# Towards An Optimal Solution to Place Bistatic Radars
for Belt Barrier Coverage with Minimum Cost

Tu N. Nguyen1, Bing-Hong Liu2, My T. Thai3, and Ivan Djordjevic4
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1Department of Computer Science, Kennesaw State University,
Marietta, GA 30060, USA
2Department of Electronic Engineering, National Kaohsiung University of
Science and Technology, Kaohsiung, Taiwan
3 Department of Computer and Information Science and Engineering, University
of Florida, Gainesville, FL 32612, USA
4 Department of Electrical and Computer Engineering, University of Arizona,
Tucson, AZ 85721, USA

###### Abstract

With the rapid growth of threats, sophistication and diversity in the manner
of intrusion, traditional belt barrier systems are now faced with a major
challenge of providing high and concrete coverage quality to expand the
guarding service market. Recent efforts aim at constructing a belt barrier by
deploying bistatic radar(s) on a specific line regardless of the limitation on
deployment locations, to keep the width of the barrier from going below a
specific threshold and the total bistatic radar placement cost is minimized,
referred to as the Minimum Cost Linear Placement (MCLP) problem. The existing
solutions are heuristic and their validity is tightly bound by the barrier
width parameter that these solutions only work for a fixed barrier width
value. In this work, we propose an optimal solution, referred to as the
Opt$\\_$MCLP, for the “open MCLP problem” that works for full range of the
barrier width. Through rigorous theoretical analysis and experimentation, we
demonstrate that the proposed algorithms perform well in terms of placement
cost reduction and barrier coverage guarantee.

###### Index Terms:

Bistatic radar deployment, barrier coverage, network optimization, sensor
networks.

##  I Introduction

Images of moats remind us of the system of guarding ancient castles and the
idea is still used today, yet at a higher level. With the development of
sensing and communication technology, instead of using moat, barriers of
sensors and radars, referred to as the bistatic radar systems, have been
building to guard not only critical places but also spaces and national
borders. Many applications (e.g., typically for military to detect targets and
intruders) use a bistatic radar system that comprises at least a radar signal
transmitters and at a radar signal least receiver111For ease of exposition,
hereafter referred to as transmitter(s) and the receiver(s), respectively.,
wherein the transmitter(s) and the receiver(s) are located in a different
location, to form the barrier coverage [1, 2, 3, 4, 5, 6].

In recent years, there are efforts in the existing literature to design
barrier coverage using radar [7, 8, 9, 10, 11, 12], wherein a radar uses radio
waves to detect an object by producing the radio waves and collecting the echo
signal reflected off from the target, giving information about the object’s
location and speed. Emerging research problems in recent years are how to
enhance the quality of barrier coverage and how to efficiently deploy bistatic
radar systems while meeting quality requirements. In terms of barrier coverage
quality, one of the important aspects that reflects the quality of bistatic
radar systems is the width of the barrier coverage area. Recent efforts aim at
constructing a linear belt barrier with pre-defined width by deploying
bistatic radar(s) on a specific line such that the total bistatic radar
placement cost is minimized, referred to as the Minimum Cost Linear Placement
(MCLP) problem [7]. The sensing model of bistatic radars (Cassini oval sensing
model [7]) is in fact very complex and the shape of sensing areas are,
therefore, varied according to the variation of distance between the
transmitter(s) and the receiver(s) in the bistatic radar system. The validity
of the most recent solutions [7, 8] is unfortunately tightly bound by the
“barrier width” parameter that these solutions only work for a pre-fixed
barrier width value. Thus, they fail to solve the MCLP problem with flexible
width ranges of barrier.

Specifically, the most recent solutions proposed for the MCLP problem [7, 8]
can only work for covering a belt-shaped area having width less than or equal
to $\frac{2\zeta_{max}}{\sqrt{3}}$, namely $0\leq
2\omega\leq\frac{2\zeta_{max}}{\sqrt{3}}$, where $2\omega$ denotes the width
of the belt-shaped area and $\zeta_{max}$ is the radius of the coverage circle
centered at the location of transmitter and receiver when the transmitter and
the receiver are located at the same location (we will present how to obtain
$\zeta_{max}$ in detail in $\S$II-A). In other words, the barrier built by the
bistatic radar system using their solution cannot cover the belt-shaped with
width greater than $\frac{2\zeta_{max}}{\sqrt{3}}$, namely
$\frac{2\zeta_{max}}{\sqrt{3}}<2\omega<2\zeta_{max}$222$\zeta_{max}$ is the
radius of the coverage circle centered at the location of transmitter and
receiver when the transmitter and the receiver are located at the same
location, therefore, the maximum width of the belt-shaped area in the MCLP
problem has to be less than $2\zeta_{max}$. In this work, we seek to design an
optimal algorithm for the “open MCLP problem” that works for full range of the
barrier width, to achieve maximum coverage for the barrier. In addition, the
existing solutions [7] proposed for the MCLP problem are still “heuristic”
(not an optimal solution) because boundary conditions are not considered. The
main contributions of this paper are summarized as follows.

  * •

We investigate the problem in both cases when $0\leq
2\omega\leq\frac{2\zeta_{max}}{\sqrt{3}}$ and
$\frac{2\zeta_{max}}{\sqrt{3}}<2\omega<2\zeta_{max}$. We propose an optimal
algorithm – dubbed the Opt$\\_$MCLP – to find the optimal solution for the
“open MCLP problem” that works for full range of the barrier width ($0\leq
2\omega<2\zeta_{max}$).

  * •

We provide rigorous theoretical analyses to demonstrate the correctness of the
proposed optimal solution.

  * •

Extensive simulations are conducted to demonstrate the performance of the
Opt$\\_$MCLP for the MCLP problem. The obtained results show that the
Opt$\\_$MCLP provides a significantly higher performance than the existing
method.

Organization: The remaining sections of this paper are organized as follows.
The remaining sections of this paper are organized as follows. The basic
mathematical notations and system model are initially introduced, and simple
examples are used to present the key ideas behind the proposed work in $\S$II.
An optimal solution, termed the Opt$\\_$MCLP is proposed for the full range of
the MCLP problem in $\S$III. Simulations are evaluated in Section IV, and the
paper is concluded in Section V.

##  II Preliminaries

###  II-A System Model and Problem Definition

A bistatic radar is composed of at least a transmitter and a receiver that are
separated and often located at different positions. In a bistatic radar, the
transmitter $t$ is responsible for producing and propagating the radio waves
and the receiver $r$ can detect an object (target) $x$ using the echo signal
reflected off from $x$ if the received signal-to-noise ratio (SNR) is not less
than a threshold $\gamma$. For any target $x$ and a bistatic radar paired by
transmitter $t$ and receiver $r$, the SNR of the radar signal that is sent
from $t$, reflected by $x$, and received by $r$ can be obtained [13] as
follows:

$SNR(t,x,r)=\frac{K}{d^{2}(t,x)\cdot d^{2}(x,r)},$ (1)

where $d(t,x)$ (or $d(x,r)$) denotes the Euclidean distance between $t$ (or
$r$) and $x$; and $K$ represents a constant that is determined by a bistatic
radar’s physical characteristics, such as the antenna’s power gain and the
transmission power. When the minimum threshold $\gamma$ is given, for any pair
of transmitter $t$ and receiver $r$, the possible locations of targets $x$
with $SNR(t,x,r)=\gamma$ can be characterized by the locus of points such that
the product of the distances to $t$ and $r$, namely $d(x,t)\cdot d(x,r)$, is
the constant equal to $\zeta^{2}_{max}$, where $\zeta_{max}$ is a constant and
denotes $\sqrt[4]{\frac{K}{\gamma}}$. The locus of points $x$, which will be a
closed curve or a pair of closed curves, is known as a Cassini oval [14] as
depicted in Fig. 1. For any target $y$ within the Cassini oval, the product of
the distances from $y$ to $t$ and $r$ is not greater than
$\sqrt{\frac{K}{\gamma}}$, that is, $d(y,t)\cdot d(y,r)\leq\zeta^{2}_{max}$;
and therefore, we have that $SNR(t,y,r)\geq\gamma$ and $y$, therefore, can be
detected by $r$. Hereafter, a point $z$ in the plane is said to be covered by
a bistatic radar paired by transmitter $t$ and receiver $r$ if $z$ is within
the area surrounded by the Cassini oval with focal points at $t$ and $r$.

When $\zeta_{max}$ is given, the shape of the Cassini oval with focal points
at transmitter $t$ and receiver $r$ is determined by the distance between $t$
and $r$, that is, $d(t,r)$. Four shape types of the Cassini oval depicted in
Fig. 1 are listed as follows with different range of $d(t,r)$ [7]:

  * •

an ellipse curve (Fig. 1) if $0\leq d(t,r)<\sqrt{2}\zeta_{max}$;

  * •

a waist curve (Fig. 1) if $\sqrt{2}\zeta_{max}\leq d(t,r)<2\zeta_{max}$;

  * •

a lemniscate of Bernoulli (Fig. 1) if $d(t,r)=2\zeta_{max}$;

  * •

a pair of closed curves (Fig. 1) if $d(t,r)>2\zeta_{max}$.

Figure 1: Four shape types of the Cassini oval with focal points at
transmitter $t$ and receiver $r$, where small solid and hollow circles denote
transmitters and receivers, respectively. The shape types with $0\leq
d(t,r)<\sqrt{2}\zeta_{max}$, $\sqrt{2}\zeta_{max}\leq d(t,r)<2\zeta_{max}$,
$d(t,r)=2\zeta_{max}$, and $d(t,r)>2\zeta_{max}$ are shown in (a), (b), (c),
and (d), respectively.

