mathrm{D}}^{\mathrm{dis}}(-T_{i})$ | 0
counterclockwise $\phi_{i}^{\mathrm{cc},\mathrm{dis}}$ | $\phi_{i\mathrm{A}}^{\mathrm{dis}}(-2T_{i})$ | $\phi_{i\mathrm{C}}^{\mathrm{dis}}(-T_{i})$ | $\phi_{i\mathrm{B}}^{\mathrm{dis}}(0)$ | $\phi_{i\mathrm{D}}^{\mathrm{dis}}(T_{i})$ | $\phi_{i\mathrm{A}}^{\mathrm{dis}}(2T_{i})$

The total phase change of clockwise- and counterclockwise-propagating $n_{i}$
is given by

$\phi_{i}^{\mathrm{c},\mathrm{dis}}=\phi_{i\mathrm{C}}^{\mathrm{dis}}(T_{i})+\phi_{i\mathrm{B}}^{\mathrm{dis}}(0)+\phi_{i\mathrm{D}}^{\mathrm{dis}}(-T_{i}),$
(18)
$\phi_{i}^{\mathrm{cc},\mathrm{dis}}=\phi_{i\mathrm{A}}^{\mathrm{dis}}(-2T_{i})+\phi_{i\mathrm{C}}^{\mathrm{dis}}(-T_{i})+\phi_{i\mathrm{B}}^{\mathrm{dis}}(0)+\phi_{i\mathrm{D}}^{\mathrm{dis}}(T_{i})+\phi_{i\mathrm{A}}^{\mathrm{dis}}(2T_{i}).$
(19)

The phase change of the interfered neutrons at the PD is given by

$\displaystyle\phi^{\mathrm{dis}}_{i}$
$\displaystyle\equiv\phi_{i}^{\mathrm{c}}-\phi_{i}^{\mathrm{cc}}$ (20)
$\displaystyle=-[\phi^{\mathrm{dis}}_{i\mathrm{A}}(2T_{i})+\phi^{\mathrm{dis}}_{i\mathrm{A}}(-2T_{i})]+[\phi^{\mathrm{dis}}_{i\mathrm{C}}(T_{i})-\phi^{\mathrm{dis}}_{i\mathrm{C}}(-T_{i})]+[\phi^{\mathrm{dis}}_{i\mathrm{D}}(-T_{i})-\phi^{\mathrm{dis}}_{i\mathrm{D}}(T_{i})].$

The clockwise- and counterclockwise-propagating neutrons impact the same point
on mirror B simultaneously, thus they receive identical displacement noise
from the motion of B. As a result, the signal $\phi_{i}$ after they interfere
contains no displacement noise from B. However, due to the fact that the
clockwise- and counterclockwise-propagating neutrons interact with A, C, and D
at different times, displacement noise cannot be fully eliminated.

###  3.2 Displacement noise in frequency domain

In order to cancel the displacement noise of A, C, and D, we must consider the
displacement noise in the frequency domain. In this paper, we define the
Fourier transform as follows

$f(t)\equiv\int F(\omega)e^{-i\omega t}d\omega.$ (21)

Note that lowercase letters represent variables in the time domain and
uppercase letters represent variables in the frequency domain. We can express
$\bm{d}_{j}(t)$ as

$\bm{d}_{j}(t)\equiv\int\bm{D}_{j}(\omega)e^{-i\omega t}d\omega,$ (22)

where $\bm{D}_{j}(\omega)$ is the complex amplitude of the mirror displacement
at frequency $\omega$. We can express $\phi_{ij}^{\mathrm{dis}}(t)$ in the
frequency domain as

$\Phi_{ij}^{\mathrm{dis}}(\omega)=2\bm{k}_{i}\cdot\bm{D}_{j}(\omega).$ (23)

Since the normals of the surfaces of test masses A, C, and D have only
x-components, Equation 17 can be rewritten using the unit vector in the x
direction $\bm{e_{x}}$ as

$\displaystyle\phi_{ij}^{\mathrm{dis}}(t)=\frac{2m}{\hbar}(\bm{v}_{i}(t)\cdot\bm{e_{x}})(\bm{d}_{j}(t)\cdot\bm{e_{x}})=\frac{2mv_{ix}}{\hbar}d_{j}(t),$
(24)

where $v_{ix}$ is the x component of $\bm{v}_{i}$. Although the z component of
$\bm{k}_{i}$ changes with time due to gravity, it does not affect
$\phi_{ij}^{\mathrm{dis}}(t)$, which is proportional to the inner product of
$\bm{k}_{i}$ and $\bm{D}_{j}$. We treat $\bm{k}_{i}$ as a time-invariant
variable because the calculation of $\Phi_{ij}(\omega)$ (in Equation 23) does
not require consideration of the time variation of $\bm{k}_{i}$. Considering
the velocity of the neutrons at the time they impact each test mass, the
$\omega$ component of Equation 20 is given by

$\displaystyle\begin{split}\Phi^{\mathrm{dis}}_{i}(\omega)&=-2\bm{k}_{i}\cdot[\bm{D}_{\mathrm{A}}(e^{-i2\omega
T_{i}}-e^{i2\omega T_{i}})\\\ &+\bm{D}_{\mathrm{C}}(e^{-i\omega
T_{i}}-e^{i\omega T_{i}})-\bm{D}_{\mathrm{D}}(e^{i\omega T_{i}}-e^{-i\omega
T_{i}})].\end{split}$ (25)

####  3.2.1 Method of cancelling displacement noise of mirrors C and D

First, we consider the displacement noise of mirrors C and D in the frequency
domain. Table 2 shows the displacement noise of C and D in the interferometer
signals of the neutrons.

