# Rotational superradiance with Bogoliubov dispersion

Sam Patrick  School of Mathematical Sciences, University of Nottingham,
Nottingham, NG7 2FD, United Kingdom

###### Abstract

Rotational superradiance affects the dynamics of many rotating systems in
nature, through either stimulated or spontaneous extraction of energy and
angular momentum. By now, this process is well-studied in the relativistic
setting, where systems are intrinsically dispersion-free. In many condensed
matter systems, however, dispersion is an unavoidable aspect of the
description for the short wavelength modes. For these systems, how might one
expect superradiance to be modified? In this work, an answer to this question
is provided using an illustrative example. The scattering of linear
excitations of a Bose-Einstein condensate are studied in the presence of a
rotating, draining vortex flow using the full Bogoliubov dispersion relation.
It is shown that dispersion suppresses the extraction of energy and angular
momentum, firstly, by decreasing the superradiant bandwidth, and secondly, by
preventing high-angular momentum modes from superradiating.

##  1 Introduction

Rotational superradiance is a fundamental process involving the amplification
of waves, with implications for the dynamics of many rotating systems in
nature. In general relativity, it allows for the extraction of energy and
angular momentum from rotating black holes, which will eventually shed almost
all of their angular momentum through sponteneous emission [1]. Recently, it
has been demonstrated that this kind of amplification also occurs in condensed
matter systems [2]. One of the fundamental differences with relativity is that
the atomic nature of condensed matter systems gives rise to non-linear
dispersion for short wavelengths. The aim of this work will be to show how
superradiance is altered by a quartic (Bogoliubov) modification to the
dispersion relation.

In general, the term superradiance is used to describe the energy enhancement
of radiation, which allows incident waves to extract energy from the system
they scatter with (see [3, 4] for a review). It is perhaps most famously
associated with energy extraction from rotating black holes, and has played a
central role in fashioning modern understanding of these elusive cosmic
entities [5, 6, 7]. Superradiance is sometimes called the wave equivalent of
the Penrose process [8], where a black hole loses mass and angular momentum by
absorbing particles with negative energy. Indeed, it was along this last line
of enquiry that the principles of black hole thermodynamics were established
[9], leading ultimately to Hawking’s discovery of black hole evaporation [10].

Since Unruh’s demonstration that certain fluids have the capacity to mimick
features of spacetime [11] (which subsequently developed into the field of
analogue gravity [12]), there has been a surge of interest in understanding
superradiance around fluid flows (see e.g. [13, 14, 15]). The prototypical
example of a superradiating fluid system is a rotating draining vortex flow.
In the uniform density approximation, the most general solution of the
irrotational and incompressiblity conditions gives the draining bathtub (DBT)
model for the vortex [16, 17]. This system shares many features with Kerr
black holes, in particular, the existence of a horizon and an ergoregion [18].

The study of such analogue systems is well-motivated from several angles.
Firstly, they can be used to test gravitational phenomena under controlled
laboratory settings. This is important since direct detection of many
interesting processes (in particular, astrophysical Hawking radiation and
superradiance) is beyond the scope of existing technology. Secondly, analogue
systems can probe how certain phenomena behave under modifications to the
governing theory. For example, it was demonstrated in [19, 20] using modified
dispersion relations that the Hawking effect does not rely on the high energy
behaviour of the theory, an important realisation given the lack of knowledge
concerning physics below the Planck scale. Finally, analogue systems are
perfectly poised to mediate the transfer of techniques and ideas between
different fields of physics. This is exemplified by recent findings that the
relaxation of draining vortices can be understood using the physics of light-
rings [21], a concept routinely applied in relativity [22]. Whilst analogue
gravity is usually geared toward providing new input into gravitational
physics, this demonstrates that analogue gravity really is a two way street.

To date, experimental efforts to detect superradiance have focussed on
classical fluids. The first direct detection of rotational superradiance was
performed using surface waves in a water tank experiment containing a draining
vortex [2]. Although the analogy to black hole physics using surface waves is
mathematically precise only in shallow water, and most modelling efforts at
the time had dealt solely with this regime, the amplification was in fact
measured closer to the deep water regime where the system is strongly
dispersive. Following on from this, a theoretical basis for superradiance in
dispersive systems was outlined in [23], and it was shown that amplification
is also expected in deep water. Recently, proposals to observe superradiance
of acoustic beams from rotating absorbers were outlined in [24, 25], and the
first experimental results were obtained in [26].

Another promising analogue gravity system involves the linear excitations of a
Bose-Einstein condensate (BEC) [27, 28]. Indeed, a series of experiments on
one dimensional BECs have successfully tested various aspects of the Hawking
effect [29, 30, 31]. The existence of vortices in BECs also raises the
possibility of observing superradiance there. After Pitaevskii’s original
treatment of the line vortex in a Bose gas [32], vortices in BECs have
received much attention in the literature (see, e.g. [33] for a review). Two
important distinctions with the classical case are that, in a BEC, the fluid
is truly irrotational (in classical systems this is only an approximation) and
the circulation of the flow is quantised. This opens up the tantalising
possibility of studying gravitational phenomena in an effective spacetime
which displays quantum behaviour. It is tempting to go further and say that
this line of research may even provide insight into the quantum nature of
gravity.

DBT superradiance in a BEC has been studied in [34, 35, 36, 37, 38] and the
potential difficulty in experimentally realising draining flows has led some
authors to consider superradiance in purely rotating systems [39, 40]. With
exception of [40, 41] which dealt with non-draining flows, the effects of
short wavelength dispersion are usually ignored on the grounds that the
quantum pressure term (which gives rise to the modified dispersion) remains
small on suitably large length scales. Within this approximation, often called
the hydrodynamic approximation, the equations of motion become second order in
spatial derivatives, which means that one benefits from all the standard
techniques used to study superradiance based on the properties of second order
differential equations. The problem with this approximation is that systems
containing horizons naturally probe the short-wavelength behaviour of the
system, since out-going waves become increasingly blue-shifted as they are
traced back toward their place of origin. Hence, a fully consistent treatment
requires the inclusion of dispersion. It will be the aim of the coming
sections to develop such a treatment.

##  2 Objectives

The main tool used in this work will be a combination of multiple scale
analysis with matched asymptotic techniques. These methods, often brought
under the umbrella term of WKB approximations [42], provide an intuitive
picture of wave scattering by recasting the problem in terms of effective
particles, with accuracy improving in the limit of short wavelengths. This
approach has been shown to yield accurate predictions for high angular
momentum modes when compared with laboratory experiments [21]. In fact, the
method also provides a good indication of the general behaviour of the
scattering coefficients even for low angular momentum modes [23].

There are several questions this paper will aim to address. Firstly, the well-
known condition for black hole superradiance is,

$\omega<m\Omega_{h}^{\mathrm{rot}},$ (1)

where $\omega$ and $m$ determine the wave energy and angular momentum
respectively, and $\Omega_{h}^{\mathrm{rot}}$ is the rotational velocity of
the spacetime on the horizon in radians per second. The same condition applies
in fluid systems in the non-dispersive approximation, with
$\Omega_{h}^{\mathrm{rot}}$ now playing the role of the rotational fluid
velocity. It was shown in [23] that a similar condition holds for the
amplification of deep water gravity waves around the DBT. The first task here
will be to establish equivalent of (1) for the Bogoliubov dispersion relation.

The second aim will be to investigate the behaviour of the reflection
coefficient, following the method established in [23]. To do this, the
solutions of the dispersion relation will be classified into separated
categories represented by a “Feynman” diagram, which depict the mode
trajectories through phase space. Each category corresponds to a particular
structure in the transfer matrix, which is used to compute the scattering
coefficients. This will allow for a comparison of the amount of amplification
between the dispersive and non-dispersive cases.

Finally, the system will be quantized following the procedure outline in [43]
and the spontaneous radiation of energy and angular momentum due to
amplification of vacuum fluctuations studied. Note that since only scattering
in the presence of real turning points (defined later) of the dispersion
relation is considered, the rates of energy and angular momentum loss include
only the contributions of superradiant modes. In reality, there will also be a
(nearly) thermal spectrum of modes emitted by the vortex, resulting from the
additional scattering which is not captured by the turning points. This
thermal emission is simply the Hawking effect, which has been well-studied in
the context of analogue models of gravity, see e.g. [44, 45], and could
feasibly be incorporated into the present framework using the more thorough
scattering treatments described therein.

The remainder of this paper is structured as follows. In Section 3, the
equations governing the background and excitations of a BEC are introduced. In
Section 4, the WKB method is used to find approximate solutions to the wave
equation, and it is shown that scattering between different WKB modes can be
estimated by computing the amount of mode mixing around turning points. These
techniques are then applied in Sections 5 and 6 to characterise the different
scattering possibilities in the non-dispersive and dispersive cases
respectively. In Section 7, the spontaneous emission of superradiant modes due
to vacuum fluctuations is computed, and the rates of energy and angular
momentum loss from the vortex compared for the non-dispersive and dispersive
cases. Finally, the relevance of these findings are discussed in Section 8.