When multiple transmitters and receivers are deployed in an area, transmitters
and receivers can be paired to form multiple bistatic radars. Here,
transmitters are assumed to operate with orthogonal frequency such that mutual
interferences at a receiver can be avoided [15]. Therefore, each receiver can
be paired with different transmitters to form bistatic radars by switching the
frequency. In addition, because receivers can receive the radar signal sent
from a transmitter, multiple receivers can also be paired with the same
transmitter. Because multiple bistatic radars are often used to detect
targets, an area is said to be covered by the set of transmitters $T$ and the
set of receivers $R$ hereafter, if for any point $z$ in the area, $z$ is
covered by at least one bistatic radar paired by a transmitter $t\in T$ and a
receiver $r\in R$.

Let $c_{t}$ and $c_{r}$ be the placement/deployment costs of a transmitter and
a receiver, respectively. Also let $Area$ be a belt-shaped (rectangle) area
with length $L$ and width $W$, where $L\geq W>0$. In addition, when a
transmitter and a receiver are located at the same location, the shape of the
generated Cassini oval will be a circle centered at the transmitter/receiver
with radius $\zeta_{max}$ [14]. While we are given $\zeta_{max}$, $c_{t}$,
$c_{r}$, and $Area$ with width less than $2\zeta_{max}$, the Minimum Cost
Linear Placement (MCLP) problem is to deploy a set of transmitters $T$ and a
set of receivers $R$ on the line that goes through the middle points of the
shorter sides of the $Area$, such that the $Area$ is fully covered and the
total placement cost of bistatic radar(s), namely
$c_{t}\times|T|+c_{r}\times|R|$, is minimized, where $|T|$ and $|R|$ denote
the cardinalities of $T$ and $R$, respectively.

Figure 2: Examples of covering a rectangle area by using one transmitter and
two receivers (best viewed in color)

.

For the MCLP problem, let $2\omega$ be the width of the $Area$, and $\beta$
denote $\frac{\zeta_{max}}{\omega}$. Although solutions for the MCLP problem
are proposed in [7], the solutions only work for the case of
$\beta\geq\sqrt{3}$, that is, $\frac{\zeta_{max}}{\omega}\geq\sqrt{3}$
($2\omega\leq\frac{2\zeta_{max}}{\sqrt{3}}$). In addition, the width of the
$Area$ in the MCLP problem is always less than $2\zeta_{max}$, which implies
that these solutions are not valid for the case of
$\frac{2\zeta_{max}}{\sqrt{3}}<2\omega<2\zeta_{max}$ ($1<\beta<\sqrt{3}$).
This motivates us to explore an optimal solution for the MCLP problem that
especially works for full range of the coverage $Area$ width ($0\leq
2\omega<2\zeta_{max}$).

##  III The Optimal Solution for the Minimum Cost Linear Placement (MCLP)
Problem

By Fig. 1, we have that when transmitter $t$ and receiver $r$ are close
enough, the area covered by $t$ and $r$ will be an ellipse or waist shape. In
order to cover a rectangle area with width $2\omega$, the distance between the
upper and the lower parts of the ellipse (or the waist) curve has to be not
less than $2\omega$. Take Fig. 3, for example, where
$\omega=\frac{\zeta_{max}}{\sqrt{3}}$. When the covered area is a waist shape
as shown in Fig. 3, because the upper and the lower parts of the curve
generated by transmitter $t$ and receiver $r$ are symmetric, and
$d(E,F)=d(t,D)=d(r,C)=\frac{\zeta_{max}}{\sqrt{3}}$, $t$ and $r$ can cover a
rectangle area with width $2\omega$. Similar example with an ellipse shape is
shown in Fig. 3. This motivates us to find an optimal deployment of a
transmitter and a receiver such that the width of the covered rectangle can be
satisfied and its length can be maximized. As the examples in Fig. 3, when a
rectangle is fully covered by transmitter $t$ and receiver $r$, $t$ (or $r$)
may be at a distance of $\theta$ from the closest vertical boundary of the
covered rectangle. By the observation, we show that transmitter $t$ and
receiver $r$ with $d(t,r)=\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$ can
cover a rectangle with width $2\omega$ and length
$\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$; and that transmitter $t$ and
receiver $r$ with $d(t,r)=\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{\prime
2}}{\omega^{2}}+1}-1}-\theta^{\prime}$ can cover a rectangle with width
$2\omega$ and maximum length
$\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{\prime
2}}{\omega^{2}}+1}-1}+\theta^{\prime}$ by Lemmas 1-2, where $\theta^{\prime}$
denotes the $\theta$ in $[0,\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}]$
having maximum value of
$\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{2}}{\omega^{2}}+1}-1}+\theta$ and
can be obtained by Newton’s method [16]. In addition, Lemma 3 shows that the
length of the rectangle with width $2\omega$ covered by one transmitter and
any number of receivers is at most
$2\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$. Note that we can use one
transmitter $t$ and two receivers, $r_{1}$ and $r_{2}$, by the sequence
$(r_{1},t,r_{2})$, to cover a rectangle with width $2\omega$ and length
$2\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$ by Lemma 2, as the example in
Fig. 4.

Figure 3: Examples of deploying transmitter $t$ and receiver $r$, where
$\omega=\frac{\zeta_{max}}{\sqrt{3}}$. The distances between $r$ and the
boundary in (a) and (b) are 0 and $\theta$, respectively.

###### Lemma 1

When $1<\beta<\sqrt{3}$ and a rectangle with width $2\omega$ is covered by
transmitter $t$ and receiver $r$, $d(t,r)+\theta$ is maximized if and only if
$d(t,r)=\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$ and $\theta=0$, where
$\theta$ denotes the distance between $t$ (or $r$) and the closest vertical
boundary of the rectangle. In addition, $d(t,r)+2\theta$ is maximized if and
only if $d(t,r)=\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{\prime
2}}{\omega^{2}}+1}-1}-\theta^{\prime}$, where $\theta^{\prime}$ denotes the
$\theta$ in $[0,\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}]$ having maximum
value of
$\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{2}}{\omega^{2}}+1}-1}+\theta$.

###### Proof:

Without loss of generality, let $\Box ABCD$ be the rectangle covered by $t$
and $r$, as the rectangle in Fig. 3 or Fig. 3. Because the upper and the lower
parts (or, the left and the right parts) of the curve generated by $t$ and $r$
are symmetric, the proof suffices to show that the cases hold when the point
$B$ is covered. When the point $B$ is exactly covered by $t$ and $r$, we have
that $d(t,B)\cdot d(r,B)=\zeta^{2}_{max}$, which implies that
$\sqrt{(d(t,r)+\theta)^{2}+\omega^{2}}\cdot\sqrt{\theta^{2}+\omega^{2}}=\zeta^{2}_{max}$.
We thus have that
$d(t,r)=\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{2}}{\omega^{2}}+1}-1}-\theta$.
Therefore,
$d(t,r)+\theta=\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{2}}{\omega^{2}}+1}-1}$.
Because $\omega$ and $\beta$ are constants, $d(t,r)+\theta$ is maximized if
and only if $\theta=0$. When $\theta=0$,
$d(t,r)+\theta=d(t,r)=\omega\sqrt{\beta^{4}-1}=\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$.
In addition,
$d(t,r)+2\theta=\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{2}}{\omega^{2}}+1}-1}+\theta$.
It is clear that $d(t,r)+2\theta$ is maximized if and only if $\theta\geq 0$.
Assume that $\theta=\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}+\epsilon$,
where $\epsilon>0$. Because $d(t,r)\geq 0$ and $1<\beta<\sqrt{3}$, we have
that $d(t,B)\cdot
d(r,B)=\sqrt{(d(t,r)+\theta)^{2}+\omega^{2}}\cdot\sqrt{\theta^{2}+\omega^{2}}$
$\geq(\sqrt{\theta^{2}+\omega^{2}})^{2}$
$=(\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}+\epsilon)^{2}+\omega^{2}$
$=\beta^{2}\zeta^{2}_{max}+2\epsilon\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}+\epsilon^{2}$
$>\zeta^{2}_{max}$. This constitutes a contradiction because the point $B$ has
to be covered, that is, $d(t,B)\cdot d(r,B)\leq\zeta^{2}_{max}$. Therefore, we
have that $\epsilon\leq 0$, that is,
$\theta\leq\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$. Because $\theta\geq
0$, we have that $d(t,r)+2\theta$ is maximized if and only if
$0\leq\theta\leq\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$. Let
$\theta^{\prime}$ be the $\theta$ in
$[0,\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}]$ having maximum value of
$\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{2}}{\omega^{2}}+1}-1}+\theta$. We
have that $d(t,r)+2\theta$ is maximized if and only if
$d(t,r)+2\theta^{\prime}=\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{\prime
2}}{\omega^{2}}+1}-1}+\theta^{\prime}$, that is,
$d(t,r)=\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{\prime
2}}{\omega^{2}}+1}-1}-\theta^{\prime}$, which completes the proof. ∎

###### Lemma 2

When $1<\beta<\sqrt{3}$, the transmitter $t$ and the receiver $r$ that are at
a distance of $\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$ apart can cover a
rectangle with width $2\omega$ and length equal to
$\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$. In addition, the transmitter $t$
and the receiver $r$ that are at a distance of
$d(t,r)=\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{\prime
2}}{\omega^{2}}+1}-1}-\theta^{\prime}$ apart can cover a rectangle with width
$2\omega$ and length equal to $d(t,r)+2\theta^{\prime}$, where
$\theta^{\prime}$ is defined in Lemma 1.