Table 2: Displacement noise of the test masses C and D Test masses | C | D
---|---|---
clockwise $\Phi_{i}^{\mathrm{c}}$ | $2\bm{k}_{i}\cdot\bm{D}_{\mathrm{C}}(\omega)e^{-i\omega T_{i}}$ | $2\bm{k}_{i}\cdot\bm{D}_{\mathrm{D}}(\omega)e^{i\omega T_{i}}$
counterclockwise $\Phi_{i}^{\mathrm{cc}}$ | $2\bm{k}_{i}\cdot\bm{D}_{\mathrm{C}}(\omega)e^{i\omega T_{i}}$ | $2\bm{k}_{i}\cdot\bm{D}_{\mathrm{D}}(\omega)e^{-i\omega T_{i}}$
Interferometer Signal $\Phi_{i}$ | $2\bm{k}_{i}\cdot\bm{D}_{\mathrm{C}}(\omega)(e^{-i\omega T_{i}}-e^{i\omega T_{i}})$ | $2\bm{k}_{i}\cdot\bm{D}_{\mathrm{D}}(\omega)(e^{i\omega T_{i}}-e^{-i\omega T_{i}})$

$\Phi_{i}^{\mathrm{dis}}(\omega)$ contains the displacement noise of mirrors C
and D. In order to normalize the interferometer signals with different neutron
velocities, we divide $\Phi^{\mathrm{dis}}_{i}(\omega)$ by
$v_{i}^{\mathrm{h}}/\sqrt{2}c$ and we obtain

$\displaystyle\Phi^{\mathrm{dis}^{\prime}}_{i}(\omega)\equiv\frac{\sqrt{2}c}{v_{i}^{\mathrm{h}}}\Phi^{\mathrm{dis}}_{i}(\omega)$
$\displaystyle=-\frac{4mc}{\hbar}[D_{\mathrm{A}}(\omega)(-i\sin{2\omega
T_{i}})+D_{\mathrm{C}}(\omega)(-i\sin{\omega
T_{i}})+D_{\mathrm{D}}(\omega)(-i\sin{\omega T_{i}})].$ (26)

Here, we take into account $n_{1}$ and $n_{2}$. Since the ratio of the
magnitudes of the displacement noise caused by C and D in
${\Phi^{\mathrm{dis}^{\prime}}_{1}(\omega)}$ and
${\Phi^{\mathrm{dis}^{\prime}}_{2}(\omega)}$ is $\sin{\omega
T_{1}}:\sin{\omega T_{2}}$, they can be eliminated by the following signal
processing:

$\displaystyle\Phi^{\mathrm{dis}}_{12}(\omega)\equiv\alpha_{1}{\Phi^{\mathrm{dis}^{\prime}}_{1}(\omega)}-\alpha_{2}{\Phi^{\mathrm{dis}^{\prime}}_{2}(\omega)},$
(27) $\displaystyle\alpha_{1}=\sin{\omega T_{2}},\quad\alpha_{2}=\sin{\omega
T_{1}}.$ (28)

We can explain the fact that $\Phi_{12}^{\mathrm{dis}}(\omega)$ has no
displacement noise from C and D with a phasor diagram as shown in Figure 5.
The details of the calculations are as follows:

$\displaystyle\begin{split}\Phi^{\mathrm{dis}}_{12}(\omega)&=-\frac{4mc}{\hbar}[D_{\mathrm{A}}(-i\sin{\omega
T_{2}}\sin{2\omega T_{1}})+(D_{\mathrm{C}}+D_{\mathrm{D}})(-i\sin{\omega
T_{1}}\sin{\omega T_{2}})]\\\ &+\frac{4mc}{\hbar}[D_{\mathrm{A}}(-i\sin{\omega
T_{1}}\sin{2\omega T_{2}})+(D_{\mathrm{C}}+D_{\mathrm{D}})(-i\sin{\omega
T_{1}}\sin{\omega T_{2}})],\end{split}$
$\displaystyle=-i\frac{4mc}{\hbar}D_{\mathrm{A}}(\sin{\omega
T_{1}}\sin{2\omega T_{2}}-\sin{\omega T_{2}}\sin{2\omega T_{1}}).$ (29)

Similarly, we can obtain $\Phi^{\mathrm{dis}}_{34}(\omega)$, that has no
displacement noise from C and D as

$\displaystyle\Phi^{\mathrm{dis}}_{34}(\omega)$
$\displaystyle\equiv\alpha_{3}{\Phi^{\mathrm{dis}^{\prime}}_{3}(\omega)}-\alpha_{4}{\Phi^{\mathrm{dis}^{\prime}}_{4}(\omega)},$
$\displaystyle=-i\frac{4mc}{\hbar}D_{\mathrm{A}}(\sin{\omega
T_{3}}\sin{2\omega T_{4}}-\sin{\omega T_{4}}\sin{2\omega T_{3}}).$ (30)
$\displaystyle\alpha_{3}$ $\displaystyle=\sin{\omega
T_{4}},\quad\alpha_{4}=\sin{\omega T_{3}}.$ (31)

Figure 5: Phasor diagram that illustrates the cancelation of displacement
noise from mirrors C and D. Each arrow represents the displacement noise that
the neutrons receive with the length of the arrows indicating the amplitude of
the displacement noise, and the direction indicating the phase of the
displacement noise. The solid arrow represents the displacement noise of
mirror C, and the dashed arrow represents the displacement noise of mirror D.
The color of the arrows correspond to the source of the displacement noise.
The red line represents the displacement noise of mirror C or D in
$\Phi_{1}^{\mathrm{c},\mathrm{dis}}(\omega)$. The blue line represents the
displacement noise in $\Phi_{1}^{\mathrm{c},\mathrm{dis}}(\omega)$, while the
pink and purple lines represent the displacement noise in
$\Phi_{2}^{\mathrm{c},\mathrm{dis}}(\omega)$ and
$\Phi_{2}^{\mathrm{c},\mathrm{dis}}(\omega)$, respectively. The green and
yellow lines represent the displacement noise in
$\Phi_{1}^{\mathrm{dis}}(\omega)$ and $\Phi_{2}^{\mathrm{dis}}(\omega)$,
respectively. Finally, the black line represents the displacement noise at
$t=0$.