##  3 Set-up

In a BEC, the mean-field condensate wavefunction $\Psi(t,\mathbf{x})$
satisfies the Gross-Pitaevskii equation (GPE),

$i\hbar\partial_{t}\Psi=-\frac{\hbar^{2}}{2M}\nabla^{2}\Psi+V(\mathbf{x})\Psi+g|\Psi|^{2}\Psi=0,$
(2)

where $\mathbf{x}$ is the position on a 2D plane, $M$ is the mass of the
particles in the condensate, $V$ is an external trapping potential and $g$ is
the interaction parameter dependent only on 2-point collisions between
particles [33]. The GPE can be derived from the following action,

$\mathcal{S}_{\mathrm{GPE}}=\int
dtd^{2}\mathbf{x}\left[\frac{i\hbar}{2}\left(\dot{\Psi}\Psi^{*}-\dot{\Psi}^{*}\Psi\right)-\frac{\hbar^{2}}{2M}\bm{\nabla}\Psi\cdot\bm{\nabla}\Psi^{*}-V|\Psi|^{2}-\frac{1}{2}g|\Psi|^{4}\right],$
(3)

where overdot denotes the derivative with respect to time and $\Psi^{*}$ is
the complex conjugate of the wavefunction.

Under the Madelung transform,

$\Psi(t,\mathbf{x})=\sqrt{\rho(t,\mathbf{x})}e^{-i\Theta(t,\mathbf{x})/\hbar},$
(4)

the GPE reduces to the equations for an irrotational fluid flow,

$\displaystyle\partial_{t}\rho+\bm{\nabla}\cdot(\rho\mathbf{v})=$
$\displaystyle\ 0,$ (5a)
$\displaystyle\tfrac{1}{2}M\mathbf{v}^{2}+g\rho+V(\mathbf{x})+V_{\mathrm{Q}}=$
$\displaystyle\ \mu,$ (5b)

with $\rho$ the fluid density, $\mathbf{v}=\bm{\nabla}\Phi$ the velocity field
and $\Phi=-\Theta/M$ the velocity potential. The fluid is assumed stationary
so that $\partial_{t}\Theta=\mu=\mathrm{const}$, where $\mu$ is the chemical
potential associated with the removal of a particle from the condensate.
Equations (5a) and (5b) are almost identical to the classical equations except
for the addition of the quantum pressure term,

$V_{\mathrm{Q}}=-\frac{\hbar^{2}}{2M}\frac{\nabla^{2}\sqrt{\rho}}{\sqrt{\rho}}.$
(6)

Note also that (5a) is simply the conserved current associated with the
internal symmetry of (3). The conserved quantity $N=\int
d^{2}\mathbf{x}~{}\rho$ gives the number of particles in the condensate.

###  3.1 Velocity field

Consider now the form of the velocity field. I will assume that $\mathbf{v}$
is independent $\theta$ and that the density is approximately uniform, i.e
$\rho\approx\mathrm{const}$. In this case, (5a) becomes
$\bm{\nabla}\cdot\mathbf{v}=0$ and (5b) implies
$\bm{\nabla}\times\mathbf{v}=0$. The unique velocity profile for a $\theta$
independent fluid is then,

$\mathbf{v}=-\frac{D}{r}\vec{\mathbf{e}}_{r}+\frac{C}{r}\vec{\mathbf{e}}_{\theta},$
(7)

where $C$ and $D$ are constants. In this work, I will be interested in
draining profiles, hence $D$ is taken to be positive. $C$ is the circulation
parameter which can be either positive or negative depending on the direction
of rotation (here I choose $C>0$). Since $\Psi$ must be periodic in $\theta$
to satisfy the boundary conditions, $C$ must be of the form,

$C=\hbar\ell/M,$ (8)

where $\ell$ is an integer called the winding number. This is the well-known
result that circulation in a BEC is quantised [32]. The flow profile in (7) is
known as the DBT in the literature. Note that vortices with winding number
higher than $\ell=1$ are usually unstable [46] and in fact, this instability
has been argued to be related to the presence of an ergoregion (the same
mechanism responsible for superradiance) [40]. Stabilisation mechanisms, e.g.
via trapping potentials, have however been demonstrated [47, 48].

In classical fluids, the angular component
$v_{\theta}=\vec{\mathbf{e}}_{\theta}\cdot\mathbf{v}$ in (7) is often used as
an idealisation of realistic velocity profiles [21]. In the present case,
however, it is the true (and only) form of the angular velocity profile for an
axisymmetric system. For the radial profile
$v_{r}=\vec{\mathbf{e}}_{r}\cdot\mathbf{v}$, one can imagine pumping atoms out
of the system near $r=0$ at a rate $\dot{N}=-\rho D$ (the possibility of
experimentally realising such a configuration has been discussed in e.g. [49,
50]). In order to keep $N$ fixed, one could then resupply atoms at the same
rate at the outer edge of the condensate.

###  3.2 Fluctuations

Now consider fluctuations of the condensate density and phase,

$\rho\to\rho(1+\eta),\quad\Phi\to\Phi+\phi.$ (9)

Linearising (5a) and (5b) in the constant density approximation yields,

$\begin{split}D_{t}\phi+c^{2}\eta-\Lambda\nabla^{2}\eta=&\ 0,\\\
D_{t}\eta+\nabla^{2}\phi=&\ 0,\end{split}$ (10)

where $D_{t}=\partial_{t}+\mathbf{v}\cdot\bm{\nabla}$ is the material
derivative and the constants $c$ and $\Lambda$ are given by,

$c=\sqrt{g\rho/M},\quad\Lambda=\hbar^{2}/4M^{2}.$ (11)

For $\Lambda=0$, the system is non-dispersive and all wavelengths will
propagate at the same speed $c$. When $\Lambda\neq 0$, shorter-wavelengths
travel faster than $c$ and the system becomes “superluminally” dispersive
111The name derives from the analogy with relativity where $c$ represents the
speed of light.. Since the system is invariant under a rescaling by two
parameters, I will set $c=D=1$ from here on. The background is then completely
characterised by choosing $C$ and $\Lambda$.

Note that the equations in (10) can be derived by minimising the action,

$\mathcal{S}=\int dtd^{2}\mathbf{x}\left[\tfrac{1}{2}\phi
D_{t}\eta-\tfrac{1}{2}\eta
D_{t}\phi-\tfrac{1}{2}\eta^{2}-\tfrac{1}{2}(\bm{\nabla}\phi)^{2}-\tfrac{1}{2}\Lambda(\bm{\nabla}\eta)^{2}\right],$
(12)

where the term in square brackets is the Lagrangian density $\mathcal{L}$.

###  3.3 Conserved currents

By Noether’s theorem, symmetries of the action give rise to conserved currents
[51]. In particular, the transformation,

$\phi\to\phi+\delta\phi,$ (13)

and similarly for $\eta$, is called a symmetry if the corresponding change in
the Lagrangian can be written in the form
$\delta\mathcal{L}=\partial_{t}f+\bm{\nabla}\cdot\mathbf{F}$, since this
leaves $\mathcal{S}$ invariant. When the equations of motion are satisfied,
$\delta\mathcal{L}$ is given by,

$\delta\mathcal{L}=\partial_{t}\left(\frac{\partial\mathcal{L}}{\partial\dot{\phi}}\delta\phi+\frac{\partial\mathcal{L}}{\partial\dot{\eta}}\delta\eta\right)+\bm{\nabla}\cdot\left(\frac{\partial\mathcal{L}}{\partial\bm{\nabla}\phi}\delta\phi+\frac{\partial\mathcal{L}}{\partial\bm{\nabla}\eta}\delta\eta\right).$
(14)

Combining these two forms for $\delta\mathcal{L}$, gives the following
conservation law,

$\partial_{t}\rho+\bm{\nabla}\cdot\mathbf{J}=0,$ (15)

where the components of the current are given by,

$\begin{split}\rho[\phi]=&\
\frac{\partial\mathcal{L}}{\partial\dot{\phi}}\delta\phi+\frac{\partial\mathcal{L}}{\partial\dot{\eta}}\delta\eta-f,\\\
\mathbf{J}[\phi]=&\
\frac{\partial\mathcal{L}}{\partial\bm{\nabla}\phi}\delta\phi+\frac{\partial\mathcal{L}}{\partial\bm{\nabla}\eta}\delta\eta-\mathbf{F}.\end{split}$
(16)

This $\rho$ (which is the time component of the current) is not to be confused
with the density defined earlier.

Due to the $t$ and $\theta$ independence of $\mathbf{v}$, the action will be
invariant under $t$ and $\theta$ translations. The corresponding conservation
laws are the conservation of energy and angular momentum respectively. In what
follows, I will be particularly interested in the radial components of these
currents. These are,

$J_{E}^{r}[\phi]=\left(-\tfrac{1}{2}v_{r}\eta-\partial_{r}\phi\right)\partial_{t}\phi+\left(\tfrac{1}{2}v_{r}\phi-\Lambda\partial_{r}\eta\right)\partial_{t}\eta,$
(17)

for the energy current and,

$J_{L}^{r}[\phi]=\left(-\tfrac{1}{2}v_{r}\eta-\partial_{r}\phi\right)\partial_{\theta}\phi+\left(\tfrac{1}{2}v_{r}\phi-\Lambda\partial_{r}\eta\right)\partial_{\theta}\eta,$
(18)

for the angular momentum current.