###### Proof:

The proof has to show that S1) if
$d(t,r)=\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$, $t$ and $r$ can fully
cover a rectangle with width $2\omega$ and length
$\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$, and that S2) if
$d(t,r)=\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{\prime
2}}{\omega^{2}}+1}-1}-\theta^{\prime}$, $t$ and $r$ can fully cover a
rectangle with width $2\omega$ and length
$\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{\prime
2}}{\omega^{2}}+1}-1}+\theta^{\prime}$, where $\theta^{\prime}$ denotes the
$\theta$ in $[0,\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}]$ having maximum
value of
$\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{2}}{\omega^{2}}+1}-1}+\theta$. The
proof of S2 is omitted here due to the similarity.

For S1, because $\frac{\zeta_{max}}{\sqrt{3}}<\omega<\zeta_{max}$, we have
that
$d(t,r)=\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}=\sqrt{\frac{\zeta^{4}_{max}-\omega^{4}}{\omega^{2}}}$
$<\sqrt{\frac{\zeta^{4}_{max}-(\frac{\zeta_{max}}{\sqrt{3}})^{4}}{(\frac{\zeta_{max}}{\sqrt{3}})^{2}}}$
$=\sqrt{\frac{8\zeta^{2}_{max}}{3}}<2\zeta_{max}$, which implies that the area
covered by $t$ and $r$ will be an ellipse or waist shape by the results in
Fig. 1 [7]. Let $\Box ABCD$ be a rectangle with width $2\omega$ and length
$\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$, as the rectangle in Fig. 3. Due
to the fact that the upper and the lower parts of the curve generated by $t$
and $r$ are symmetric, the proof of S1 suffices to show that $\Box trCD$ is
within the area covered by $t$ and $r$. Let $\overrightarrow{tD^{\prime}}$ (or
$\overrightarrow{rC^{\prime}}$) be the line from $t$ (or, from $r$),
perpendicular to $\overline{tr}$, and intersected by the lower part of the
generated curve at point $D^{\prime}$ (or $C^{\prime}$). Also let the curve
$\widetilde{D^{\prime}C^{\prime}}$ be the lower part of the curve generated by
$t$ and $r$ with endpoints $D^{\prime}$ and $C^{\prime}$. Because the curve
generated by $t$ and $r$ is an ellipse or waist shape, and the left and the
right parts of the generated curve are symmetric, we have that the minimum
distance between $\overline{tr}$ and $\widetilde{D^{\prime}C^{\prime}}$ is the
minimum value of $d(D^{\prime},\overline{tr})$, $d(C^{\prime},\overline{tr})$,
and $d(E^{\prime},\overline{tr})$, where $d(D^{\prime},\overline{tr})$,
$d(C^{\prime},\overline{tr})$, and $d(E^{\prime},\overline{tr})$ denote the
minimum distance from $D^{\prime}$, $C^{\prime}$, and $E^{\prime}$,
respectively, to $\overline{tr}$, and $E^{\prime}$ denotes the midpoint of
$\widetilde{D^{\prime}C^{\prime}}$. We thus also have that $\Box trCD$ is
fully covered if $d(D^{\prime},\overline{tr})\geq d(D,\overline{tr})=\omega$,
$d(C^{\prime},\overline{tr})\geq d(C,\overline{tr})=\omega$, and
$d(E^{\prime},\overline{tr})\geq d(E,\overline{tr})=\omega$, that is, the
points $C$, $D$, and $E$ have to be covered by $t$ and $r$, where the point
$E$ is the midpoint of $\overline{CD}$. For the point $D$, we have that
$d(t,D)\cdot d(r,D)=\omega\cdot\sqrt{d^{2}(r,t)+d^{2}(t,D)}$
$=\zeta^{2}_{max}$, implying that $D$ is covered by $t$ and $r$. Similarly,
the point $C$ is also covered by $t$ and $r$. For the point $E$, we assume
that $E$ is not covered by $t$ and $r$. That is, $d(t,E)\cdot
d(r,E)=(\sqrt{(\frac{d(t,r)}{2})^{2}+\omega^{2}})^{2}>\zeta^{2}_{max}$, and
thus, we have that $3\omega^{4}-4\omega^{2}\zeta^{2}_{max}+\zeta^{4}_{max}>0$,
implying that $(3\omega^{2}-\zeta^{2}_{max})(\omega^{2}-\zeta^{2}_{max})>0$.
We have that $\omega<\frac{\zeta_{max}}{\sqrt{3}}$ or $\omega>\zeta_{max}$,
which constitutes a contradiction because
$\frac{\zeta_{max}}{\sqrt{3}}<\omega<\zeta_{max}$. Therefore, the points $C$,
$D$, and $E$ are covered by $t$ and $r$, and $\Box ABCD$ is fully covered by
$t$ and $r$. This completes the proof of S1, and thus, the proof of the lemma
is also completed. ∎

###### Lemma 3

When $1<\beta<\sqrt{3}$ and a transmitter $t$ is given, the maximum length of
the rectangle with width $2\omega$ covered by $t$ and any number of receivers
is equal to $2\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$.

###### Proof:

Assume that there exists a sequence $\Psi=(R^{n_{p_{0}}},t,R^{n_{p_{1}}})$
deployed on a line such that the length of the rectangle covered by $\Psi$,
denoted by $\ell(\Psi)$, is greater than
$2\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$, where $n_{p_{0}},n_{p_{1}}\geq
0$ and $R^{n_{p_{i}}}$ denotes the set of $n_{p_{i}}$ receivers. By Lemmas
1-2, we have that the maximum length of the covered rectangle from $t$ to one
side boundary is at most $\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$. This
implies that
$2\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}\geq\ell(\Psi)>2\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$,
which constitutes a contradiction. This thus completes the proof. ∎

Figure 4: Example of deploying one transmitter with two receivers when
$\omega=\frac{\zeta_{max}}{\sqrt{3}}$.

Let $I^{m}_{n}$ be the sequence $(r_{1},t_{1},r_{2},t_{2},\ldots,r_{m},t_{m})$
if $n=m$, and be the sequence
$(r_{1},t_{1},r_{2},t_{2},\ldots,r_{m},t_{m},r_{m+1})$ if $n=m+1$. The
proposed placement for $1<\beta<\sqrt{3}$, termed the Rotated Placement
hereafter, is to place $m$ transmitters and $n$ receivers following the
sequence $I^{m}_{n}$ sequentially, where the distance between a transmitter
and its adjacent receiver is $\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$ by
Lemma 2, as the examples in Fig. 5.

Figure 5: Examples of deploying $m$ transmitters and $n$ receivers by the
Rotated Placement, where $\varphi$ denotes
$\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$.

The procedure MinCost$\\_$RP is designed to find a sequence of transmitters
and receivers with minimum cost to cover a rectangle area $Area$ with length
$\ell$ and width $2\omega$. By Lemma 1-2, if
$\ell\leq\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{\prime
2}}{\omega^{2}}+1}-1}+\theta^{\prime}$, the $Area$ can be covered by one
transmitter and one receiver. Otherwise, by Lemma 1-3, one transmitter coupled
with two receivers can be used to cover a rectangle with length
$2\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$ that is called a unit hereafter.
The procedure MinCost$\\_$RP is then to see at least how many units are
required. The necessary numbers of transmitters and receivers are calculated
and stored in $numTx$ and $numRx$, respectively. When the $Area$ is not fully
covered, additional one transmitter is added if the length of the remaining
uncovered part of the $Area$ is not greater than the half length of a unit;
and otherwise, additional one transmitter and one receiver are added. Finally,
by the obtained $numTx$ and $numRx$, the Rotated Placement is used to generate
the optimal solution $I^{numTx}_{numRx}$, which is proved in Theorem 1.

###### Theorem 1

When $1<\beta<\sqrt{3}$, the procedure MinCost$\\_$RP can find an optimal
solution to cover a rectangle area $Area$ with length $\ell$ and width
$2\omega$.

###### Proof:

By Lemma 1-2, we have that deploying one transmitter and one receiver to cover
the $Area$ is an optimal solution if
$\ell\leq\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{\prime
2}}{\omega^{2}}+1}-1}+\theta^{\prime}$. In addition, because it is easy to
verify that the difference between $numTx$ and $numRx$ in the procedure
MinCost$\\_$RP is at most one, the proof suffices to show that the solution
generated by the procedure MinCost$\\_$RP has a minimum cost. Three cases,
including C1) $\ell=2k\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$, C2)
$2k\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}<\ell\leq(2k+1)\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$,
and C3)
$(2k+1)\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}<\ell<(2k+2)\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$,
are considered, where $k\geq 0$ and
$\ell>\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{\prime
2}}{\omega^{2}}+1}-1}+\theta^{\prime}$. Because the proofs of C2 and C3 are
similar to that of C1, the proofs of C2 and C3 are omitted.