####  3.2.2 Method of canceling displacement noise of A

In this subsection, we will focus on the displacement noise of BS A in the
frequency domain. In the Sagnac-type neutron DFI configuration, the clockwise-
propagating neutrons always pass through A and are thus not affected by its
displacement. However, the counterclockwise neutrons accumulate the
displacement noise of A twice, at $t=-2T_{i}$ and $t=2T_{i}$. If A is
displaced by $\bm{d}_{\mathrm{A}}$ at $t=-2T_{i}$, the counterclockwise
neutron propagation distance increases, leading to an advance in the neutron
phase. Conversely, if A is displaced by $\bm{d}_{\mathrm{A}}$ at $t=2T_{i}$,
the counterclockwise neutron propagation distance decreases, resulting in a
delay in the neutron phase. As a result, the sign of the displacement noise of
A is opposite for $t=-2T_{i}$ and $t=2T_{i}$. The impact of the displacement
noise of A on $\Phi_{i}^{\mathrm{dis}}(\omega)$ can be expressed as

$\Phi_{i}^{\mathrm{dis}}(\omega)=-2\bm{k}_{i}\cdot\bm{D}_{\mathrm{A}}(\omega)(e^{-i2\omega
T_{i}}-e^{i2\omega T_{i}}).$ (32)

Similar to the method for canceling the displacement noise of mirrors C and D
in Equation 27, the displacement noise of A can be cancelled by combining two
signals containing the displacement noise of A. If we cancel the displacement
noise of A using $\Phi_{12}^{\mathrm{dis}}(\omega)$ and
$\Phi_{34}^{\mathrm{dis}}(\omega)$, which contain noise components from A
only, we can obtain a signal $\Phi_{\mathrm{SagnacDFI}}(\omega)$ that has all
displacement noise of the Sagnac-type DFI canceled:

$\Phi^{\mathrm{dis}}_{\mathrm{SagnacDFI}}(\omega)=\beta_{1}\Phi_{12}^{\mathrm{dis}}(\omega)-\beta_{2}\Phi_{34}^{\mathrm{dis}}(\omega)=0,$
(33)

where

$\displaystyle\beta_{1}=\sin 2\omega T_{3}\sin\omega T_{4}-\sin 2\omega
T_{4}\sin\omega T_{3}\mathrm{and}$ $\displaystyle\quad\beta_{2}=\sin 2\omega
T_{1}\sin\omega T_{2}-\sin 2\omega T_{2}\sin\omega T_{1}.$ (34)

####  3.2.3 Sagnac effect in Sagnac-type neutron DFI

In the Sagnac-type neutron DFI, neutrons undergo the Sagnac effect and
experience a phase change when they are detected after circling the
interferometer, if the interferometer rotates with a certain angular frequency
$\omega_{0}$. From the perspective of the inertial system, the rotation of the
interferometer can be expressed by the displacement of each test mass.
However, in the DFI, all displacement noise is canceled. Thus, the signals
obtained from a Sagnac-type neutron DFI, $\Phi_{\mathrm{SagnacDFI}}(\omega)$,
do not include the Sagnac effect.

##  4 Gravitational wave signal of neutron DFI

###  4.1 Gravitational wave signal in a Sagnac-type neutron DFI

Let us consider the Klein-Gordon equation for a particle with mass $m$,

$\displaystyle\left(-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}+\nabla^{2}-\frac{m^{2}c^{2}}{\hbar^{2}}\right)\phi(t,\bm{x})=0.$ (35)

The wave function at time $t$ and position $\bm{x}$ with wavenumber vector
$\bm{k}$ and angular frequency of neutrons $\Omega$ is

$\displaystyle\phi_{0}(t,\bm{x})=e^{i(\Omega t-\bm{k}\cdot\bm{x})}.$ (36)

Due to the effect of GWs, Equation 36 becomes

$\phi(t,\bm{x})=\phi_{0}(t,\bm{x})\left[1+i\phi^{\mathrm{gw}}((t,\bm{x})\right].\\\
$ (37)

In the TT gauge, if the GWs propagate through flat spacetime, we can write

$ds^{2}=-(cdt)^{2}+\left[\eta_{ab}+h_{ab}(t,\bm{x})\right]dx^{a}dx^{b}~{}(a,b=1,2,3).$
(38)

In this coordinate system, from Equation 35 we obtain

$\displaystyle\frac{1}{c^{2}}\left(\Omega+\frac{\partial\phi^{\mathrm{gw}}}{\partial
t}\right)^{2}-k_{a}k_{b}\left(1-h^{ab}\right)^{2}$ $\displaystyle=0.$ (39)

The leading order term in $h^{ab}$ is

$\displaystyle\frac{\partial\phi^{\mathrm{gw}}}{\partial t}$
$\displaystyle=-\frac{h^{ab}k_{a}k_{b}c^{2}}{2\Omega}$ (40)
$\displaystyle\approx-\frac{h^{ab}k_{a}k_{b}\hbar}{2m}.$ (41)

Here, assuming that the wavenumber vector is sufficiently small, we can make
the following approximation,

$\frac{\Omega^{2}}{c^{2}}\approx\frac{m^{2}c^{2}}{\hbar^{2}}.$ (42)

As mentioned in Section 2, $n_{i}$ propagate through the interferometer on the
trajectories represented by Equations 1-13. Here we reset the time, that is,
we set the time to be $t_{0}$ when the $n_{i}$ cross the surface of the
mirror. The wavenumber vectors of the neutrons during propagation are given by

$\displaystyle\bm{k}_{i\mathrm{AC}}(t)$
$\displaystyle=\frac{m}{\hbar}(-\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},-\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},v_{i}^{\mathrm{v}}-g(t-t_{0})),$
(43) $\displaystyle\bm{k}_{i\mathrm{AD}}(t)$
$\displaystyle=\frac{m}{\hbar}(\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},-\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},v_{i}^{\mathrm{v}}-g(t-t_{0}))~{},$
(44) $\displaystyle\bm{k}_{i\mathrm{CB}}(t)$
$\displaystyle=\frac{m}{\hbar}(\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},-\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},v_{i}^{\mathrm{v}}-g(t-t_{0}))~{},$
(45) $\displaystyle\bm{k}_{i\mathrm{DB}}(t)$
$\displaystyle=\frac{m}{\hbar}(-\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},-\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},v_{i}^{\mathrm{v}}-g(t-t_{0})),$
(46) $\displaystyle\bm{k}_{i\mathrm{BD}}(t)$
$\displaystyle=\frac{m}{\hbar}(\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},-v_{i}^{\mathrm{v}}-g(t-t_{0}-4T_{i})),$
(47) $\displaystyle\bm{k}_{i\mathrm{BC}}(t)$
$\displaystyle=\frac{m}{\hbar}(-\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},-v_{i}^{\mathrm{v}}-g(t-t_{0}-4T_{i})),$
(48) $\displaystyle\bm{k}_{i\mathrm{DA}}(t)$
$\displaystyle=\frac{m}{\hbar}(-\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},-v_{i}^{\mathrm{v}}-g(t-t_{0}-4T_{i})),\mathrm{and}$
(49) $\displaystyle\bm{k}_{i\mathrm{CA}}(t)$
$\displaystyle=\frac{m}{\hbar}(\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},\frac{1}{\sqrt{2}}v_{i}^{\mathrm{h}},-v_{i}^{\mathrm{v}}-g(t-t_{0}-4T_{i})).$
(50)