###  3.4 Mode decomposition

Due to the symmetry of background, it is beneficial decompose the fields
$\phi$ and $\eta$ into their different frequency $\omega$ and azimuthal $m$
components. In this paper, I will work with the following notation,

$\phi=\sum_{\lambda}\left(\alpha_{\lambda}\varphi_{\lambda}+\alpha^{*}_{\lambda}\varphi^{*}_{\lambda}\right),\qquad\eta=\sum_{\lambda}\left(\alpha_{\lambda}n_{\lambda}+\alpha^{*}_{\lambda}n^{*}_{\lambda}\right),$
(19)

with $\lambda$ denoting a particular $\omega,m,j$ triplet. The field modes
are,

$\varphi_{\lambda}\equiv\varphi_{\omega
mj}(t,\theta,r)=\tilde{\varphi}_{j}(\omega,m,r)e^{im\theta-i\omega t},$ (20)

and similarly for $n_{\lambda}$. The $\mathbb{C}$-fields $\varphi$ and
$\varphi^{*}$ are often called positive and negative frequency components
respectively. The $\alpha_{\lambda}$ are constant amplitudes multiplying the
$\mathbb{C}$-fields, which need to be taken in a symmetric combination due to
the fact that $\phi$ and $\eta$ are both real. The sum over $\lambda$ is short
for,

$\sum_{\lambda}=\sum_{m,j}\int d\omega,$ (21)

where the integral runs from $\omega\in[0,\infty)$ and the azimuthal sum is
over $m\in(-\infty,\infty)$. Finally, $\tilde{\varphi}_{j}$ is a particular
solution to the radial equations of motion, which are obtained by substituting
(20) into (10),

$\begin{split}-i\left(\omega-\frac{mC}{r^{2}}\right)\tilde{\varphi}_{j}+\frac{1}{r}\partial_{r}\tilde{\varphi}_{j}+\left(1+\frac{\Lambda
m^{2}}{r^{2}}\right)\tilde{n}_{j}-\frac{\Lambda}{r}\partial_{r}\tilde{n}_{j}-\Lambda\partial_{r}^{2}\tilde{n}_{j}=&\
0,\\\
-i\left(\omega-\frac{mC}{r^{2}}\right)\tilde{n}_{j}+\frac{1}{r}\partial_{r}\tilde{n}_{j}-\frac{m^{2}}{r^{2}}\tilde{\varphi}_{j}+\frac{1}{r}\partial_{r}\tilde{\varphi}_{j}+\partial_{r}^{2}\tilde{\varphi}_{j}=&\
0.\end{split}$ (22)

When $\Lambda=0$, these combine into a single second order ordinary
differential equation and one will have $j=1,2$. Conversely, for $\Lambda\neq
0$, (22) has four independent solutions, i.e. $j=1,2,3,4$.

Due to the linearity of the equations of motion (10), each $\lambda$ component
evolves independently and can therefore be considered separately. Similarly,
the positive and negative frequency parts will also evolve independently. The
Lagrangian governing the individual field modes is,

$\begin{split}\mathcal{L}_{\mathbb{C}}=\tfrac{1}{2}\big{(}\tfrac{1}{2}\varphi_{\lambda}^{*}D_{t}n_{\lambda}&\
+\tfrac{1}{2}\varphi_{\lambda}D_{t}n^{*}_{\lambda}-\tfrac{1}{2}n^{*}_{\lambda}D_{t}\varphi_{\lambda}-\tfrac{1}{2}n_{\lambda}D_{t}\varphi^{*}_{\lambda}\\\
&\
-n_{\lambda}n^{*}_{\lambda}-\bm{\nabla}\varphi_{\lambda}\cdot\bm{\nabla}\varphi^{*}_{\lambda}-\Lambda\bm{\nabla}n_{\lambda}\cdot\bm{\nabla}n^{*}_{\lambda}\big{)}.\end{split}$
(23)

Applying Noether’s theorem for the internal symmetry
$\varphi_{\lambda}\to\varphi_{\lambda}e^{-i\alpha}$ (and also for
$\varphi^{*}_{\lambda},n_{\lambda},n^{*}_{\lambda}$) one finds the
conservation of the norm current, whose components are,

$\begin{split}\rho_{N}[\varphi]=&\
\frac{i}{2}\left(\varphi_{\lambda}n^{*}_{\lambda}-n_{\lambda}\varphi^{*}_{\lambda}\right),\\\
\mathbf{J}_{N}[\varphi]=&\
\frac{i}{2}\Big{\\{}\mathbf{v}\left[\varphi_{\lambda}n^{*}_{\lambda}-n_{\lambda}\varphi^{*}_{\lambda}\right]+\varphi_{\lambda}\bm{\nabla}\varphi^{*}_{\lambda}-(\bm{\nabla}\varphi_{\lambda})\varphi^{*}_{\lambda}\\\
&\qquad+\Lambda\left[n_{\lambda}\bm{\nabla}n^{*}_{\lambda}-(\bm{\nabla}n_{\lambda})n^{*}_{\lambda}\right]\Big{\\}}.\end{split}$
(24)

This motivates the definition of the following inner product of two functions
(which solve the equations of motion),

$(\varphi_{\lambda_{1}},\varphi_{\lambda_{2}})=\frac{i}{2}\int
d^{2}\mathbf{x}\left(\varphi_{\lambda_{1}}n^{*}_{\lambda_{2}}-n_{\lambda_{1}}\varphi^{*}_{\lambda_{2}}\right).$
(25)

Since this is independent of $t$, the following quantity is conserved
radially,

$\begin{split}W[\varphi_{\lambda_{1}},\varphi_{\lambda_{2}}]=&\
\frac{i}{2}r\Big{\\{}v_{r}\left[\varphi_{\lambda_{1}}n^{*}_{\lambda_{2}}-n_{\lambda_{1}}\varphi^{*}_{\lambda_{2}}\right]+\varphi_{\lambda_{1}}\partial_{r}\varphi^{*}_{\lambda_{2}}-(\partial_{r}\varphi_{\lambda_{1}})\varphi^{*}_{\lambda_{2}}\\\
&\qquad+\Lambda\left[n_{\lambda_{1}}\partial_{r}n^{*}_{\lambda_{2}}-(\partial_{r}n_{\lambda_{1}})n^{*}_{\lambda_{2}}\right]\Big{\\}}.\end{split}$
(26)

##  4 WKB solutions

In this section, I drop the subscript $\lambda$ to avoid complicating the
notation. It will be restored in later sections where it is necessary. For the
velocity profile in (7), the equations of motion (10) do not admit closed form
solutions. However, if the fluctuations vary on a scale $k^{-1}$ which is much
smaller that the scale $L$ over which $\mathbf{v}$ changes, one can define a
small parameter $\epsilon=1/kL\ll 1$ and write,

$\varphi=\mathcal{A}(\mathbf{x},t)\exp\left(\frac{iS(\mathbf{x},t)}{\epsilon}\right),\quad
n=\mathcal{B}(\mathbf{x},t)\exp\left(\frac{iS(\mathbf{x},t)}{\epsilon}\right),$
(27)

where $\mathcal{A}$ and $\mathcal{B}$ are local amplitudes and $S$ is the
phase. The solution is obtained by substituting (27) in (10) and solving order
by order in $\epsilon$. In practice, the first two orders are usually all that
is needed to obtain a good approximation. The approximation improves as the
wavelength decreases and $\epsilon$ becomes smaller. In what follows, this
will be the case for large $m$.

###  4.1 Dispersion relation

At $\mathcal{O}(\epsilon^{0})$, the equations of motion (10) give the
Hamilton-Jacobi equation,

$\left(\partial_{t}S+\mathbf{v}\cdot\bm{\nabla}S\right)^{2}-(\bm{\nabla}S)^{2}-\Lambda(\bm{\nabla}S)^{4}=0.$
(28)

Identifying the frequency and wavevector through,

$\omega=-\partial_{t}S,\qquad\mathbf{k}=\bm{\nabla}S,$ (29)

with $k=||\mathbf{k}||$, the Hamilton-Jacobi equation is equivalent to the
dispersion relation,

$\Omega^{2}\equiv\left(\omega-\mathbf{v}\cdot\mathbf{k}\right)^{2}=k^{2}+\Lambda
k^{4},$ (30)

which determines the relationship between the local values of $\omega$ and
$\mathbf{k}$ when $\mathbf{v}$ is varying. Note that for the case of
$\mathbf{v}=0$, (30) is the Bogoliubov dispersion relation originally derived
in [52]. Using this notation, (10) can be used to write a leading order
relation between the amplitudes,

$\mathcal{B}=i\Omega f^{-1}\mathcal{A},\quad f=1+\Lambda k^{2}.$ (31)

Since the dispersion relation is quadratic in $\omega$, it has two branches,

$\omega^{\pm}_{D}=\mathbf{v}\cdot\mathbf{k}\pm\sqrt{k^{2}+\Lambda k^{4}},$
(32)

with $\omega^{+}_{D}$ the upper branch and $\omega^{-}_{D}$ the lower branch.
The group velocity defines the direction of travel of a mode and is given by,

$\bm{v}_{g}=\bm{\nabla}_{\mathbf{k}}\omega=\mathbf{v}\pm\mathbf{k}\frac{1+2\Lambda
k^{2}}{\sqrt{k^{2}+\Lambda k^{4}}}.$ (33)

This also determines the direction in which energy is carried.