1:procedure MinCost$\\_$RP($\omega$, $\ell$)

2: if $\ell\leq\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{\prime
2}}{\omega^{2}}+1}-1}+\theta^{\prime}$, where $\theta^{\prime}$ is defined in
Lemma 1 then

3: Let $I$ be the sequence $(r,t)$ with
$d(t,r)=\omega\sqrt{\frac{\beta^{4}}{\frac{\theta^{\prime
2}}{\omega^{2}}+1}-1}-\theta^{\prime}$

4: $minValue$ $\leftarrow$ $c_{t}+c_{r}$

5: return $(minValue,I)$

6: end if

7:
$numTx\leftarrow\left\lfloor\frac{\ell}{2\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}}\right\rfloor$

8: $numRx\leftarrow numTx+1$

9: $remainingLen\leftarrow\ell\ (\mathrm{mod}\
2\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}})$

10: if $remainingLen>0$ then

11: $numTx\leftarrow numTx+1$

12: if
$\lceil\frac{remainingLen}{\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}}\rceil=2$
then

13: $numRx\leftarrow numRx+1$

14: end if

15: end if

16: Let $I^{numTx}_{numRx}$ be the sequence generated by the Rotated Placement
with $numTx$ transmitters and $numRx$ receivers

17: $minValue$ $\leftarrow$ $numTx\times c_{t}+numRx\times c_{r}$

18: return $(minValue,I^{numTx}_{numRx})$

19:end procedure

For C1, let $I^{m_{1}}_{n_{1}}$ and $\Psi^{m_{2}}_{n_{2}}$ be the sequences
obtained by the procedure MinCost$\\_$RP and the optimal solution,
respectively. Assume that $I^{m_{1}}_{n_{1}}$ is not an optimal solution. This
implies that $\varsigma(I^{m_{1}}_{n_{1}})>\varsigma(\Psi^{m_{2}}_{n_{2}})$,
where $\varsigma(I^{m_{1}}_{n_{1}})$ (or $\varsigma(\Psi^{m_{2}}_{n_{2}})$)
denotes the placement cost of $I^{m_{1}}_{n_{1}}$ (or $\Psi^{m_{2}}_{n_{2}})$,
and $m_{1}\neq m2$ or $n_{1}\neq n_{2}$. Because
$\ell=2k\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$, we have that $m_{1}=k$
and $n_{1}=k+1$ by the procedure MinCost$\\_$RP. By Lemma 3, due to the fact
that one transmitter with receivers can cover a rectangle with length at most
$2\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$, at least
$\left\lfloor\frac{2k\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}}{2\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}}\right\rfloor=k$
transmitters are required to cover the $Area$. Because $m_{1}=k$, we have that
C1.1) $m_{2}>k$, or C1.2) $m_{2}=k$ and $n_{2}<n_{1}=k+1$. Due to the fact
that the proof of C1.2 is similar to that of C1.1, the proof of C1.2 is
omitted here. For C1.1, if $m_{2}>k$, because
$\varsigma(I^{m_{1}}_{n_{1}})>\varsigma(\Psi^{m_{2}}_{n_{2}})$, we have that
$c_{t}\times k+c_{r}\times(k+1)>c_{t}\times(k+y)+c_{r}\times n_{2}$, where
$m_{2}=k+y$ and $y\geq 1$. Because $y\geq 1$ and $c_{t}\geq c_{r}$, we have
that $n_{2}<(k+1)-\frac{yc_{t}}{c_{r}}\leq k$. Due to the fact that the
covered area is the same when receivers and transmitters are swapped, by Lemma
3, we have that one receiver with transmitters can cover a rectangle with
length at most $2\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$. In addition,
because $n_{2}<k$, that is, $n_{2}\leq k-1$, the rectangle with length at most
$2(k-1)\sqrt{\beta^{2}\zeta^{2}_{max}-\omega^{2}}$ can be covered by $n_{2}$
receivers with transmitters, which implies that the $Area$ cannot be covered
by $\Psi^{m_{2}}_{n_{2}}$. This constitutes a contradiction, and completes the
proof of C1.1. This thus completes the proof of the theorem. ∎

For an $Area$ with width $W=2\omega$ and length $L=\ell$, the algorithm
Opt$\\_$MCLP is designed as shown in Algorithm 1 to combine the function
ComputeMinCost, which is proposed in [7] and used for
$2\omega\leq\frac{2\zeta_{max}}{\sqrt{3}}$, with the procedure MinCost$\\_$RP
and used for $\frac{2\zeta_{max}}{\sqrt{3}}<2\omega<2\zeta_{max}$.

Algorithm 1 Opt$\\_$MCLP($\omega$, $\ell$)

1:$S\leftarrow\emptyset$; $minValue\leftarrow\infty$

2:if $\omega\leq\frac{\zeta_{max}}{\sqrt{3}}$ then

3: Use ComputeMinCost function in [7] to get the total cost $minValue$ and the
placement sequence $S$ by given $\omega$ and $\ell$

4:else

5: $(minValue,S)\leftarrow$ MinCost$\\_$RP ($\omega$, $\ell$)

6:end if

7:return $(minValue,S)$

##  IV Performance Evaluation

Figure 6: The reduced cost ratio when $\zeta_{max}=2$, $\mu=0.1$, $W$ ranges
from $0.1$ to $2.3$, and $\alpha$ ranges from $1$ to $100$. The values of $L$
are $10$ in (a) and $100$ in (b), respectively.

Figure 7: The total placement cost when $\zeta_{max}=2$, $\mu=0.1$, $W$ ranges
from $0.1$ to $3.9$, and $\alpha$ ranges from $1$ to $100$. The values of $L$
are $10$ in (a) and $100$ in (b), respectively.

Here, simulations developed by C++ were used to evaluate the performance of
the Opt$\\_$MCLP for the MCLP problem. In the simulations, $\mu$ and $c_{r}$
were set to $0.1$ and $1$, respectively. In addition, $W$, $L$, $\zeta_{max}$,
and $\alpha$ were set from $0.1$ to $10$, from $10$ to $100$, from $1$ to $4$,
and from $1$ to $100$, respectively, where $\alpha$ denoted
$\frac{c_{t}}{c_{r}}$. Let the reduced cost ratio of $B$ compared with $A$ be
$\frac{c_{A}-c_{B}}{c_{A}}$, where $A$ and $B$ were solutions of the MCLP
problem and $c_{A}$ (or, $c_{B}$) denoted the total placement cost required by
$A$ (or, $B$). For the MCLP problem, because transmitters and receivers had to
be deployed on a line to form a linear barrier, the Algorithm 2 in [7] with
one linear barrier was compared in terms of the reduced cost ratio in
$\S$IV-A. Because the Algorithm 2 proposed in [7] can be only used for
covering a rectangle area with width less than or equal to
$\frac{2\zeta_{max}}{\sqrt{3}}$, $W$ was set from $0.1$ to $2.3$ when
$\zeta_{max}=2$. In addition, due to the fact that the width of the $Area$ had
to be less than $2\zeta_{max}$ in the MCLP problem, we also evaluated the
total placement cost of the Opt$\\_$MCLP in $\S$IV-A when $\zeta_{max}=2$ and
$W$ ranged from $0.1$ to $3.9$.

###  IV-A The MCLP Problem

Let the reduced cost ratio of the Opt$\\_$MCLP($\alpha=k$) denote the reduced
cost ratio of the Opt$\\_$MCLP compared with the Algorithm 2 in [7] having one
linear barrier when $\alpha=k$. Fig. 6 and Fig. 6 show the reduced cost ratios
of the Opt$\\_$MCLP($\alpha=1$ and Opt$\\_$MCLP($\alpha=100$), respectively,
when the values of $L$ are equal to $10$, $50$, and $100$, respectively. It is
clear that the Opt$\\_$MCLP provides better performance than the Algorithm 2
proposed in [7] with one linear barrier. In addition, the Opt$\\_$MCLP reduces
the placement cost by up to $49\%$ in comparison with the other method.

Let the total placement cost of the Opt$\\_$MCLP($\alpha=k$) denote the total
placement cost of the Opt$\\_$MCLP when $\alpha=k$. Fig. 7 and Fig. 7 show the
total placement costs of the Opt$\\_$MCLP($\alpha=1$) and
Opt$\\_$MCLP($\alpha=100$), respectively, when the values of $L$ are equal to
$10$, $50$, and $100$, respectively. It is clear that in Fig. 7 or Fig. 7, the
total placement cost of the Opt$\\_$MCLP increases with the increasing of
$\alpha$. This is because the cost of a transmitter increases when $\alpha$
increases. In addition, it is also clear that the greater the value of $W$,
the higher the total placement cost required by the Opt$\\_$MCLP($\alpha=1$),
Opt$\\_$MCLP($\alpha=10$), or Opt$\\_$MCLP($\alpha=100$). This is due to the
fact that transmitters and receivers are closely deployed in order to cover
the $Area$ with greater width, and therefore, more transmitters and receivers
are required. Moreover, when the value of $L$ increases from $10$ to $100$,
the total placement cost of the Opt$\\_$MCLP($\alpha=1$),
Opt$\\_$MCLP($\alpha=10$), or Opt$\\_$MCLP($\alpha=100$) increases because
more area has to be covered.

##  V Conclusion

The sensing model of bistatic radars (Cassini oval sensing model [7]) is in
fact very complex and the shape of sensing areas are, therefore, varied
according to the variation of distance between the transmitter(s) and the
receiver(s) in the bistatic radar system. In this paper, we study the MCLP
problem for constructing a belt barrier with minimum placement cost. For the
MCLP problem with $0\leq 2\omega\leq\frac{2\zeta_{max}}{\sqrt{3}}$, a
function, termed the MinCost$\\_$RP, is proposed to find an optimal solution.
In addition, for $\frac{2\zeta_{max}}{\sqrt{3}}<2\omega<2\zeta_{max}$, an
optimal solution, termed the Opt$\\_$MCLP, is proposed. Theoretical analysis
is also provided for proving the optimality of the Opt$\\_$MCLP. Simulation
results show that the Opt$\\_$MCLP has a significantly lower placement cost
than the existing solution.

## References

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# Hydrodynamics of nonlinear gauge-coupled quantum fluids

Y. Buggy  L.G. Phillips  P. Öhberg  SUPA, Institute of Photonics and Quantum
Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom

###### Abstract

By constructing a hydrodynamic canonical formalism, we show that the
occurrence of an arbitrary density-dependent gauge potential in the meanfield
Hamiltonian of a Bose-condensed fluid invariably leads to nonlinear flow-
dependent terms in the wave equation for the phase, where such terms arise due
to the explicit dependence of the mechanical flow on the fluid density. In
addition, we derive a canonical momentum transport equation for this class of
nonlinear fluid and obtain an expression for the stress tensor. Further, we
study the hydrodynamic equations in a particular nonlinear fluid, where the
effective gauge potential results from the introduction of weak contact
interactions in an ultracold dilute Bose gas of optically-addressed two-level
atoms. In the Cauchy equation of mechanical momentum transport of the
superfluid, two non-trivial terms emerge due to the density-dependent vector
potential. A body-force of dilation appears as a product of the gauge
potential and the dilation rate of the fluid, while the stress tensor features
a canonical flow pressure term given by the inner-product of the gauge
potential and the canonical current density. By numerical simulation, we
illustrate an interesting effect of the nonlinear gauge potential on the
groundstate wavefunction of a superfluid in the presence of a foreign
impurity. We find that the groundstate adopts a non-trivial local phase, which
is antisymmetric under reversal of the gauge potential. The phase profile
leads to a canonical-flow or phase-flow dipole about the impurity, resulting
in a skirting mechanical flow. As a result, the pressure becomes asymmetric
about the object and the condensate undergoes a deformation.