The phase change of $\phi^{\mathrm{gw}}_{i}$ caused by GWs is given by

$\dfrac{\partial\phi^{\mathrm{gw}}_{i}}{\partial
t}\approx-\frac{h^{ab}k_{ia}k_{ib}}{2m}\hbar.$ (51)

Therefore, the phase change induced by the GWs during propagation between
points A and C is

$\phi_{i\mathrm{DB}}^{\mathrm{gw}}(t)=-\frac{\hbar}{2m}\int_{t}^{t+T_{i}}h^{ab}(t^{\prime},\bm{x}_{\mathrm{DB}}(t^{\prime}))k_{i\mathrm{DB}a}(t^{\prime})k_{i\mathrm{DB}b}(t^{\prime})dt^{\prime}.$
(52)

Defining $H^{ab}(\omega)$ as the Fourier transform of the GWs at angular
frequency $\omega$,

$h^{ab}(t,\bm{x}(t))=\int H^{ab}(\omega)e^{-i\omega t}d\omega.$ (53)

If we expand $h^{ab}(t)$ at the position $\bm{x}(t)$, we obtain the second
term of order $\bm{k}_{\mathrm{gw}}\cdot\bm{x}(t)$. However, this second term
is negligible because the typical size of a Sagnac-type DFI is $x\sim
100\,\mathrm{m}$, and $k_{\mathrm{gw}}x\sim x/\lambda_{\mathrm{gw}}$ is much
smaller for GWs at 1Hz [14]. For simplicity, we ignore the
$\bm{x}(t)$-dependence of $H^{ab}(\omega)$. In this paper, we calculate the
response of a Sagnac-type neutron DFI to cross-mode GWs traveling in the
z-axis direction. The Fourier transform of the GW signal received by a neutron
with velocity $v_{i}$ is:

$\Phi_{i\mathrm{DB}}^{\mathrm{gw}}(\omega)\approx-\frac{\hbar}{2m}\\{P_{0}(\omega)k_{i\mathrm{DB}a}k_{i\mathrm{DB}b}H^{ab}(\omega)\\},$
(54)

where

$\displaystyle P_{0}(\omega)$
$\displaystyle=-\frac{i}{\omega}\left(1-e^{-i\omega T_{i}}\right).$ (55)

When GWs of amplitude $h^{ab}$, polarization angle $\psi$, and angle of
incidence $(\theta,\varphi)$ arrive, the corresponding rotation matrix is

$\mathcal{R}=\begin{pmatrix}\cos{\varphi}&\sin{\varphi}&0\\\
-\sin{\varphi}&\cos{\varphi}&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}\cos{\theta}&0&-\sin{\theta}\\\ 0&1&0\\\
\sin{\theta}&0&\cos{\theta}\end{pmatrix}\begin{pmatrix}\cos{\psi}&\sin{\psi}&0\\\
-\sin{\psi}&\cos{\psi}&0\\\ 0&0&1\end{pmatrix},$ (56)

and the GW strain $h^{{}^{\prime}}_{ab}$ is

$h^{{}^{\prime}}_{ab}=\mathcal{R}_{ap}\mathcal{R}_{bq}h_{pq}=(\mathcal{R}h\mathcal{R}^{T})_{ab}.$
(57)

Similarly, the GW signals received by the clockwise- and counterclockwise-
propagating neutrons are given by

$\displaystyle\phi_{i}^{\mathrm{c},\mathrm{gw}}(t)=\phi_{\mathrm{AD}}^{\mathrm{gw}}(t-T_{i})+\phi_{\mathrm{DB}}^{\mathrm{gw}}(t)+\phi_{\mathrm{BC}}^{\mathrm{gw}}(t+T_{i})+\phi_{\mathrm{CA}}^{\mathrm{gw}}(t+2T_{i}),$
(58)
$\displaystyle\phi_{i}^{\mathrm{cc},\mathrm{gw}}(t)=\phi_{\mathrm{AC}}^{\mathrm{gw}}(t-T_{i})+\phi_{\mathrm{CB}}^{\mathrm{gw}}(t)+\phi_{\mathrm{BD}}^{\mathrm{gw}}(t+T_{i})+\phi_{\mathrm{DA}}^{\mathrm{gw}}(t+2T_{i}).$
(59)

The Fourier transforms of Equation 58 and 59 are

$\displaystyle\Phi_{i}^{\mathrm{c},\mathrm{gw}}(\omega)=\Phi_{\mathrm{AD}}^{\mathrm{gw}}(\omega)e^{i\omega
T_{i}}+\Phi_{\mathrm{DB}}^{\mathrm{gw}}(\omega)+\Phi_{\mathrm{BC}}^{\mathrm{gw}}(\omega)e^{-i\omega
T_{i}}+\Phi_{\mathrm{CA}}^{\mathrm{gw}}(\omega)e^{-2i\omega
T_{i}}\mathrm{and}$ (60)
$\displaystyle\Phi_{i}^{\mathrm{cc},\mathrm{gw}}(\omega)=\Phi_{\mathrm{AC}}^{\mathrm{gw}}(\omega)e^{i\omega
T_{i}}+\Phi_{\mathrm{CB}}^{\mathrm{gw}}(\omega)+\Phi_{\mathrm{BD}}^{\mathrm{gw}}(\omega)e^{-i\omega
T_{i}}+\Phi_{\mathrm{DA}}^{\mathrm{gw}}(\omega)e^{-2i\omega T_{i}}.$ (61)

Also, we define the clock noise $\phi^{\mathrm{c},\mathrm{clock}}_{i}(t)$ and
$\phi^{\mathrm{cc},\mathrm{clock}}_{i}(t)$ and their Fourier transforms
$\Phi^{\mathrm{c},\mathrm{clock}}_{i}(\omega)$ and
$\Phi^{\mathrm{cc},\mathrm{clock}}_{i}(\omega)$ as