As (28) is a first order partial differential equation, its solution can be
obtained by first splitting into a system of ordinary differential equations
and solving these for characteristic curves. These characteristics can be
found from an effective Hamiltonian $\mathcal{H}$ which, using (32), can be
expressed concisely as,

$\mathcal{H}=-\frac{1}{2}(\omega-\omega_{D}^{+})(\omega-\omega_{D}^{-}).$ (34)

The characteristics are obtained as the solutions of Hamilton’s equations,

$\dot{x}^{\mu}=\frac{\partial\mathcal{H}}{\partial
k_{\mu}},\qquad\dot{k}_{\mu}=-\frac{\partial\mathcal{H}}{\partial x^{\mu}}$
(35)

where $x^{\mu}=(\mathbf{x},t)$, $k_{\mu}=(\mathbf{k},-\omega)$. In this
section, the overdot denotes the derivative with respect to $\tau$ which
parametrises the characteristics. Solving the system of equations (35) gives
the coordinates and the conjugate momenta in terms of the parameter $\tau$,
i.e. $x^{\mu}=x^{\mu}(\tau)$ and $k_{\mu}=k_{\mu}(\tau)$. The phase part of
$\varphi$ in (27) can then be reconstructed by integrating (29) along the
different trajectories. In addition to (35), the solutions are required to
satisfy the Hamiltonian constraint,

$\mathcal{H}=0,$ (36)

which guarantees that they lie on one of the two branches of the dispersion
relation (30).

The analysis can be simplified by specifying to the $t$ and $\theta$
independent system introduced in Section 3.1. In polar coordinates, the wave
vector has components,

$\mathbf{k}=(p,m/r),\qquad k=\sqrt{p^{2}+m^{2}/r^{2}},$ (37)

where $p$ is the radial wave vector. By Hamilton’s equations (35), $\omega$
and $m$ are fixed for a given mode, hence, the only variables appearing in the
effective Hamiltonian are $r$ and $p$. The equation $\mathcal{H}=0$ can then
be solved directly for $p=p(r)$, thereby circumventing the need to introduce a
parameter $\tau$ and solve (35) for $r=r(\tau)$ and $p=p(\tau)$. The highest
power of $p$ in $\mathcal{H}$ will determine the number of solutions that
exist. From here on, these solutions will be labelled $p^{l}$ and throughout
this work, an upper index will be used to indicate that a particular quantity
is associated to the $l$ solution of the dispersion relation 222Except when
discussing the branches of the dispersion relation in (32) in which case the
superscript $\pm$ indicates the upper and lower branches respectively..

###  4.2 Transport equation

At $\mathcal{O}(\epsilon^{1})$, the equations of motion (10) give a transport
equation for the amplitude,

$\partial_{t}(f^{-1}\Omega\mathcal{A}^{2})+\bm{\nabla}\cdot(\bm{v}_{g}f^{-1}\Omega\mathcal{A}^{2})=0,$
(38)

which can be solved for $\mathcal{A}$ using the solutions of the Hamilton-
Jacobi equation (28). This equation describes how the amplitude evolves
adiabatically along the characteristics. Using the $t$ and $\theta$ symmetric
system of Section 3.1, the amplitude is simply,

$\mathcal{A}=|qr|^{-\frac{1}{2}}\mathcal{N},\qquad q\equiv
q(r,p^{l})=f^{-1}(r,p^{l})\mathcal{H}^{\prime}(r,p^{l}),$ (39)

where $\mathcal{N}$ is a constant. I have also used
$\vec{\mathbf{e}}_{r}\cdot\bm{v}_{g}\Omega=\mathcal{H}^{\prime}$ where the
prime denotes derivative with respect to $p$. Hence, the general expression
for the radial part of the mode becomes,

$\tilde{\varphi}_{j}=\sum_{l}|q^{l}r|^{-\frac{1}{2}}\mathcal{N}^{l}_{j}e^{i\int
p^{l}dr},$ (40)

where the sum over $l$ accounts for the fact that a given solution of the
radial equation may be a combination of WKB modes, and the constants
$\mathcal{N}^{l}_{j}$ will be different for each of the $j$ independent
solutions.

The norm of an individual $l$ WKB mode is obtained from (24) as,

$\rho_{N}[\tilde{\varphi}_{j}^{l}]=\frac{\Omega^{l}}{f^{l}}|\tilde{\varphi}_{j}^{l}|^{2},$
(41)

and the energy density is simply the same quantity multiplied by $\omega$.
Since $f>0$ for propagating waves, the $\omega>0$ modes with negative energy
are those which lie on the $\omega_{D}^{-}$ of the dispersion relation where
$\Omega<0$.

Next, inserting the full expression (40) into (26) gives,

$\begin{split}W[\tilde{\varphi}_{j},\tilde{\varphi}_{j^{\prime}}]=&\
\frac{1}{2}\sum_{l,l^{\prime}}\Bigg{[}v_{r}\left(\frac{\Omega^{l}}{f^{l}}+\frac{\Omega^{l^{\prime}}}{f^{l^{\prime}}}\right)+(p^{l}+p^{l^{\prime}})\left(1+\Lambda\frac{\Omega^{l}\Omega^{l^{\prime}}}{f^{l}f^{l^{\prime}}}\right)\Bigg{]}\frac{\mathcal{N}^{l}_{j}{\mathcal{N}^{l^{\prime}}_{j^{\prime}}}^{*}}{|q^{l}q^{l^{\prime}}|^{\frac{1}{2}}}e^{i\int(p^{l}-p^{l^{\prime}})dr},\end{split}$
(42)

where the sum is performed over all pairings of modes contained in the
different solution. In this notation, $\tilde{\varphi}_{j^{\prime}}$ can be a
different independent solution to the radial equation for the same $m,\omega$,
as encoded by the different set of coefficients
$\mathcal{N}^{l^{\prime}}_{j^{\prime}}$.

Since (26) is constant in $r$ by definition, and the phase term will cause
oscillations if $l\neq l^{\prime}$, the factor in square brackets must vanish
for these cases. This is proven in [45] for the case of weakly dispersive
gravity waves, which also obey a quartic dispersion relation. Using (30), the
factor in square brackets simplifies for $l=l^{\prime}$ and one finds,

$W[\tilde{\varphi}_{j},\tilde{\varphi}_{j^{\prime}}]=\sum_{l}\mathrm{sgn}(q^{l})\mathcal{N}^{l}_{j}{\mathcal{N}^{l}_{j^{\prime}}}^{*}=\mathrm{const}.$
(43)

This is the key relation from which one can deduce the existence of
superradiance in the system, and is equivalent to the energy current up to a
factor of $\omega$.

###  4.3 Mode mixing

It is important to note that if the WKB solutions are everywhere valid, then
each mode will evolve adiabatically along $r$ without exchanging energy with
any of the others (in this case, (43) is trivially satisfied owing to
constancy of the $\mathcal{N}_{j}^{l}$). The locations where the WKB solutions
break down thus play an important role in determining the amount of energy
exchanged between modes (or mode mixing). The key assumption underlying WKB is
a slowly varying amplitude compared to the phase, hence, the worst possible
violation of the approximation occurs when the amplitude suddenly diverges.
Using (39), one can see that this occurs if $\mathcal{H}^{\prime}=0$ somewhere
in the system. From Hamilton’s equations (35), this is equivalent to
$\dot{r}=0$. In other words, these are the locations where an analogous
classical particle with energy-momentum relation (30) comes to a halt and
reverses it’s direction, i.e. the classical turning points. Denoting these
locations $r_{tp}$, they are found by solving the simultaneous equations,

$\mathcal{H}_{tp}=0,\qquad\partial_{p}\mathcal{H}_{tp}=0,$ (44)

where the subscript ${tp}$ denotes that a quantity has been evaluated on a
turning point. Solving these equations yields the pair $(r_{tp},p_{tp})$, i.e.
the location of the turning point and the local momentum there.