††preprint: APS/123-QED

##  I Introduction

In classical mechanics, the interaction of charged particles with the
electromagnetic field can be completely described in terms of the force fields
$\mathbf{E}$ and $\mathbf{B}$. The electromagnetic potentials $\phi$ and
$\mathbf{A}$ on the other hand, enter merely as auxiliary mathematical
quantities bearing no physical significance. The situation is drastically
different in quantum physics: quantisation of a classical theory proceeds from
knowledge of the canonical momenta, and it is the energies and momenta which
are the central quantities determining the phases of quantum wavefunctions. As
a result, charged particles couple directly to the electromagnetic potentials
in the quantum theory, where the form of this coupling notably leads to the
Aharanov-Bohm effect and the local gauge invariance of quantum mechanics. The
implications of the fundamental role played by the potentials Aharonov and
Bohm (1959), have since led to a diverse range of intriguing physical effects.
These arise through the interplay between particle-particle interactions and
applied fields. Although the weak field behaviour of gauge-coupled systems is
well described by linear response theory, large perturbing field values do not
generally allow for a meaningful first order expansion Mattuck (2012). As the
field is gradually increased, the ordering of the system changes abruptly at
certain critical values and a variety of physical phenomena become associated
with each intensity range Fazekas (1999): from paramagnetic effects Stoner
(1935), to the quantum Hall Klitzing  _et al._ (1980); Yennie (1987); Prange
and Girvin (1987) and spin quantum Hall Kane and Mele (2005a, b); Fu  _et al._
(2007); Bernevig  _et al._ (2006) effects observed in two-dimensional electron
systems. This notably led to the classification of symmetry protected
topological phases of matter Thouless  _et al._ (1982); Chen  _et al._ (2013)
and paved the way for the implementation of topological insulators Qi and
Zhang (2011); Hasan and Kane (2010), illustrating the range of intriguing
phenomena which emerge in gauge-coupled many-body systems.

The charge neutrality of Bose-condensed atomic systems seemingly restricts the
discovery of exotic states of matter of this kind. However, the versatility,
controllability and robust character of ultracold quantum gases, have since
allowed for the possibility of simulating artificial gauge potentials for
charge-neutral systems. These are generally engineered through combined
interactions, such that a system exhibits spatially varying local eigenstates
Dalibard  _et al._ (2011); Goldman  _et al._ (2014). In other words, the
action of a gauge potential can be mimicked by imparting a geometric phase
onto the wavefunction Berry (1984); Peskin and Tonomura (1989); Dalibard  _et
al._ (2011); Goldman  _et al._ (2014). In this regard, the elucidation of the
geometrical nature of the Aharonov-Bohm phase Berry (1984) was a landmark in
understanding magnetism in quantum mechanics. Local eigenstates can be induced
in a variety of different ways. Initial attempts exploited the equivalence of
the Lorentz and Coriolis forces, by stirring the condensate with a focused
laser beam in a magnetic trap Madison  _et al._ (2000), a technique that
quickly led to the observation of vortex lattices Abo-Shaeer  _et al._ (2001).
More recent implementations have relied almost exclusively on dressing the
bare atomic states using light-matter interactions. For instance, a two-photon
Raman scheme Lin and Spielman (2016) was employed in a series of experiments
to engineer both electric Lin  _et al._ (2011a) and magnetic Lin  _et al._
(2009) synthetic force fields, as well as synthetic spin-orbit coupling Lin
_et al._ (2011b), spin Hall effect Beeler  _et al._ (2013) and partial waves
Williams  _et al._ (2012). Atomic light-dressing has also opened up the
possibility of generating non-Abelian vector potentials with non-commuting
components. These can be implemented for atoms with degenerate eigenstates,
and generally emerge when coupling to a laser field produces a degenerate
subspace of dressed states Dalibard  _et al._ (2011). In addition, efforts
have been made to extend the first generation of synthetic potentials - whose
space and time dependence are prescribed externally and unaffected by particle
motion - and endow these with dynamical properties Lin and Spielman (2016).
For instance, it was shown Edmonds  _et al._ (2013) how the introduction of
weak collisional interactions in an ultracold dilute Bose gas of optically
addressed two-level atoms, gives rise to a nonlinear effective vector
potential $\mathbf{A}(\rho)$ acting on the condensate, where
$\left|\mathbf{A}\right|$ is modulated by the density of the atomic gas.
Density-dependent gauge potentials have also been proposed Greschner  _et al._
(2014) in spin-dependent optical lattices, by combining periodically modulated
interactions and Raman-assisted hopping. In fact, such a potential was
recently experimentally implemented by modulating the interaction strength in
synchrony with lattice shaking Clark  _et al._ (2018), resulting in a density-
dependent hopping amplitude.

It is the emergence of such nonlinear gauge potentials which has motivated the
present study. From a hydrodynamical point of view this is an interesting
situation because the kinetic energy density becomes nonlinear in the fluid
density. Thus, flow depends explicitly on the density profile of the fluid,
where the magnitude of flow of a volume element typically increases as the
element shrinks. In this paper, we investigate the fluid stress and body-
forces which emerge in a superfluid subject to a nonlinear gauge potential.
The paper comprises two parts. In section II, we construct a general formalism
and demonstrate how the occurrence of an arbitrary density-dependent gauge
potential in the meanfield Hamiltonian acting on a macroscopic wavefunction,
invariably produces flow-dependent terms in the wave equation for the phase.
We also derive a canonical momentum transport equation for this class of
nonlinear fluid and obtain an expression for the stress tensor. In section
III, we apply our results from section II to a particular type of nonlinear
fluid, whose microscopic model is well-established. We find that the nonlinear
gauge potential gives rise to two non-trivial terms in the mechanical momentum
transport equation of the superfluid, in the form of a canonical flow pressure
and a body-force of dilation. Finally, by numerical simulation, we illustrate
an interesting implication of the nonlinear gauge potential on the ground
state phase profile of an inhomogeneous superfluid.

##  II General formalism for nonlinear gauge-coupled quantum fluids

###  II.1 Hydrodynamic canonical formalism

Let us assume that an effective density-dependent vector potential
$\mathbf{A}\left(\rho\right)$ and effective scalar potential
$\eta\left(\rho\right)$, emerge in the meanfield Lagrangian

$L_{MF}=\int
d^{3}\mathbf{r}\left[\psi^{*}\left(i\hbar\partial_{t}-\hat{H}_{MF}\right)\psi\right],$
(1)

governing the dynamics of a macroscopic wavefunction $\psi$. Since a gauge
potential enters the meanfield Hamiltonian, $\hat{H}_{MF}$, in a manner
consistent with the minimal substitution,
$\hat{\mathbf{p}}\rightarrow\hat{\mathbf{p}}-\mathbf{A}$, our study pertains
to the class of quantum fluids described by

$\hat{H}_{MF}=\frac{\left(\hat{\mathbf{p}}-\mathbf{A}\left(\rho\right)\right)^{2}}{2m}+\eta\left(\rho\right)+V\left(\mathbf{r},t\right),$
(2)

where we have included an external potential, $V$. Further, we shall consider
a particular microscopic model leading to a $\hat{H}_{MF}$ of the above form,
but presently, let us construct a hydrodynamic formalism for the more general
case. In accordance with Eq. (1), the Lagrangian density of the nonlinear
field, may be presented as

$\mathcal{L}=\frac{i\hbar}{2}\left(\psi^{*}\dot{\psi}-\dot{\psi}^{*}\psi\right)\\\
-\frac{1}{2m}\left[\left(\hat{\mathbf{p}}-\mathbf{A}\right)\psi\right]^{*}\left[\left(\hat{\mathbf{p}}-\mathbf{A}\right)\psi\right]-\rho\eta-\rho
V,$ (3)

where we have denoted partial differentiation with respect to time by a dot
and performed the transformation
$\mathcal{L}\rightarrow\mathcal{L}-i\hbar\partial_{t}\left(\psi^{*}\psi\right)/2$,
in order for $\mathcal{L}$ to be real.
Perhaps the most conspicuous feature of a density-dependent gauge potential,
is the occurrence of a flow nonlinearity in the wave equation for $\psi$. One
may gain insight into the appearance of such a term, by casting the meanfield
description into a hydrodynamical form. To do so, we write the macroscopic
wavefunction in the Madelung Madelung (1926), or polar form

$\psi=\sqrt{\rho}e^{i\frac{\theta}{\hbar}},\quad\quad\psi^{*}=\sqrt{\rho}e^{-i\frac{\theta}{\hbar}},$
(4)

and treat $\rho$ and $\theta$ as the independent field variables. In terms of
these, the Lagrangian density (3), assumes the form

$\mathcal{L}=-\rho\left(\dot{\theta}+\frac{1}{2}mv^{2}+\eta+V\right)-\mathcal{Q},$
(5)

where, $v=\left|\mathbf{v}\right|$, is the magnitude of the mechanical flow,
or gauge-covariant flow

$\mathbf{v}=\mathbf{u}-\mathbf{A}/m,$ (6)