$\displaystyle\phi^{\mathrm{c},\mathrm{clock}}_{i}(t)=\phi^{\mathrm{cc},\mathrm{clock}}_{i}(t)=\frac{mc^{2}}{\hbar}\left(\tau_{\mathrm{A}}(t+2T_{i})-\tau_{\mathrm{A}}(t-2T_{i})\right),$
(62)
$\displaystyle\Phi^{\mathrm{c},\mathrm{clock}}_{i}(\omega)=\Phi^{\mathrm{cc},\mathrm{clock}}_{i}(\omega)=\frac{mc^{2}}{\hbar}\left(e^{-2\omega
T_{i}}\tau_{\mathrm{A}}(\omega)-e^{2\omega
T_{i}}\tau_{\mathrm{A}}(\omega)\right),$ (63)

The phase of the $n_{i}$ are given by

$\displaystyle\Phi_{i}^{\mathrm{c}}(\omega)=\Phi_{i}^{\mathrm{c},\mathrm{gw}}(\omega)+\Phi_{i}^{\mathrm{c},\mathrm{dis}}(\omega)+\Phi_{i}^{\mathrm{c},\mathrm{clock}}(\omega),$
(64)
$\displaystyle\Phi_{i}^{\mathrm{cc}}(\omega)=\Phi_{i}^{\mathrm{cc},\mathrm{gw}}(\omega)+\Phi_{i}^{\mathrm{cc},\mathrm{dis}}(\omega)+\Phi_{i}^{\mathrm{cc},\mathrm{clock}}(\omega).$
(65)

The GW signal that the $n_{i}$ receive during propagation between A and C is
given by

$\displaystyle\Phi^{\mathrm{gw}}_{i\mathrm{DB}}$
$\displaystyle\approx-\frac{\hbar}{2m}\\{P_{0}(\omega)k_{\mathrm{DB}_{a}}k_{\mathrm{DB}_{b}}H^{ab}(\omega)\\},$
(66)
$\displaystyle=ih^{\mathrm{gw}}\frac{m}{2\hbar\omega}(v_{i}^{\mathrm{h}})^{2}(1-e^{-i\omega
T_{i}}),$

where $h^{\mathrm{gw}}$ is the GW amplitude. Because of the difference in
neutron propagation direction, GW signals that neutrons receive during
propagation along each side of the Sagnac-type neutron DFI are expressed as

$\displaystyle\Phi^{\mathrm{gw}}_{i\mathrm{AD}}=-\Phi^{\mathrm{gw}}_{i\mathrm{DB}},\quad\Phi^{\mathrm{gw}}_{i\mathrm{BC}}=-\Phi^{\mathrm{gw}}_{i\mathrm{DB}},\quad\Phi^{\mathrm{gw}}_{i\mathrm{CA}}=\Phi^{\mathrm{gw}}_{i\mathrm{DB}}.$
(67)

Substituting Equation 66 and 67 into Equation 58,
$\Phi^{\mathrm{c},\mathrm{gw}}_{i}(\omega)$ can be written explicitly as

$\displaystyle\Phi^{\mathrm{c},\mathrm{gw}}_{i}(\omega)$
$\displaystyle=\Phi^{\mathrm{gw}}_{i\mathrm{DB}}(-e^{i\omega
T_{1}}+1-e^{-i\omega T_{1}}+e^{-2i\omega T_{1}})$
$\displaystyle=\Phi^{\mathrm{gw}}_{i\mathrm{DB}}[-(e^{i\omega
T_{i}}+e^{-i\omega T_{i}})+e^{-i\omega T_{i}}(e^{i\omega T_{i}}+e^{-i\omega
T_{i}})]$ $\displaystyle=\Phi^{\mathrm{gw}}_{i\mathrm{DB}}[2(-1+e^{-i\omega
T_{i}})\cos(\omega T_{i})]$
$\displaystyle=ih^{\mathrm{gw}}\frac{m}{\hbar\omega}(v_{i}^{\mathrm{h}})^{2}(1-e^{-i\omega
T_{i}})(-1+e^{-i\omega T_{i}})\cos{\omega T_{i}},$
$\displaystyle=ih^{\mathrm{gw}}\frac{m}{\hbar\omega}(v_{i}^{\mathrm{h}})^{2}e^{-i\omega
T_{i}}(2i\sin{\frac{\omega T_{i}}{2}})^{2}\cos{\omega T_{i}}$
$\displaystyle=-4ih^{\mathrm{gw}}\frac{m}{\hbar\omega}(v_{i}^{\mathrm{h}})^{2}e^{-i\omega
T_{i}}(1-\cos{\omega T_{i}})\cos{\omega T_{i}}.$ (68)

The sign of the GW signal from the z direction received by the neutrons
depends on the velocity of the neutrons in the xy-plane. The sign of the GW
signal received by neutrons traveling along the same path in the opposite
direction is opposite. Thus,
$\Phi^{\mathrm{c},\mathrm{gw}}_{i}(\omega)=-\Phi^{\mathrm{cc},\mathrm{gw}}_{i}(\omega)$.
By combining signals for displacement noise cancellation in
$\Phi_{i}^{\mathrm{c},\mathrm{gw}}(\omega)$ and
$\Phi_{i}^{\mathrm{cc},\mathrm{gw}}(\omega)$, the GW signal
$\Phi_{\mathrm{SagnacDFI}}^{\mathrm{gw}}$ contained in
$\Phi_{\mathrm{SagnacDFI}}$ is calculated as

$\displaystyle\Phi^{\mathrm{gw}}_{i}(\omega)=\Phi^{\mathrm{c},\mathrm{gw}}_{i}(\omega)-\Phi^{\mathrm{cc},\mathrm{gw}}_{i}(\omega)=2\Phi^{\mathrm{c},\mathrm{gw}}_{i}(\omega).$
(69)

Thus,

$\displaystyle\Phi^{\mathrm{gw}}_{12}(\omega)$
$\displaystyle=\alpha_{1}\frac{\sqrt{2}c}{v_{1}}\Phi^{\mathrm{gw}}_{1}(\omega)-\alpha_{2}\frac{\sqrt{2}c}{v_{2}}\Phi^{\mathrm{gw}}_{2}(\omega),\quad\Phi^{\mathrm{gw}}_{34}(\omega)=\alpha_{3}\frac{\sqrt{2}}{c}{v_{3}}\Phi^{\mathrm{gw}}_{3}(\omega)-\alpha_{4}\frac{\sqrt{2}}{c}{v_{4}}\Phi^{\mathrm{gw}}_{4}(\omega),$
(70)

and

$\displaystyle\begin{split}\Phi^{\mathrm{gw}}_{\mathrm{SagnacDFI}}(\omega)=\beta_{1}\Phi^{\mathrm{gw}}_{12}(\omega)-\beta_{2}\Phi^{\mathrm{gw}}_{34}(\omega).\end{split}$
(71)