The turning points also have a simple interpretation in terms of the
dispersion relation. Using (34), the conditions in (44) are equivalent to,

$\omega=\omega_{D}^{\pm}(r_{tp},p_{tp}),\qquad\partial_{p}\omega_{D}^{\pm}(r_{tp},p_{tp})=0,$
(45)

and thus, the turning points are the extrema of the dispersion relation in the
$p$ direction. It is then easy to see why the $r_{tp}$ are related to mode
mixing. Consider two $p^{l}$ which are initially distinct solutions of the
dispersion relation. As $r$ is varied (and the $\omega_{D}^{\pm}$ change
shape) the two solutions can approach one another if there is an extremum in
between them. When both solutions sit on the extremum, they have equal $p$ and
moving past the turning point, the two modes move off in the complex plane. In
other words, a turning point converts two real solutions of (36) into complex
solutions, and in doing so facilitates an interaction between them.

To overcome the breakdown of WKB at turning points, there is an established
technique in the literature based on a matched asymptotic expansion. This
method is described fully in e.g. [23, 53]. The spirit of the calculation is
to expand $\mathcal{H}$ around the turning point, promote this to a wave
equation and then write down an exact solution (which turns out to be a
combination of Airy functions). Next, one looks at the asymptotic form of the
solution far away from the turning point and notices that this is simply a
particular combination of WKB modes. However, if the asymptotic solution is
approached rapidly then one can simply compare the WKB amplitudes at the
turning point itself. This method improves as $m$ increases since the argument
of the Airy function grows with $m$, which means it’s asymptotic value becomes
a better approximation closer to the turning point. The matrix which relates
the WKB modes either side of $r_{tp}$ is,

$\begin{pmatrix}A^{\mathrm{R}}\\\
A^{\mathrm{L}}\end{pmatrix}=T\begin{pmatrix}A^{\downarrow}\\\
A^{\uparrow}\end{pmatrix},\qquad
T=e^{\frac{i\pi}{4}}\begin{pmatrix}1&-\frac{i}{2}\\\
-i&\frac{1}{2}\end{pmatrix},$ (46)

when the modes are real for $r<r_{tp}$ and complex for $r>r_{tp}$ and,

$\begin{pmatrix}A^{\uparrow}\\\
A^{\downarrow}\end{pmatrix}=\widetilde{T}\begin{pmatrix}A^{\mathrm{R}}\\\
A^{\mathrm{L}}\end{pmatrix},\qquad\widetilde{T}=e^{\frac{i\pi}{4}}\begin{pmatrix}\frac{1}{2}&-\frac{i}{2}\\\
-i&1\end{pmatrix},$ (47)

when the modes are complex for $r<r_{tp}$ and real for $r>r_{tp}$. Here, the
propagating modes $\mathrm{R}$ and $\mathrm{L}$ are defined so that
$p^{\mathrm{R}}>p^{\mathrm{L}}$. The complex modes are defined so that
$\uparrow$ is the one which grows in the direction of increasing $r$ and
$\downarrow$ decays.

###  4.4 Transfer matrix

To relate the WKB amplitudes in the asymptotic regions of the flow, one can
define an $M\times M$ matrix (where $M$ is the number of modes in the system)
called the transfer matrix, $\mathcal{M}$. Before writing down $\mathcal{M}$,
it will be instructive to establish some preliminaries.

Firstly, since I will ultimately be interested in relations between the
different mode amplitudes as determined by (26) (which includes a factor of
$r$ out the front) it is useful to define a new set of WKB modes,

$R(r)=\sum_{l}A^{l}(r)e^{i\int p^{l}(r)dr},$ (48)

which are related to those in (40) through $R(r)=\sqrt{r}\tilde{\varphi}(r)$
(note that I have dropped the subscript $j$, denoting the solution to the
radial equation, since subscripts in this section will be used to indicate the
$r$ location where a quantity is evaluated). The $\sqrt{r}$ factors out the
part of the amplitude which increases simply due to the fact that a wave
moving in the direction of decreasing $r$ gets focussed onto a smaller disk.

Now, define a column vector $\mathbf{A}$, which consists of the WKB amplitudes
$A^{l}$, and a row vector $\mathbf{P}$, containing the WKB phases $e^{i\int
p^{l}dr}$. Let’s say that we know the full details of the amplitudes and
phases at a point $r_{b}$ and we want to transport this solution to another
point $r_{a}<r_{b}$ where the WKB approximation holds everywhere along the
path. First, the full solution at $r_{b}$ is given by
$R_{b}=\mathbf{P}_{b}\cdot\mathbf{A}_{b}$. Then, defining the factor,

$\mathcal{F}^{l}_{ab}=\left|\frac{q^{l}_{b}}{q^{l}_{a}}\right|^{\frac{1}{2}}\exp\left(-i\int^{r_{b}}_{r_{a}}p^{l}dr\right),$
(49)

the amplitudes can be transported as,

$\mathbf{A}_{a}=\mathrm{diag}(\mathcal{F}_{ab}^{l})\mathbf{A}_{b},$ (50)

so that the solution at $r_{a}$ can be defined with respect to original phase
vector through $R_{a}=\mathbf{P}_{b}\cdot\mathbf{A}_{a}$.

The transfer matrix $\mathcal{M}$ relates the mode amplitudes in the
asymptotic regions,

$\mathbf{A}_{0}=\mathcal{M}\mathbf{A}_{\infty},$ (51)

where in defining these regions, it suffices (at the considered level of
approximation) to find two locations $r_{0,\infty}$ such that there are no
turning points for $r<r_{0}$ or $r>r_{\infty}$, since the energy content of
the different modes beyond these points is then fixed. With this definition,
one can use the solution at $r_{\infty}$, i.e.
$R_{\infty}=\mathbf{P}_{\infty}\cdot\mathbf{A}_{\infty}$, to deduce the same
at $r_{0}$, i.e. $R_{0}=\mathbf{P}_{\infty}\cdot\mathbf{A}_{0}$ with
$\mathbf{A}_{0}$ given above. To construct $\mathcal{M}$, one performs a
series of matrix multiplications using (46), (47) and (50) (see [53] for an
explicit example of this). However, due to the way that $T$ and
$\widetilde{T}$ act on the amplitude vectors, the situation is a bit different
if there are two turning points (say $r_{a}<r_{b}$) where two real modes are
converted into complex modes and then back into real modes. In this case, the
amplitudes of the interacting modes are related via,

$\begin{pmatrix}A_{a}^{\mathrm{R}}\\\
A_{a}^{\mathrm{L}}\end{pmatrix}=\mathcal{N}_{ab}\begin{pmatrix}A_{b}^{\mathrm{R}}\\\
A_{b}^{\mathrm{L}}\end{pmatrix},$ (52)

with,

$\begin{split}\mathcal{N}_{ab}=&\
\mathcal{F}_{ab}^{\downarrow}\begin{bmatrix}1+\tfrac{1}{4}f_{ab}^{2}&i\left(1-\tfrac{1}{4}f_{ab}^{2}\right)\\\
-i\left(1-\tfrac{1}{4}f_{ab}^{2}\right)&1+\tfrac{1}{4}f_{ab}^{2}\end{bmatrix},\\\
f_{ab}=&\
\exp\left(-\int^{r_{b}}_{r_{a}}\mathrm{Im}[p^{\downarrow}]dr\right),\end{split}$
(53)

where the $\downarrow$ mode is the complex solution of the dispersion relation
which decays with increasing $r$ between the two turning points (see [53] for
details).

For the scattering problems considered in this work, it turns out that (52)
contains all the necessary physics to compute the amount of superradiance.
This is because, in the two mode case (i.e $\Lambda=0$) one can define
$r_{0}=r_{a}$ and $r_{\infty}=r_{b}$ and then (52) is equivalent to the full
transfer matrix. In the four mode case (i.e $\Lambda\neq 0$) the modes
decouple into two pairs (i.e. $\mathcal{M}$ becomes block diagonal) and the
computation of the scattering coefficients proceeds identically to the two
mode case.

##  5 Non-dispersive modes in the DBT

Figure 1: An example of the branches of the dispersion relation (32) for $m=1$
and $C=0.2$. Two $r$ locations are shown; one outside the horizon (black
curves) and one inside (red curves). All modes inside the horizon are in-going
since the gradient of both branches (in $p$) is everywhere negative. This
particular set of parameters corresponds to Type $\mathrm{II}^{0}$ scattering,
defined further down.

In the case where $\Lambda=0$, the equations of motion (10) can be recast as a
Klein-Gordon equation for the fluctuations of a scalar field propogating
through an effective $(2+1)$-dimensional spacetime. This is the conventional
way that superradiance in the DBT is studied and has been discussed on many
occasions in the literature, e.g. [14, 54, 34]. Hence, I will not reproduce
the analysis here, opting instead to infer the important properties of the
system (in particular, the existence of a horizon and superradiance) directly
from the dispersion relation. This approach has the benefit that it can be
easily generalised to the dispersive case, where the description in terms of
an effective metric is no longer available.