$\mathbf{u}=\mbox{\boldmath$\nabla$}\theta/m$ is the canonical flow, or phase
flow, and

$\mathcal{Q}=\frac{\hbar^{2}}{8m\rho}\left(\mbox{\boldmath$\nabla$}\rho\right)^{2},$
(7)

is a quantum energy density contribution. We make the distinction between the
canonical flow and the gauge flow. In the canonical flow, we include the total
flow which can be accounted for locally by a phase twist in a suitable gauge,
whereas the gauge flow denotes the flow contribution from $\mathbf{A}$ which
can not be absorbed in the phase without destroying the form of the dynamical
equations of the fluid. The fact that it is not possible to gauge away
density-dependent vector potentials, will be covered elsewhere.
Since $\mathcal{L}$ is linear in the field velocities and the linear form
appears as $-\rho\dot{\theta}$, the field components $\rho$ and $\theta$ play
the role of conjugate variables Jackiw (1993); Buggy and Öhberg (2019),
governed by the canonical field equations

$\displaystyle\dot{\rho}\left(\mathbf{r}\right)$ $\displaystyle=\frac{\delta
H}{\delta\theta\left(\mathbf{r}\right)},$ (8)
$\displaystyle\dot{\theta}\left(\mathbf{r}\right)$
$\displaystyle=-\frac{\delta H}{\delta\rho\left(\mathbf{r}\right)},$ (9)

where the Hamiltonian and Lagrangian densities, are related by

$\mathcal{L}=-\rho\dot{\theta}-\mathcal{H},$ (10)

such that

$\mathcal{H}=\rho\left(\frac{1}{2}mv^{2}+\eta+V\right)+\mathcal{Q}.$ (11)

Note that the derivatives on the right hand side of Eqs. (8) and (9) are
functional, or variational derivatives, e.g. $\delta
H/\delta\rho=\partial\mathcal{H}/\partial\rho-\mbox{\boldmath$\nabla$}\cdot\left(\partial\mathcal{H}/\partial\left(\mbox{\boldmath$\nabla$}\rho\right)\right)$.
Inserting the Hamiltonian density (11) into the canonical field Eqs. (8) and
(9), yields, respectively, the wave equations

$\displaystyle\partial_{t}$
$\displaystyle\rho+\mbox{\boldmath$\nabla$}\cdot\left(\rho\mathbf{v}\right)=0,$
(12) $\displaystyle\partial_{t}$
$\displaystyle\theta+\frac{1}{2}mv^{2}+\Phi\left(\rho,\mathbf{u}\right)+V+Q=0,$
(13)

where the density-dependence of the kinetic term within the brackets of Eq.
(11) leads to an additional nonlinear flow-dependent term in the wave equation
for the phase, such that

$\Phi=-\rho\mathbf{v}\cdot\frac{\partial\mathbf{A}}{\partial\rho}+\eta+\rho\frac{\partial\eta}{\partial\rho},$
(14)

and

$Q=-\frac{\hbar^{2}}{2m}\frac{\nabla^{2}\sqrt{\rho}}{\sqrt{\rho}},$ (15)

is the quantum potential, which emerges due to the quantum energy density
$\mathcal{Q}$ in Eq. (11). Equation (12) expresses the conservation of mass
while Eq. (13) takes the form of a quantum Hamilton-Jacobi equation (QHJE),
the gradient of which expresses the conservation of mechanical momentum.
In summary, when a fluid is subject to a nonlinear vector potential, the flow
$\mathbf{v}$ in Eq. (6) depends explicitly on the density of the fluid and the
kinetic energy density $\kappa=\rho mv^{2}/2$ becomes nonlinear in $\rho$.
Thus, the change $\delta\kappa$ in an infinitesimal volume due to
$\delta\rho$, is not determined simply by the kinetic energy $mv^{2}/2$ of the
volume as it is typically, since $\delta\kappa=\left(mv^{2}/2+\rho
m\mathbf{v}\cdot\partial\mathbf{v}/\partial\rho\right)\delta\rho$. As a
result, a nonlinear flow-dependent term enters the wave equation for the
phase. This feature is intrinsic to systems whose effective Hamiltonian (2)
features a density-dependent vector potential $\mathbf{A}\left(\rho\right)$.

###  II.2 Canonical momentum transport equation

In the following section, we derive a canonical momentum transport equation
for the nonlinear fluid and investigate the implications of the nonlinear
gauge potential on the stress tensor. To do so, recall that the dynamical
state of the matter-field is completely specified by the stress-energy-
momentum tensor

$T_{\mu\nu}=-\sum_{\phi=\rho,\theta}\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}\partial_{\nu}\phi+\delta_{\mu\nu}\mathcal{L},$
(16)

while the transport equations governing energy-flow and momentum-flow, follow
from the conservation law

$\partial_{\mu}T_{\mu\nu}=\partial_{\nu}\mathcal{L},$ (17)

where we have adopted a relativistic-like notation with $\mu=0,1,2,3$. The
Lagrangian density of the field, is given by Eq. (5). Alternatively,
$\mathcal{L}$ may be cast in terms of the fields and their spatial
derivatives, by substituting the QHJE (13) into Eq. (5), which yields

$\mathcal{L}=-\frac{\hbar^{2}}{4m}\nabla^{2}\rho+\rho\left(\Phi-\eta\right).$
(18)

Rendering $\mathcal{L}$ into this form is essential for evaluating the
components of the field stress tensor, $T_{ij}$. The stress-energy-momentum
tensor from Eq. (16), characterises the dynamical state of the field by
specifying the energy density, the momentum density, and the currents
associated with both of these quantities. The energy density of the field, is

$-T_{00}=-\rho\dot{\theta}-\mathcal{L}=\mathcal{H},$ (19)

where the last equality follows from Eq. (10). The energy current density,
takes the form

$-T_{k0}=D\dot{\rho}w_{k}-\rho\dot{\theta}v_{k}\equiv\mathcal{S}_{k},$ (20)

where

$w_{k}=-\frac{D}{\rho}\nabla_{k}\rho,$ (21)

is the osmotic velocity Nelson (1966); Wyatt (2006); Faris (2014) and
$D=\hbar/\left(2m\right)$ is the quantum diffusion coefficient. The canonical
momentum density, reads

$T_{0k}=\rho\nabla_{k}\theta=\rho mu_{k}\equiv\mathcal{P}_{k}.$ (22)

The canonical momentum current density or stress tensor $T_{jk}$ of the field,
is found to be

$T_{jk}=\rho m\left(w_{j}w_{k}+v_{j}u_{k}\right)+\delta_{jk}\mathcal{L}.$ (23)

Note the distinction between $v_{j}$ and $u_{k}$ in the above equation. The
interpretation of Eqs. (20) and (23) as the respective current densities of
the quantities defined in Eqs. (19) and (22), follows from the conservation
law in Eq. (17), which separates out into an equation of continuity of energy

$\partial_{t}\mathcal{H}+\nabla_{i}\mathcal{S}_{i}=-\frac{\partial\mathcal{L}}{\partial
t},$ (24)

and an equation of continuity of momentum

$\partial_{t}\mathcal{P}_{k}+\nabla_{j}T_{jk}=\frac{\partial\mathcal{L}}{\partial
x_{k}}.$ (25)

Let us cast the above equation into a Cauchy form Aris (2012) and describe the
momentum transport in the reference frame of the fluid. Substituting Eqs. (22)
and (23) into (25) and making use of the continuity of fluid mass, leads to
the following canonical momentum transport equation in the fluid frame:

$m\rho\left(\frac{\partial}{\partial
t}+\mathbf{v}\cdot\mbox{\boldmath$\nabla$}\right)u_{k}=\frac{\partial\mathcal{L}}{\partial
x_{k}}+\nabla_{j}\Pi_{jk},$ (26)

where

$\Pi_{jk}=-T_{jk}+m\rho v_{j}u_{k},$ (27)

is the fluid stress tensor. Notice the difference in sign convention used for
$\Pi_{jk}$ and $T_{jk}$ in the transport equations (26) and (25). In addition,
$\Pi_{jk}$ and $T_{jk}$ differ by a flow-stress term, $m\rho v_{j}u_{k}$, as a
result of the relative motion between the fluid and field frames. The
$\Pi_{jk}$ define a linear map between the surface normal vectors and the
forces acting on these, such that the stress tensor of a fluid may be written
in the form Landau (1959)

$\Pi_{jk}=-P\delta_{jk}+\sigma_{jk},$ (28)

where $P$ is the fluid pressure associated with normal forces and the
$\sigma_{jk}$ account for shearing forces. Note that $P$ represents the
pressure of an infinitesimal volume element which flows with the fluid, and
not at a fixed point of space. As such, for a typical fluid, $P$ is
independent of the flow profile of the fluid. However, for a nonlinear gauge-
coupled fluid, this is no longer the case. Indeed, upon substituting
expressions (18) and (23) into Eq. (27), we find that the stress tensor of the
fluid takes the form of Eq. (28), where

$\sigma_{jk}=-m\rho w_{j}w_{k},$ (29)

is the quantum stress tensor, while the fluid pressure is equivalent to the
Lagrangian density of the field from Eq. (18), such that

$P=-\frac{\hbar^{2}}{4m}\nabla^{2}\rho+\rho\left(\Phi-\eta\right).$ (30)

Hence, by virtue of Eq. (14), the fluid pressure in a nonlinear gauge-coupled
quantum fluid depends explicitly on the flow profile of the fluid.