###  4.2 Gravitational wave signal of a single, two-velocity neutron DFI

Similar to the GW signals in the Sagnac-type neutron DFI, the GW signals in
the single, two-velocity neutron DFI
$\Phi_{\mathrm{SingleDFI}}^{\mathrm{gw}}$, are obtained as follows:

$\displaystyle\Phi_{\mathrm{SingleDFI}}^{\mathrm{gw}}=ih^{\mathrm{gw}}\frac{m}{\hbar\omega}[\gamma_{1}(v_{1\mathrm{SingleDFI}}^{\mathrm{h}})^{2}(1-e^{-i\omega
T_{1}})^{2}-\gamma_{2}(v_{2\mathrm{SingleDFI}}^{\mathrm{h}})^{2}(1-e^{-i\omega
T_{2}})^{2}],$ (72)

where

$\displaystyle\gamma_{1}=\frac{\sqrt{2}c\sin{\omega
T_{2}}}{v_{1\mathrm{SingleDFI}}^{\mathrm{h}}}\mathrm{and}\quad\gamma_{2}=\frac{\sqrt{2}c\sin{\omega
T_{1}}}{v_{2\mathrm{SingleDFI}}^{\mathrm{h}}}.$ (73)

##  5 Sensitivity

DFI signal, as its name implies, is free from displacement noise. Therefore,
the sensitivity of a DFI to GW signals is limited by neutron shot noise. If
the flux of the neutrons $n_{i}$ is $F_{i}$, then their shot noise $N_{i}$ is
given by

$N_{i}=\frac{1}{\sqrt{F_{i}}}.$ (74)

In a neutron DFI, a signal without any displacement noise is obtained by
combining signals from neutrons with four velocities. Since
$N_{1},N_{2},N_{3}$, and $N_{4}$ are all independent of each other, the shot
noise $N_{\mathrm{SagnacDFI}}$ after DFI combination is given by

$N_{\mathrm{SagnacDFI}}=\sqrt{\left(\alpha_{1}\beta_{1}\frac{\sqrt{2}c}{v_{1}^{\mathrm{h}}}N_{1}\right)^{2}+\left(\alpha_{2}\beta_{1}\frac{\sqrt{2}c}{v_{2}^{\mathrm{h}}}N_{2}\right)^{2}+\left(\alpha_{3}\beta_{2}\frac{\sqrt{2}c}{v_{3}^{\mathrm{h}}}N_{3}\right)^{2}+\left(\alpha_{4}\beta_{2}\frac{\sqrt{2}c}{v_{4}^{\mathrm{h}}}N_{4}\right)^{2}}.$
(75)

Similarly, the shot noise of a single, two-velocity neutron DFI is given by
[15]

$\displaystyle
N_{\mathrm{SingleDFI}}=\sqrt{2(\gamma_{1}N_{1})^{2}+2(\gamma_{2}N_{2})^{2}}.$
(76)

Note that the factor of 2 in $\gamma_{1}N_{1}$ and $\gamma_{2}N_{2}$ comes
from the fact that a single, two-velocity neutron DFI uses two neutron beams
with velocities $v_{1}$ and $v_{2}$. In the GW detection, the GW signal is
expressed in the unit of $/\sqrt{\mathrm{Hz}}$ as,

$S_{\mathrm{DFI}}(\omega)=|\Phi_{\mathrm{DFI}}^{\mathrm{gw}}|\omega^{1/2}.$
(77)

Setting $S_{\mathrm{DFI}}/N_{\mathrm{DFI}}$ = 1, we obtain the amplitude
spectral density of shot noise limited sensitivity to GW amplitude
$h_{\mathrm{n}}$ of the Sagnac-type neutron DFI in $/\sqrt{\mathrm{Hz}}$.

$\displaystyle h_{\mathrm{n}}(\omega)$ $\displaystyle\equiv
H_{\mathrm{n}}(\omega)\omega^{1/2}$
$\displaystyle=\frac{N_{\mathrm{DFI}}(\omega)}{\Phi^{\mathrm{norm}}_{\mathrm{DFI}}(\omega)},$
(78)

where $\Phi^{\mathrm{norm}}_{\mathrm{DFI}}(\omega)$ is the GW signal
normalized by the GW amplitude $h^{\mathrm{gw}}$,

$\displaystyle\Phi^{\mathrm{norm}}_{\mathrm{DFI}}(\omega)\equiv\frac{S_{\mathrm{DFI}}(\omega)}{h^{\mathrm{gw}}}.$
(79)

In this paper, we optimize the sensitivity by varying the velocity of neutrons
incident on the DFI. We have calculated the sensitivity for various
combinations of neutron velocities in the range of $v_{i}=75\,\mathrm{m/s}$ to
$100\,\mathrm{m/s}$, where similar fluxes are expected. Note that the neutron
velocities were varied at $1\,\mathrm{m/s}$ intervals, and we assume a
constant neutron flux at all velocities ($F_{i}=10^{13}\,/\mathrm{s}$). Table
3 shows the most sensitive combinations of neutron velocities.

Table 3: Neutron Velocity i | $v_{i}\,\mathrm{m/s}$ | $v_{i}^{\mathrm{h}}\,\mathrm{m/s}$ | $v_{i}^{\mathrm{v}}\,\mathrm{m/s}$
---|---|---|---
1 | 93.31 | 87.00 | 33.74
2 | 75.00 | 69.00 | 29.39
3 | 77.04 | 77.00 | 2.59
4 | 100.00 | 100.00 | 0

Let us compare the sensitivity of a Sagnac-type neutron DFI and a single, two-
velocity neutron DFI with the same interferometer size ($L$ =
$100\,\mathrm{m}$), four-velocity neutron sources, and the same neutron flux.
These sensitivities are optimized for the incident neutron velocities. The
sensitivity of the single, two-velocity neutron DFI is calculated with
$v_{1\mathrm{SingleDFI}}=75\,\mathrm{m/s}$ and
$v_{2\mathrm{SingleDFI}}=100\,\mathrm{m/s}$.