The effective Hamiltonian for this case is,

$\mathcal{H}=\tfrac{1}{2}\left[(1-1/r^{2})p^{2}-2\tilde{\omega}p/r-\tilde{\omega}^{2}+m^{2}/r^{2}\right],$
(54)

where I have introduced the frequency in the rotating frame,

$\tilde{\omega}=\omega-mC/r^{2}.$ (55)

This has roots which are labelled $l\in\\{+,-\\}$, and are given by,

$p^{\pm}=\frac{\tilde{\omega}/r\pm\sqrt{-V}}{1-1/r^{2}},$ (56)

with $V$ defined by,

$V=-\tilde{\omega}^{2}+(1-1/r^{2})m^{2}/r^{2}.$ (57)

The solutions are labelled such that $\mathrm{Re}[p^{+}]>\mathrm{Re}[p^{-}]$
outside of the horizon (to be defined shortly). In particular, the $+$ mode is
radially out-going (i.e. $\vec{\mathbf{e}}_{r}\cdot\bm{v}_{g}^{+}>0$) as
$r\to\infty$, whereas the $-$ mode is in-going (i.e.
$\vec{\mathbf{e}}_{r}\cdot\bm{v}_{g}^{-}<0$). Note that when $V>0$, the two
roots become complex with identical real parts, and equal and opposite
imaginary parts. In this region they are labelled
$l\in\\{\uparrow,\downarrow\\}$, where the $\uparrow$ mode is the one which
grows with increasing $r$, $\mathrm{Im}[p^{\uparrow}]<0$, and the $\downarrow$
mode is the one which decays with increasing $r$,
$\mathrm{Im}[p^{\downarrow}]>0$. An example of the functions $p^{\pm}(r)$ can
be found in Fig. 2 of [23], which will shortly be represented in a “Feynman”
diagrammatic form.

###  5.1 The horizon

Consider now the branches of the dispersion relation in (32) as a function of
$p$ at a given $r$ (see e.g. Fig. 1). The intersection of a line of constant
$\omega$ with the branches gives the two roots, and by (33), the gradient
$\partial_{p}\omega_{D}^{\pm}$ at these points gives the local group velocity
of the modes. As $r\to\infty$, the $-$ is always in-going with
$\partial_{p}\omega_{D}^{+}(p^{-})<0$, whereas the $+$ mode is out-going with
$\partial_{p}\omega_{D}^{+}(p^{+})>0$. However, approaching the origin, both
modes have $\partial_{p}\omega_{D}^{+}(p^{\pm})<0$ and are therefore in-going.
The transition between these two scenarios occurs as the two branches of the
dispersion relation rotate clockwise in the $(p,\omega)$ plane and one of the
modes is sent to $p^{\pm}\to\pm\infty$. Looking at the expression for $p$ in
(56), this occurs where the denominator is zero, which occurs for
$r=r_{h}\equiv 1$ (or in dimensional units $r_{h}=D/c$). This location (the
horizon) is the boundary of the region inside of which there are no modes
which escape to spatial infinity.

Figure 2:  An illustration of the scattering of the two non-disperive modes
through the $(r,p)$ phase space in the form of “Feynman” diagrams ($r$ on the
horizonal and $p$ on the vertical). Only the real part of $p$ is shown, and
increases moving up the diagram.

###  5.2 Scattering types

The different scattering possibilities can be classified using a similar
scheme to that developed in [23]. In the non-dispersive case, there are three
different scattering types which can be represented using phase space
diagrams. The diagrams involve a schematic illustration of the real part of
$p^{\pm}(r)$ through the $(r,p)$ phase space and take on a similar form to
Feynman diagrams. The important features of these diagrams are the number of
turning points, the modes which interact there and the asymptotics of the
modes. For any possible combination of the wave and background parameters,
i.e. $\omega,m$ and $C$, the scattering outcome falls into one of the three
categories. Which category it falls into depends on the size of $\omega$
relative to the characteristic frequencies of the system, which are defined
now.

The simplest possible outcome is that each WKB mode evolves adiabatically
across the system without interacting with the other, i.e. there are no
turning points. In this case, the $+$ mode diverges at $r_{h}$ whereas the $-$
mode is regular there. This will be called Type $\mathrm{I}^{0}$
scattering333Note, the superscript $0$ indicates the non-dispersive case, i.e.
$\Lambda=0$. In the dispersive case, the superscript $+$ will be used for
$\Lambda>0$..

The next possibility is that the the modes have the same asymptotics as the
previous case, but now there is an interaction between both modes somewhere
for $r>r_{h}$. Due to the asymptotics, there must be two turning points; one
to convert the real modes at large $r$ into complex modes, and a second to
convert complex modes back into real modes near the horizon. In this
intermediate region, the complex nature of $p$ leads to an exponential fall
off of the amplitude, i.e. the modes tunnel between the turning points. This
case will be called Type $\mathrm{II}^{0}$ scattering.

The transition between Types $\mathrm{I}^{0}$ and $\mathrm{II}^{0}$ occurs
when the two turning points meet at a single location. On the dispersion
relation, this means that $p^{+}$ and $p^{-}$ become equal momentarily before
departing back in the direction they came from. At this point (which is the
well known light-ring $r_{lr}$ from black hole physics [22]) the condition
$\partial_{r}\omega_{D}^{+}=0$ is also satisfied. Using (34), the conditions
for this location are,

$\mathcal{H}_{lr}=0,\qquad\partial_{r}\mathcal{H}_{lr}=0,\qquad\partial_{p}\mathcal{H}_{lr}=0,$
(58)

which yields a triplet $(r_{lr},p_{lr},\omega_{lr})$. These conditions imply a
relation between $r_{lr}$ and $p_{lr}$,

$p_{lr}r_{lr}=B_{\pm}\equiv mC\pm\sqrt{m^{2}C^{2}+m^{2}},$ (59)

where the $+$ sign is for the upper branch and $-$ sign for the lower one.
Note that this relation also holds for $\Lambda\neq 0$. The light-ring is
given by,

$r^{\pm}_{lr}=\pm(B_{\pm}^{2}+m^{2})^{\frac{1}{2}}/B_{\pm},$ (60)

and the light-ring frequency by,

$\omega^{\pm}_{lr}=mC/r_{lr}^{2}+(1-1/r_{lr}^{2})B_{\pm}.$ (61)

Since I consider $\omega>0$ modes, only the light-ring on the upper branch is
required and I will therefore set $r_{lr}=r^{+}_{lr}$ and
$\omega_{lr}=\omega^{+}_{lr}$ from here on. Note that $r_{lr}$ is independent
of $m$ whereas $\omega_{lr}$ scales linearly with $|m|$.

In the final possibility, the turning point structure is the same as the last
case but now the $p^{+}$ mode is regular at $r_{h}$ with the $p^{-}$ mode
diverging there. By considering how the dispersion relation evolves with $r$,
e.g. in Fig. 1, it is easy to convince oneself that this occurs when the modes
are on the lower branch just outside the horizon. To identify the relevant
frequency controlling when this occurs, consider the following. For a
tunnelling mode to re-emerge on either $\omega_{D}^{+}$ or $\omega_{D}^{-}$,
the two branches must have extrema. However, the extrema vanish at $r_{h}$
since the right (left) part of the upper (lower) branch asymptotes to,

$\omega_{\star}=mC/r_{h}^{2}\equiv m\Omega_{h}^{\mathrm{rot}}.$ (62)

Hence, for $\omega>\omega_{\star}$, the modes will be on the upper branch just
outside of $r_{h}$, whereas for $\omega<\omega_{\star}$ they will be on the
lower branch. Note that $\omega_{\star}<\omega_{lr}$ for all $m$ and $C$. This
$\omega_{\star}$ is in fact the well-known threshold frequency for
superradiance introduced in (1). I will show precisely why this condition
implies superradiance in the next section. For now, it serves as a condition
for the final type of scattering, which I call Type $\mathrm{III}^{0}$.

There is another convenient way to understand the different scattering
possibilities by plotting the evolution of the extrema of $\omega_{D}^{\pm}$
with r. To do this, one solves $\partial_{p}\omega_{D}^{\pm}=0$ to find a
relation $p=p_{ex}(r)$, where $+p_{ex}$ gives the local momentum at the
extrema on the upper branch and $-p_{ex}$ gives the same on the lower branch.
Then, the value of $\omega$ on the extrema is given by,

$\omega^{\pm}(r)\equiv\omega_{D}^{\pm}(r,\pm
p_{ex}(r))=mC/r^{2}\pm\sqrt{(1-1/r^{2})m^{2}/r^{2}}.$ (63)

The turning points can be understood as the intersection of these curves with
a line of $\omega=\mathrm{const}$. Then $\omega_{lr}$ is simply the extremum
of $\omega_{+}$ (for $\omega>0$) in the radial direction and
$\omega_{\star}=\omega^{\pm}(r_{h})$. These curves are illustrated for a
particular value of $m$ and $C$ in Fig. 3. Note finally that since the
function $V$ defined in (57) can be written,

$V=-(\omega-\omega^{+})(\omega-\omega^{-}),$ (64)

the turning points correspond to the zeros of $V$. Hence, $V$ can be thought
of as an effective potential barrier.