##  III The nonlinear gauge-coupled superfluid

###  III.1 The origin of the nonlinear gauge potential

The formalism outlined in the previous section was general, in the sense that
the nonlinear gauge potential was viewed as an arbitrary function of the
density. We now turn our attention to a specific model yielding an effective
gauge-coupled Hamiltonian. In particular, it has been shown Edmonds  _et al._
(2013) that the introduction of weak contact-interactions in an optically-
addressed dilute Bose gas of two-level atoms, leads to a density-modulated
gauge potential acting on the condensate. To see this, we begin by noting that
the microscopic Hamiltonian of the dilute cloud, may be written as

$\hat{H}=\hat{H}_{Kin}+\hat{H}_{LM}+\hat{H}_{IP},$ (31)

where $\hat{H}_{Kin}$ is the single-particle kinetic energy operator,
$\hat{H}_{LM}$ describes the light-matter coupling, which we treat semi-
classically, and $\hat{H}_{IP}$ is the interparticle potential

$\hat{H}_{IP}=\sum_{i<j}\left(\sum_{a,b=1,2}g_{ab}\delta(\mathbf{r}_{i}-\mathbf{r}_{j})|\lambda_{a}^{(i)}\lambda_{b}^{(j)}\rangle\langle\lambda_{a}^{(i)}\lambda_{b}^{(j)}|\right)\otimes\mathbb{1}_{ij}.$
(32)

Here $i$ and $j$ label the atoms, $|\lambda_{a}^{(i)}\rangle$ denotes the
$a^{th}$ internal state of atom $i$ and $\mathbb{1}_{ij}$ is the identity on
the complement of the Hilbert space for particles $i$ and $j$. The coupling
constants are related to the associated scattering lengths, in the customary
form $g_{ij}=4\pi\hbar^{2}a_{ij}/m$. We also assume a constant detuning over
space, and write

$\hat{H}_{LM}=\frac{\hbar\Omega}{2}\sum_{j}\left(e^{i\phi(\mathbf{r}_{j})}|\lambda_{1}^{(j)}\rangle\langle\lambda_{2}^{(j)}|+e^{-i\phi(\mathbf{r}_{j})}|\lambda_{2}^{(j)}\rangle\langle\lambda_{1}^{(j)}|\right)\otimes\mathbb{1}_{j},$
(33)

where $\Omega$ is the Rabi frequency characterising the light-matter coupling
and $\phi$ is the phase of the laser field. By assuming that the $N$-body
wavefunction is a product of $N$ identical single-particle wavefunctions, we
obtain Butera  _et al._ (2016) a meanfield Lagrangian in the form of Eq. (1),
where $\hat{H}_{MF}$ acts on the two-component macroscopic wavefunction,
$\psi$, as

$\hat{H}_{MF}\psi\left(\mathbf{r}\right)=\left(\frac{\hat{\mathbf{p}}^{2}}{2m}+\hat{U}_{LM}+\hat{U}_{MF}\right)\psi\left(\mathbf{r}\right)$
(34)

where
$\hat{U}_{LM}=\frac{\hbar\Omega}{2}\left(\begin{array}[]{cc}0&e^{-i\phi(\mathbf{r})}\\\
e^{i\phi(\mathbf{r})}&0\end{array}\right)$, and
$\hat{U}_{MF}=\left(\begin{array}[]{cc}g_{11}\rho_{1}+g_{12}\rho_{2}&0\\\
0&g_{12}\rho_{1}+g_{22}\rho_{2}\end{array}\right)$ describes the meanfield
collisional effects, with $\rho_{i}$ representing the density of atoms
occupying the $i^{th}$ internal state. The full wavefunction may be written
$\sum_{i=\pm}\psi_{i}|\chi_{i}\rangle$, where the $|\chi_{\pm}\rangle$ denote
the eigenstates of $\hat{U}_{LM}+\hat{U}_{MF}$, the so-called interacting
dressed states. When the light-matter coupling is much stronger than the
interparticle potential, which we will assume to be the case in what follows,
these can be approximated by treating $\hat{U}_{MF}$ as a perturbation to
$\hat{U}_{LM}$. Furthermore, by preparing the atoms in a particular dressed
state, $|\chi_{+}\rangle$ say, then, within the adiabatic approximation, we
may set $\psi_{-}$ to zero and obtain a projected meanfield Lagrangian (1),
where the meanfield Hamiltonian governing the dynamics of
$\psi_{+}\equiv\psi$, is obtained in the form

$\hat{H}=\frac{\left(\hat{\mathbf{p}}-\mathbf{A}\right)^{2}}{2m}+W+\frac{\hbar\Omega}{2}+\frac{g}{2}\rho,$
(35)

where $g=\left(g_{11}+g_{22}+2g_{12}\right)/4$, and the geometric synthetic
potentials enter the effective Hamiltonian in the form of a scalar function,
$W=\frac{\hbar^{2}}{2m}|\langle\chi_{+}|\mbox{\boldmath$\nabla$}\chi_{-}\rangle|^{2}$
and a Berry connection
$\mathbf{A}=i\hbar\langle\chi_{+}|\mbox{\boldmath$\nabla$}\chi_{+}\rangle$.
Notice that the $|\chi_{\pm}\rangle$ depend on $\rho$, and $\mathbf{A}$
inherits this dependence: thus we have a density dependent synthetic gauge
potential. To first order, the synthetic potentials read

$\displaystyle W$
$\displaystyle=\frac{\left|\mathbf{A}^{(0)}\right|^{2}}{2m},$ (36)
$\displaystyle\mathbf{A}$
$\displaystyle=\mathbf{A}^{\left(0\right)}+\mathbf{a}\rho,$ (37)

where $\mathbf{A}^{0}=-\frac{\hbar}{2}\mbox{\boldmath$\nabla$}\phi$ is the
single particle contribution to the vector potential and

$\mathbf{a}=\frac{g_{11}-g_{22}}{8\Omega}\mbox{\boldmath$\nabla$}\phi,$ (38)

controls the effective strength and orientation of the gauge potential.

###  III.2 Mechanical momentum transport equation

The nonlinear gauge-coupled superfluid governed by the effective Hamiltonian
from Eq. (35), may be viewed as a particular case of quantum fluid discussed
in section II, where, in an appropriate gauge, $\eta=g\rho/2$ and
$\mathbf{A}=\mathbf{a}\rho$. Hence, the nonlinear scalar term entering the
QHJE (13), now reads

$\Phi=-\mathbf{v}\cdot\mathbf{A}+g\rho,$ (39)

where $\mathbf{v}=\mathbf{u}-\mathbf{a}\rho/m$. As such, the canonical
momentum transport equation for the superfluid is retained in the form of Eq.
(26), where the stress tensor of the fluid takes the form

$\Pi_{jk}=\sigma_{jk}-\delta_{jk}\left(-\frac{\hbar^{2}}{4m}\nabla^{2}\rho+\frac{g}{2}\rho^{2}-\mathbf{J}\cdot\mathbf{A}\right),$
(40)

where $\mathbf{J}=\rho\mathbf{v}$ and $\sigma_{jk}$ is the quantum stress
tensor from Eq. (29). Substituting the canonical flow $u_{k}$ in Eq. (26) for
the mechanical flow $v_{k}=u_{k}-a_{k}\rho/m$, and evaluating
$\partial\mathcal{L}/\partial x_{k}$ holding the fields $\rho$, $\theta$ and
their derivatives constant, leads to a Cauchy equation of mechanical momentum
transport

$m\rho\left(\frac{\partial}{\partial
t}+\mathbf{v}\cdot\mbox{\boldmath$\nabla$}\right)v_{k}=\rho
f_{k}+\nabla_{j}\Pi_{jk},$ (41)

where $\Pi_{jk}$ is given by (40) and the body-force acting on the superfluid,
takes the form

$f_{k}=-\nabla_{k}V+A_{k}\mbox{\boldmath$\nabla$}\cdot\mathbf{v}.$ (42)

Note that in order to obtain the above expression, we have used the continuity
equation (12) and relation
$\mbox{\boldmath$\nabla$}\times\mathbf{a}=\mathbf{0}$. We have also assumed
that $\mathbf{a}$ is independent of time. Typically, body-forces are
associated with external potentials whereas fluid stress is connected with
nonlinear potentials. However, in the case of a nonlinear vector potential, we
see that $\mathbf{A}\left(\rho\right)$ plays a double role in Eq. (41),
carrying implications for both $\Pi_{jk}$ and $f_{k}$.

###  III.3 Canonical flow pressure and body-force of dilation

Since the fluid pressure can be read from the diagonal components of the
stress tensor (see Eq. 28), we have

$P=-\frac{\hbar^{2}}{4m}\nabla^{2}\rho+\frac{g}{2}\rho^{2}-\mathbf{J}\cdot\mathbf{A}.$
(43)

Hence the fluid pressure depends on the overlap of the current density and the
vector potential, and as such, depends explicitly on the canonical flow,
$\mathbf{u}$, of the fluid. In other words, the fluid pressure becomes a
function of both independent dynamical variables $\rho$ and $\mathbf{u}$. One
consequence of this, is that $P$ transforms from one Galilean frame of
reference to another. In order to obtain Galilean covariant transformation
laws where the pressure remains an invariant quantity, clearly, the nonlinear
potentials will have to be transformed in some fashion. This will be covered
elsewhere. Expanding the current density in expression (43), the fluid
pressure may be written as

$P=-\frac{\hbar^{2}}{4m}\nabla^{2}\rho+\left(\frac{g}{2}+\frac{a^{2}}{m}\rho\right)\rho^{2}-\mathbf{J}_{\mathbf{u}}\cdot\mathbf{A},$
(44)

where $a=\left|\mathbf{a}\right|$ and $\mathbf{J}_{\mathbf{u}}=\rho\mathbf{u}$
is the canonical current density. We shall call the pressure term which
depends explicitly on the canonical flow, the canonical flow pressure:

$P_{\mathbf{u}}=-\mathbf{J}_{\mathbf{u}}\cdot\mathbf{A}.$ (45)

Complementing this pressure term, a nonlinear body-force enters Eq. (42) as a
result of the time-dependence of $\mathbf{A}\left(\rho\right)$, namely

$f_{k}^{d}=A_{k}\mbox{\boldmath$\nabla$}\cdot\mathbf{v},$ (46)

which may be interpreted as a body-force of dilation. This follows from the
continuity of fluid mass from Eq. (12), which can be given the form

$\mbox{\boldmath$\nabla$}\cdot\mathbf{v}=-\frac{1}{\rho}\left(\frac{\partial}{\partial
t}+\mathbf{v}\cdot\mbox{\boldmath$\nabla$}\right)\rho.$ (47)

The right hand side of the above equation represents the dilation rate of the
fluid Aris (2012). Therefore, if we track an infinitesimal volume element of
fluid as it flows, an additional body-force is exerted throughout the element
whenever the size of the volume element changes. If for instance, the element
is compressed as a result of entering an increasing surrounding local pressure
field, flow is imparted onto the whole element through $f_{k}^{d}$. This
explains how a shrinking infinitesimal volume element acquires additional
gauge-flow, which must be the result of a body-force.