Figure 6: Sensitivities of a Sagnac-type neutron DFI and a single two-velocity
neutron DFI.

Figure 6 shows that the sensitivity of the Sagnac-type neutron DFI is higher
than that of the single, two-velocity neutron DFI around $0.4\mathrm{Hz}$.
This is indicated by the dip in the Sagnac-type curve being lower than that of
the single, two-velocity curve at that frequency. The signal-to-noise ratio
(SNR) is defined as

$\displaystyle(\mathrm{SNR})^{2}\equiv
4\int^{f_{\mathrm{max}}}_{f_{\mathrm{min}}}\frac{|h^{\mathrm{gw}}(f)|^{2}}{h_{\mathrm{n}}(f)^{2}}df,$
(80)

where $f_{\mathrm{max}}=10\mathrm{Hz}$, $f_{\mathrm{min}}=10^{-2}\mathrm{Hz}$.
The ratio of SNR of the Sagnac-type neutron DFI to the single, two-velocity
DFI, $\mathrm{S}_{\mathrm{r}}$, for GW signals from binary stars proportional
to $f^{-\frac{7}{6}}$ [LABEL:] with arbitrary magnitude is

$\mathrm{S}_{\mathrm{r}}\equiv\frac{\mathrm{SNR}_{\mathrm{SagnacDFI}}}{\mathrm{SNR}_{\mathrm{SingleDFI}}}\approx
1.57.$ (81)

##  6 Discussion

First, let us consider the amount of phase change experienced by $n_{i}$ as
they propagate inside the interferometer due to GWs traveling in the z
direction. Newtrons, $n_{i}$ traverse one side of the interferometer in time,
$T_{i}$. When cross-mode GWs with a period of $2T_{i}$ pass the
interferometer, the neutrons undergo the maximum phase shift. In the Sagnac-
type neutron DFI, the neutrons are affected by GWs as they propagate through
four sides of the interferometer. In a single, two-velocity neutron DFI the
neutrons are affected by GWs only as they propagate through two sides of the
interferometer. The neutrons in the Sagnac-type neutron DFI experience twice
the phase change, $\Phi_{i}^{\mathrm{gw}}(\omega)$, compared to those in a
single two-velocity neutron DFI. Thus, the ratio of
$\Phi_{i}^{\mathrm{gw}}(\omega)$ to $N_{i}$ in the Sagnac-type neutron DFI is
twice that in a single two-velocity neutron DFI. Figure 6 shows the best
sensitivity, $h_{n}$, of the Sagnac-type neutron DFI is $5.9\times
10^{-19}\,/\sqrt{\mathrm{Hz}}$, and that of the single two-velocity neutron
DFI is $9.2\times 10^{-19}\,/\sqrt{\mathrm{Hz}}$. These magnitudes differ only
by a factor of about 1.6, due to a difference in signal processing. The
frequency (period) of the GWs that produce the largest phase change depends on
the $v_{i}^{\mathrm{h}}$ of the neutrons used to calculate the DFI signal.
Since several neutrons, $n_{i}$, (i = 1 to 4) are used, the DFI signal is
calculated by subtracting the neutron signals with different velocities from
each other, and a part of the GW signals is also subtracted. The Sagnac-type
neutron DFI with more subtractions of neutron signals loses more GW signals in
the signal processing than a single, two-velocity neutron DFI. Thus, the best
sensitivity $h_{n}$ of the Sagnac-type neutron DFI and that of the single,
two-velocity neutron DFI differ only by a factor of about 1.6.
Next, let us consider the peaks that appear in the Sagnac-type neutron DFI
sensitivity curve. These peaks have two origins. The first is that neutrons
traverse square orbits, and the second is the coefficients $\alpha_{i}$ and
$\beta_{i}$ used in signal processing. According to Equation 68,
$\Phi_{i}^{\mathrm{gw}}\propto(1-\cos{\omega T_{i}})\cos{\omega T_{i}}=0$ when
$\omega=\frac{2n\pi}{T_{i}}\quad\mathrm{or}\quad\frac{\pi}{T_{i}}(n-\frac{1}{2})(n=1,2,...)$.
This means that the sums of the GW signals with periods of $T_{i}/n$ or
$2T_{i}/(n-\frac{1}{2})$ ($n=1,2,...$) on the $n_{i}$ is zero. First, when
$n_{i}$ propagate along one side of the interferometer, cross-mode GWs with a
period of $T_{i}/n$ induce a positive (or negative) phase change and an equal
and opposite phase change. Second, the $n_{i}$ are not sensitive to GWs with a
period of $T_{i}/n$. Neutrons with $v_{i}^{\mathrm{h}}$ reflected from the
mirrors bend their propagation direction by 90 degrees every $T_{i}$. Cross-
mode GWs with a period of $2T_{i}/(n-\frac{1}{2})$ give a constant phase
change while the neutrons propagate along one side of the interferometer, and
an equal and opposite phase change while they propagate along the opposite.
Therefore, the round trip phase change induced by the cross-mode GWs with a
periods of $2T_{i}/(n-\frac{1}{2})$ is zero. Since the neutron signals
$\Phi_{i}(\omega)$ do not have a GW signal with a specific period (frequency),
the DFI is not sensitive to GWs at that frequency.

Next, let us focus on the coefficients used for signal processing $\alpha_{i}$
and $\beta_{i}$. When $\omega=\frac{n\pi}{T_{2}}$,
$\alpha_{1}\equiv\sin{\omega T_{2}}=0$. When
$\omega=\frac{n\pi}{T_{3}},\frac{n\pi}{T_{4}}$ or $\cos{\omega
T_{3}}-\cos{\omega T_{4}}=0$, $\beta_{1}\equiv\sin{2\omega T_{3}}\sin{\omega
T_{4}}-\sin{2\omega T_{4}}\sin{\omega T_{3}}=2\sin{\omega T_{3}}\sin{\omega
T_{4}}(\cos{\omega T_{3}}-\cos{\omega T_{4}})=0$. To summarize,
$\alpha_{1}\beta_{1}\Phi_{1}^{\mathrm{gw}}=0$ when

$\displaystyle\omega=\frac{2n\pi}{T_{1}},\frac{\pi}{T_{1}}(n-\frac{1}{2}),\frac{n\pi}{T_{2}},\frac{n\pi}{T_{3}},\frac{n\pi}{T_{4}}(n=1,2,...)\quad\mathrm{or}\quad\cos{\omega
T_{3}}-\cos{\omega T_{4}}=0.$ (82)