Figure 3: The value of $\omega$ at the extrema of $\omega_{D}^{\pm}$ for
$m=1$, $C=0.5$. The intersection of a line with $\omega=\mathrm{const}$ gives
the location of the turning points. Thus, Type $\mathrm{I}^{0}$ scattering
occurs above $\omega^{+}_{lr}$ and Type $\mathrm{II}^{0}$ occurs for
$\omega_{\star}<\omega<\omega^{+}_{lr}$. Below $\omega_{\star}$, positive
frequencies tunnel to the lower branch of the dispersion relation, which is
scattering Type $\mathrm{III}^{0}$. Note that since the dispersion relation is
invariant under $\\{\omega,m\\}\to\\{-\omega,-m\\}$, the scattering of the
$m=-1$ mode is described by the same plot inverted about the $r$-axis (the
signs on all the labels should also be exchanged). Thus for $m<0$, Type
$\mathrm{II}^{0}$ occurs for $0<\omega<\omega^{+}_{lr}$ and Type
$\mathrm{I}^{0}$ for $\omega>\omega^{+}_{lr}$.

###  5.3 Scattering coefficients

Since the radial equation of motion is second order in spatial derivatives,
there are two independent solutions $j=1,2$. Following [43], and using the
diagrams in Fig. 2, these are defined by the asymptotics,

$\begin{split}R_{1}\sim&\
\frac{1}{2\pi|q^{-}_{\infty}|^{\frac{1}{2}}}\times\begin{cases}e^{i\int^{r_{\infty}}p^{-}dr}+\mathcal{R}e^{i\int^{r_{\infty}}p^{+}dr},\
\ \quad r\to r_{\infty},\\\ \mathcal{T}e^{i\int^{r_{0}}p^{\mp}dr},\ \ \
\qquad\qquad\qquad r\to r_{0},\end{cases}\\\ R_{2}\sim&\
\frac{1}{2\pi|q^{\pm}_{0}|^{\frac{1}{2}}}\times\begin{cases}\mathcal{U}e^{i\int^{r_{\infty}}p^{+}dr},\qquad\quad\qquad\qquad
r\to r_{\infty},\\\
e^{i\int^{r_{0}}p^{\pm}dr}+\mathcal{V}e^{i\int^{r_{0}}p^{\mp}dr},\ \ \qquad
r\to r_{0},\end{cases}\end{split}$ (65)

where the upper sign is taken for Types $\mathrm{I}^{0}$ and $\mathrm{II}^{0}$
and the lower sign for Type $\mathrm{III}^{0}$. Here,
$\mathcal{R},\mathcal{T},\mathcal{U},\mathcal{V}$ are scattering coefficients
and the factor of $2\pi$ is there so that the incident part of the mode is
normalised in the inner-product (25) (see A). It is also understood that the
subscript on $R$ is used to specify the $j$ mode, rather than the radial
location as on other quantities.

Plugging these into (43), the scattering coefficients obey the following
relations,

$\displaystyle|q_{\infty}^{-}|(2\pi)^{2}~{}W[\tilde{\varphi}_{1},\tilde{\varphi}_{1}]=$
$\displaystyle\
q^{-}_{\infty}+q^{+}_{\infty}|\mathcal{R}|^{2}=q^{\mp}_{0}|\mathcal{T}|^{2},$
(66a)
$\displaystyle|q_{0}^{\pm}|(2\pi)^{2}~{}W[\tilde{\varphi}_{2},\tilde{\varphi}_{2}]=$
$\displaystyle\
q^{+}_{\infty}|\mathcal{U}|^{2}=q^{\pm}_{0}+q^{\mp}_{0}|\mathcal{V}|^{2},$
(66b)
$\displaystyle|q_{\infty}^{-}q_{0}^{\pm}|^{\frac{1}{2}}(2\pi)^{2}~{}W[\tilde{\varphi}_{1},\tilde{\varphi}_{2}]=$
$\displaystyle\
q^{+}_{\infty}\mathcal{R}\mathcal{U}^{*}=q^{\mp}_{0}\mathcal{T}\mathcal{V}^{*},$
(66c)

where the factors on the left hand side have been left there for later use
when evaluating the quantum currents.

Consider now the classical scattering of an in-coming wave with the vortex
(i.e. the $R_{1}$ solution). Superradiance occurs when the reflected wave
carries away more energy than the incident wave had coming in, i.e.
$q^{+}_{\infty}|\mathcal{R}|^{2}>q^{-}_{\infty}$, which by (66a) implies that
$q_{0}^{\mp}<0$ is a necessary (and sufficient) condition for superradiance.
Using $q=f^{-1}\vec{\mathbf{e}}_{r}\cdot\bm{v}_{g}\Omega$ and realising that
in $R_{1}$ the solution on the horizon is always in-going, superradiance will
occur provided $\Omega_{0}<0$. This is true for the solution which tunnels to
the lower branch of the dispersion relation, i.e. the one in Type
$\mathrm{III}^{0}$ scattering.

The condition for superradiance assumes a more familiar form if one sets
$r_{\infty}$ to be true spatial infinity and $r_{0}$ to sit just outside
$r_{h}$. Then $q_{\infty}^{+}=-q_{\infty}^{-}=\omega$ and
$q_{0}^{\mp}=-\tilde{\omega}_{h}$, and (66a) becomes,

$|\mathcal{R}|^{2}+\frac{\tilde{\omega}_{h}}{\omega}|\mathcal{T}|^{2}=1,$ (67)

which is the usual relation between scattering coefficients from black hole
physics. Thus amplification occurs for $\tilde{\omega}_{h}<0$, which
corresponds to Type $\mathrm{III}^{0}$ scattering.

To find an expression for the reflection coefficient, I will again exploit the
freedom to move the points $r_{0}$ and $r_{\infty}$ (although identical
results are found when these locations are fixed [23]). In Type
$\mathrm{I}^{0}$ scattering, there are no real turning points and thus, at the
considered level of approximation, one has $|\mathcal{R}|=0$. For the other
two cases, choose $r_{0}$ to sit just inside $r_{1}$, and $r_{\infty}$ just
outside $r_{2}$. Applying the formula in (52) and inserting the amplitudes for
$R_{1}$, the reflection coefficient is given by,

$|\mathcal{R}|=\left(\frac{1-f_{12}^{2}/4}{1+f_{12}^{2}/4}\right)^{\mathrm{sgn}(\tilde{\omega}_{h})},$
(68)

which as expected satisfies $|\mathcal{R}|<1$ for Type $\mathrm{II}^{0}$ and
$|\mathcal{R}|>1$ for Type $\mathrm{III}^{0}$. This will be plotted later on
in Fig. 10 along with the dispersive solutions.

##  6 Dispersive modes in the DBT

Figure 4: An example of the branches of the dispersion relation (32) for
$m=1$, $C=0.2$ and $\Lambda=0.08$. At large $r$ (black curves) there are only
two real solutions of the dispersion relation, $p^{\pm}$, which are the same
as those in the non-dispersive case. At small $r$ (red curves) it is possible
for the other solutions, $p^{t,b}$, to also become real.

The effective Hamiltonian for the dispersive case is given by,

$\mathcal{H}=\frac{1}{2}\left[\Lambda p^{4}+\left(1-\frac{1-2\Lambda
m^{2}}{r^{2}}\right)p^{2}-\frac{2\tilde{\omega}}{r}p-\tilde{\omega}^{2}+\frac{m^{2}}{r^{2}}\left(1+\frac{\Lambda
m^{2}}{r^{2}}\right)\right].$ (69)

Since this is a depressed quartic, there are now four different solutions
which I will label $l\in\\{+,-,t,b\\}$. The $+$ and $-$ solutions are the same
ones from the non-dispersive case; in particular, they correspond to the out-
and in-going modes as $r\to\infty$ and obey
$\mathrm{Re}[p^{+}]\geq\mathrm{Re}[p^{-}]$ everywhere. The $t$ and $b$ modes
arise due to dispersion and are defined to be those which satisfy
$p^{t,b}(r\to\infty)\in\mathbb{C}$ and
$\mathrm{Re}[p^{t}]\geq\mathrm{Re}[p^{b}]$. In many cases, these modes can
become real, propagating solutions in the vortex core. An example of this is
given in Fig. 4, where it is shown how the dispersive modes arise on the
dispersion relation. The functions $p^{l}(r)$ for the same parameters are
displayed in Fig. 5.

Figure 5: The evolution of the four modes with $r$ for the same parameters as
in Fig. 4. Only the real part of $p$ is shown. When neighbouring trajectories
intersect in phase space, mode mixing occurs. This particular diagram
corresponds to Type $\mathrm{VIII}^{+}_{\mathrm{a}}$ scattering defined later
on. Note that the trajectories of the complex $p^{t,b}$ modes at large $r$ do
not really intersect that of $p^{-}$, but rather pass around it in the complex
plane.

An important difference between the non-dispersive and dispersive cases is the
absence of a horizon in the latter. The reason for this is that, due the
$p^{4}$ term in (69), there is no longer a critical radius below which
$\partial_{p}\omega_{D}^{\pm}<0$ for all $p$ (see Fig. 4 where this behaviour
is readily apparent). Consequently, there is no blocking of out-going high
momentum modes approaching the origin. (An exception to this is when the
propagation of the $+$ and $-$ modes is prohibited in the vortex core.
However, when this happens, both in- and out-going modes are blocked rather
than just the out-going one).