###  III.4 Ground state canonical flow-dipole in an inhomogeneous superfluid

In this final section, we illustrate an interesting effect of the nonlinear
gauge potential on the ground state wavefunction of an inhomogeneous
superfluid. For the system considered here, the pair of hydrodynamic equations
(12) and (13), are equivalent to the nonlinear Schrödinger equation

$i\hbar\partial_{t}\psi=\left[\frac{\left(\hat{\mathbf{p}}-\mathbf{a}\rho\right)^{2}}{2m}-\mathbf{a}\cdot\mathbf{J}+g\rho+V\right]\psi,$
(48)

where

$\mathbf{J}=\frac{i\hbar}{2m}\left[\psi\left(\mbox{\boldmath$\nabla$}+\frac{i}{\hbar}\mathbf{A}\right)\psi^{*}-\psi^{*}\left(\mbox{\boldmath$\nabla$}-\frac{i}{\hbar}\mathbf{A}\right)\psi\right],$
(49)

is the gauge-covariant current. We consider the case of a monochromatic laser
field with constant phase twist, i.e. a plane wave, such that
$\mathbf{a}\left(\mathbf{r}\right)$ is constant. See Fig. 1 for a description
of a possible experimental realisation. The numerical integration of Eq. (48)
was achieved using the Crank-Nicholson method for a system of dimension $d=2$
with periodic boundary conditions. We present results for a condensate
populated by $N=1600$ particles in a box with side length $L=47$, comprising
$416\times 416$ points. We let the origin of the system coincide with the
center of the box and adopt Cartesian coordinates, denoting the horizontal and
vertical axes by $x$ and $y$, respectively. For our simulation, we have chosen
parameters $\left|\mathbf{a}\right|=0.73\hbar L^{2}$ and
$g=3.66\hbar^{2}L/\left(2m\right)$, and set the orientation of the gauge
potential at angle $\pi/4$ relative to the $x$-axis, e.g.
$\mathbf{\hat{a}}=\left(\mathbf{\hat{x}}+\mathbf{\hat{y}}\right)/\sqrt{2}$. To
establish an inhomogeneous ground state profile, we introduce an immobile
impurity into the system, which we model as a Gaussian potential
$V\left(\mathbf{r}\right)=20e^{-\left|\mathbf{r}\right|^{2}/2}$. The ground
state was obtained using the method of imaginary time propagation.

Figure 1: An example of an experimental setup with a superfluid cloud of atoms
with density $|\psi({\bf r})|^{2}$. An incident laser beam with constant
intensity and phase $\phi({\bf r})=kx$ where $k$ is the wave number of the
light, together with a collisionally induced detuning of the resonant laser
frequency, gives rise to the nonlinear gauge potential.

In the absence of an impurity, the ground state is that of a homogeneous
superfluid, where $\rho=\rho_{0}$ is constant. Assuming there is no superflow,
the phase is constant over space and oscillates periodically in time. However,
unlike a standard weakly-interacting superfluid, the gauge-coupled superfluid
is not at rest, but exhibits a steady current as a result of the gauge-flow
$-\mathbf{a}\rho_{0}/m$. In other words, the ground state of the homogeneous
superfluid is in a steady state of flow even though no spatial phase twists
occur in the system. Next, let us introduce the localised potential into the
system. This leads to a density-depleted region in the vicinity of the
impurity. As a consequence, the gauge-flow is no longer uniform as in the
homogeneous case, but drops in magnitude upon approaching the center of the
impurity. This introduces both non-vanishing transverse and longitudinal
components for the gauge-flow $-\mathbf{a}\rho/m$. The longitudinal component
is a significant energy expense for the system, due to the introduction of
real-time dependence into the wave-amplitude of the state.

Figure 2: Plot showing the non-trivial phase profile adopted by the ground-
state wavefunction in the vicinity of a Gaussian impurity. $17$ contours are
included, equally spaced between $-\pi/2$ and $+\pi/2$.

One may verify numerically that an initial state (with non-vanishing ground
state overlap), evolves in imaginary time in such a way that the divergence of
the gauge-covariant current approaches zero throughout space. Notice that in
order to achieve this and for the ground state density distribution to be
preserved in time, a non-trivial local phase profile must be adopted by the
groundstate wavefunction in order to compensate for the non-steady gauge-
current. In other words, the ground state of the system exhibits a non-
vanishing canonical flow. This is illustrated in FIG. 2, where we have plotted
a series of ground state phase contours in the vicinity of the object, evenly
spaced from $-\pi/2$ to $\pi/2$. The phase is antisymmetric under reversal of
the gauge potential, $\mathbf{a}\rightarrow-\mathbf{a}$. In the bottom left
half of the plot, the phase increases from $0$ to $\pi/2$ as we approach
$\left(-1.25,-1.25\right)$, whereas in the upper right half the phase
decreases from $0$ to $-\pi/2$ as we approach $\left(1.25,1.25\right)$. In
figure 3 we show vector plots of the associated ground-state canonical flow
$\mathbf{u}=\left(\psi^{*}\mbox{\boldmath$\nabla$}\psi-\psi\mbox{\boldmath$\nabla$}\psi^{*}\right)/\left(i\left|\psi\right|^{2}\right)$
and mechanical flow $\mathbf{v}=\mathbf{u}-\mathbf{a}\rho/m$.

Figure 3: Plots showing the ground state canonical flow, $\mathbf{u}$, and
mechanical flow, $\mathbf{v}=\mathbf{u}-\mathbf{A}/m$, in the vicinity of the
impurity. The magnitude of the flow in units $\hbar/\left(2mL\right)$, is
given by the arrow length$\times 0.068$.

Here, we notice that $\mathbf{u}$ takes the form of a flow-dipole, leading to
a mechanical flow field which skirts around the object. In turn, the canonical
flow-dipole has interesting implications for the fluid pressure. In figure 4,
we show the ground state canonical flow pressure and total pressure, computed
using expressions (45) and (44), respectively.

Figure 4: Plots showing the canonical flow pressure, $P_{\mathbf{u}}$, and the
fluid pressure, $P$, in the vicinity of the impurity. The pressure is given in
units $\hbar^{2}/\left(2mL^{5}\right)$ and asymmetric about the origin. The
impurity is compressed along the axis of the nonlinear gauge ($y=x$) and
stretched along the normal axis ($y=-x$).

The flow nonlinearity favours occupation (inoccupation) of the blue (red)
regions in the left image of FIG. 4, leading to an aspherical pressure about
the impurity (right image). This leads to a deformation of the condensate, as
seen from the wave-amplitude plot in FIG. 5.

Figure 5: Ground state wave-amplitude, $\left|\psi\right|$, in the vicinity of
an immobile Gaussian impurity, where $\int
d^{3}\mathbf{r}\left|\psi\right|^{2}=1$.

##  IV CONCLUSION

The hydrodynamic canonical formalism is an ideal framework for understanding
how a nonlinear gauge potential invariably leads to nonlinear flow terms in
the wave equation of a quantum fluid, these resulting from the nonlinear
density-dependence of the kinetic energy density. In turn, two non-trivial
terms emerge in the mechanical momentum-transport equation of a superfluid
subject to a density-modulated gauge potential, in the form of a canonical
flow pressure and a body force of dilation. The nonlinear gauge potential has
interesting implications for an inhomogeneous superfluid, where a nontrivial
local phase is adopted by the ground state wavefunction. The nonlinear gauge
potential also has important consequences for the Galilean covariance of the
fluid, where new transformations laws are required in order to restore the
invariance of the fluid under the transformation group. The canonical flow
pressure should also carry significant implications for the elementary
excitations of the fluid. For instance, one should no longer expect the
velocity of sound to be determined exclusively by the adiabatic compressibily,
since $P$ depends explicitly on the flow. This calls for a generalised
expression relating the velocity of sound to the fluid pressure. Finally, the
nonlinear body force of dilation should appear in the drag force acting on a
moving impurity and may be investigated numerically. For typical quantum
fluids, the drag force is determined by the configuration of the fluid density
in the vicinity of the localised object potential. In contrast, the reaction
to the body force of dilation should occur throughout the whole fluid, taking
place wherever the divergence of the velocity field is non-vanishing. Here, we
would expect the onset of vortex nucleation to depend on the relative
orientation of the gauge potential with respect to the travelling impurity.

###### Acknowledgements.

We would like to thank Manuel Valiente for helpful and interesting
discussions. Y.B acknowledges support from EPSRC CM-CDT Grant No.
EP/L015110/1. L.G.P acknowledges support from the EPSRC CM-CDT. P.Ö
acknowledges support from EPSRC grant No. EP/M024636/1.

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