Similarly, $\alpha_{1}\beta_{1}\Phi_{2}^{\mathrm{gw}}=0$ when

$\displaystyle\omega=\frac{2n\pi}{T_{2}},\frac{\pi}{T_{2}}(n-\frac{1}{2}),\frac{n\pi}{T_{1}},\frac{n\pi}{T_{3}},\frac{n\pi}{T_{4}}(n=1,2,...)\quad\mathrm{or}\quad\cos{\omega
T_{3}}-\cos{\omega T_{4}}=0.$ (83)

Thus,
$\beta_{1}(\alpha_{1}\Phi_{1}^{\mathrm{gw}}-\alpha_{2}\Phi_{2}^{\mathrm{gw}})=0$
when

$\displaystyle\omega=\frac{2n\pi}{T_{1}},\frac{2n\pi}{T_{2}},\frac{n\pi}{T_{3}},\frac{n\pi}{T_{4}}(n=1,2,...)\quad\mathrm{or}\quad\cos{\omega
T_{3}}-\cos{\omega T_{4}}=0.$ (84)

The lowest frequencies that satisfy each of these conditions derived from
$\frac{1}{T_{1}},\frac{1}{T_{2}},\frac{1}{2T_{3}},\frac{1}{2T_{4}},$ and
$\cos{\omega T_{3}}-\cos{\omega T_{4}}=0$ are respectively,

$\displaystyle
f=0.870\,\mathrm{Hz},0.69\,\mathrm{Hz},0.385\,\mathrm{Hz},0.500\,\mathrm{Hz},\quad\mathrm{and}\quad
f\approx 0.435\,\mathrm{Hz}.$ (85)

Similarly,
$\beta_{2}(\alpha_{3}\Phi_{3}^{\mathrm{gw}}-\alpha_{4}\Phi_{4}^{\mathrm{gw}})=0$
when

$\displaystyle\omega=\frac{n\pi}{T_{1}},\frac{n\pi}{T_{2}},\frac{2n\pi}{T_{3}},\frac{2n\pi}{T_{4}}(n=1,2,...)\quad\mathrm{or}\quad\cos{\omega
T_{1}}-\cos{\omega T_{2}}=0.$ (86)

The lowest frequencies that satisfy each of these conditions derived from
$\frac{1}{2T_{1}},\frac{1}{2T_{2}},\frac{1}{T_{3}},\frac{1}{T_{4}},$ and
$\cos{\omega T_{1}}-\cos{\omega T_{2}}=0$ are respectively,

$\displaystyle
f=0.435\,\mathrm{Hz},0.345\,\mathrm{Hz}(=\frac{1}{2T_{2}}),0.770\,\mathrm{Hz},1.000\,\mathrm{Hz},\quad\mathrm{and}\quad
f\approx 0.385\,\mathrm{Hz}.$ (87)

According to Equation 85 and 87, $\Phi_{\mathrm{SagnacDFI}}^{\mathrm{gw}}=0$
when

$\displaystyle
f=\frac{n}{T_{1}},\frac{n}{T_{2}},\frac{n}{T_{3}},\frac{n}{T_{4}},(n=1,2,...)$
(88)

in addition, $\Phi_{\mathrm{SagnacDFI}}^{\mathrm{gw}}\approx 0$ at the
frequencies corresponding to ($\frac{1}{2T_{1}}$ and $\cos{\omega
T_{3}}-\cos{\omega T_{4}}=0$) and ($\frac{1}{2T_{3}}$ and $\cos{\omega
T_{1}}-\cos{\omega T_{2}}=0$), respectively,

$\displaystyle f\approx 0.435\,\mathrm{Hz}\quad\mathrm{and}\quad f\approx
0.385\,\mathrm{Hz}.$ (89)

Thus, the sensitivity curve of $\Phi_{\mathrm{DFI}}^{\mathrm{gw}}(\omega)$ in
Figure 6 has the peaks at the frequencies in Equation 88 and 89.

##  7 Conclusion

In this paper, we propose a Sagnac-type neutron DFI and have confirmed
analytically that DFI can be realized by injecting neutrons into a Sagnac-type
neutron interferometer considering the influence of gravity. We have also
shown that the sensitivity of the Sagnac-type neutron DFI is superior to that
of the conventional single, two-velocity DFI. When canceling displacement
noise with multiple signals, the location and time at which the neutrons
receive the displacement noise are important. In order to cancel the
displacement noise of all test masses with four neutron signals, we adjust the
neutrons’ velocities and injection angles to satisfy the following conditions:
1) The neutron trajectories are symmetrical about the y-axis. 2) The four
neutron groups are incident on the interferometer at the same point on BS A.
3) Neutrons, $\mathrm{neutrons}_{1}$ and $\mathrm{neutrons}_{2}$
($\mathrm{neutrons}_{3}$ and $\mathrm{neutrons}_{4}$) hit the same points on
$\mathrm{C}_{1}$ and $\mathrm{C}_{2}$ ($\mathrm{D}_{1}$ and $\mathrm{D}_{2}$)
in Figure 4, respectively. Additionally, we optimize the incident neutron
velocities to achieve the best sensitivity for both the Sagnac-type neutron
DFI and the single, two-velocity neutron DFI.

One of the technical challenges we face in realizing a Sagnac-type neutron DFI
is the neutron reflection angle. Currently, mirrors with a reflection angle of
only a few degrees have been created using current technology. However, there
is no fundamental principle that limits the reflection angle. Thus, it is
possible to develop a mirror with a large reflection angle, such as 45
degrees. Another practical issue that must be considered when demonstrating
neutron DFI experimentally is the accuracy of the instrument. The sensitivity
of neutron DFI is affected by factors such as the accuracy of the distance
between test masses, the reflectivity of the mirrors, and the accuracy of the
displacement noise cancellation. For these reasons, developing such a
technique is challenging, but we believe it is feasible. A more detailed study
of these technical issues will be the subject of future work.

## Acknowledgments

We would like to thank R. L. Savage for English editing. This work was
supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI
Grant Number JP19K21875. A. N. is supported by JSPS KAKENHI Grant Nos.
JP23K03408, JP23H00110, and JP23H04893.

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