###  6.1 Scattering types

The scattering possibilities for $\Lambda>0$ will now be classified in a
similar manner to those in Section 5.2. Dispersion significantly enhances the
number of possible outcomes compared with the non-dispersive case. Hence,
instead of discussing each case individually, I will instead provide a
parameter space plot in Fig. 9 to illustrate the parameter ranges associated
with the different types of scattering. The parameter space is divided up by
three important frequencies which I discuss now.

####  6.1.1 Light-ring

Similarly to the non-dispersive case, the light-ring frequency provides a
boundary in parameter space above which the $+$ and $-$ modes decouple. Using
the conditions in (58) along with $p_{lr}r_{lr}=B_{\pm}$ from (59), the
location of the light-ring (on $\omega_{D}^{+}$) is,

$r_{lr}=\sqrt{\frac{B_{+}^{2}+m^{2}}{2B_{+}^{2}}}\left(1+\sqrt{1-4\Lambda
B_{+}^{2}}\right)^{\frac{1}{2}}\left(1-4\Lambda
B_{+}^{2}\right)^{\frac{1}{4}}.$ (70)

The light-ring momentum and frequency are then immediately given by
$p_{lr}=B_{+}/r_{lr}$ and $\omega_{lr}=\omega_{D}^{+}(r_{lr},p_{lr})$.

For the following discussion, it is useful to visualise the light-rings as the
extrema of the $\omega^{\pm}$ curves (which are defined in the same way as in
the non-dispersive case, i.e. the value of $\omega$ at the extrema of
$\omega_{D}^{\pm}$). I now define the following critical parameters,

$C_{0}=\left|\frac{1-4\Lambda
m^{2}}{4\Lambda^{\frac{1}{2}}m}\right|,\qquad\Lambda_{c}=1/4m^{2},$ (71)

which play a key role in characterising the scattering. To give some intuition
about the significance of these parameters, I discuss below their influence on
the scattering of $m>0$ modes. By the symmetry
$\\{\omega,m\\}\to\\{-\omega,-m\\}$ of the dispersion relation, they will have
a similarly important role for scattering of $m<0$ modes.

Firstly, for $\Lambda<\Lambda_{c}$, the value of $C$ relative to $C_{0}$
determines when there is a light-ring on the upper branch. For $C<C_{0}$, the
light-ring is real and positive; for $C=C_{0}$, $r_{lr}=0$ and for $C>C_{0}$,
$r_{lr}$ is complex, i.e. there is no extremum on $\omega^{+}$. The absence of
a light-ring has very interesting consequences for scattering. In particular,
it means that strong rotation suppresses the propagation of the $+$ and $-$
modes in the vortex core, forcing them to tunnel all the way down to $r=0$.

Now consider $\Lambda>\Lambda_{c}$. In this case, there is no light-ring on
the upper branch of the dispersion relation for any value of $C$. The value of
$C_{0}$ now determines whether there is an extremum on $\omega^{-}$; in
particular, it is absent for $C<C_{0}$. Although we are not interested in the
light-ring frequency on the lower branch (since only the $\omega>0$ modes are
considered) it’s absence is important since it implies that the $+$ and $-$
modes can no longer tunnel to the lower branch of the dispersion relation. The
reason for this is that at large $r$, the $m^{2}/r^{2}$ term in
$\omega_{D}^{\pm}$ will initially cause the two branches to separate. However,
if $\partial_{r}\omega_{D}^{-}$ is nowhere satisfied, the lower branch will
continue moving towards lower values of $\omega$ approaching $r=0$, thereby
making it impossible for positive frequency modes to reach this branch. Hence,
superradiance (which relies on tunnelling to the $\omega_{D}^{-}$ branch) is
impossible in this regime. However, above $C_{0}$ superradiance will still
occur. The implication of this is that above $\Lambda_{c}$, each $m>0$ mode
has a minimum possible rotation below which it cannot superradiate. Note that
all of the properties discussed above can be deduced using the example
$\omega^{\pm}$ curves given in Fig. 6.

Figure 6: The value of $\omega$ at the extrema of $\omega_{D}^{\pm}$ for four
different cases. The intersection of a line of $\omega=\mathrm{const}$ with
these curves gives the location of the turning points on the upper and lower
branches of the dispersion relation. Hence one can read these plots to explain
why the different scattering types occur in different frequency ranges in Fig.
9. The $\omega^{\pm}$ are shown here for $m>0$. However, due to the symmetry
$\\{\omega,m\\}\to\\{-\omega,-m\\}$ of the dispersion relation, one can simply
invert the plots through the horizontal axis (and exchange the $+$ and $-$
labels) to understand the $m<0$ side of the parameter space. Note that in the
top left panel, as $C$ is increased $\omega_{lr}^{+}$ eventually passes inside
the $\omega^{-}$ curve, which corresponds to passing the point where
$\omega_{lr}=\omega_{eq}$ in the top left panel of Fig. 9. The specific
parameters $[m,C,\Lambda]$ used to produce these plots were $[1,0.5,0.01]$
(top left), $[1,0.5,2.5]$ (top right), $[1,0.2,0.5]$ (bottom left) and
$[1,0.5,0.5]$ (bottom right).

####  6.1.2 Threshold frequency

As seen in the non-dispersive case, the onset of superradiance is signalled by
the tunnelling of the $+$ and $-$ modes from $\omega_{D}^{+}$ at large $r$ to
$\omega_{D}^{-}$ at small $r$. Let’s start by assuming that there exists a
threshold frequency $\omega_{\star}$ that governs when this occurs. For
$\Lambda=0$, it was simple to see what this frequency should be since we only
had to look for when two modes appeared on the lower branch of the dispersion
relation outside of the horizon. In the present case, this criterion is not
sufficient, since it is now possible for propagating modes to exist on the
upper and lower branches of the dispersion relation simultaneously (as
demonstrated by Fig. 4). Hence, in the dispersive case, one must make sure
that it is indeed the $+$ and $-$ modes which tunnel to $\omega_{D}^{-}$ and
not the $t$ and $b$ modes.

To find a necessary condition for this, consider the following argument. Using
only the dispersion relation, one can find an example where $p^{\pm}$ tunnel
to $\omega_{D}^{-}$ and one where they do not simply by testing different
values of $\omega,m,C,\Lambda$. The trajectories of the four $p^{l}(r)$
through the complex $p$-plane in these examples would look like those shown in
Fig. 7. The difference between these two cases is that $p^{\pm}$ and $p^{t,b}$
bounce off each other in opposite directions in $p$-plane as $r$ is varied.
This deflection is centred on a saddle point of $\mathcal{H}$ in the complex
plane and in the limit that $\omega=\omega_{\star}$, the modes undergo a head
on collision at the point $p_{\star}$ (and also at the point given by it’s
complex conjugate). The task at hand then is to find the expression for these
points.

To do this, let us write $p=x+iy$ and $\mathcal{H}(p)=U+iW$, so that,

$\begin{split}U=&\
\frac{1}{2}\Big{[}\Lambda(x^{4}-6x^{2}y^{2}+y^{4})+\left(1+2\Lambda
m^{2}/r^{2}-1/r^{2}\right)(x^{2}-y^{2})\\\
&\qquad\qquad\qquad-2\tilde{\omega}x/r-\tilde{\omega}^{2}+\left(1+\Lambda
m^{2}/r^{2}\right)m^{2}/r^{2}\Big{]},\\\ W=&\
-\tilde{\omega}y/r+\left(1+2\Lambda m^{2}/r^{2}-1/r^{2}\right)xy+2\Lambda
xy(x^{2}-y^{2}).\end{split}$ (72)

Using the fact that $p_{\star}$ is a saddle point, one has the condition that
all $x,y$ derivatives of $U,W$ must vanish, but since $\mathcal{H}(p)$ is
holomorphic, the Cauchy-Riemann relations are satisfied and two of these
conditions give redundant information. When the saddle point is a solution to
the dispersion relation, $U$ and $W$ also vanish. Solving these four
conditions yields,

$r_{\star}=(1+2\Lambda^{\frac{1}{2}}m)^{\frac{1}{2}},\quad\omega_{\star}=mC/r_{\star}^{2},\quad
p_{\star}=i(m^{2}+(r_{\star}^{2}-1)/2\Lambda)^{\frac{1}{2}}/r_{\star}.$ (73)

For small $\Lambda$, the non-dispersive behaviour of the threshold frequency
is recovered at low $m$ whereas at high $m$, $\omega_{\star}$ tends to a
constant value of $C/2\Lambda^{\frac{1}{2}}$.

Figure 7: The trajectory of the four $p$ solutions to the dispersion relation
through the complex $p$-plane, for $\omega<\omega_{\star}$ (panel A) and
$\omega>\omega_{\star}$ (panel B). The curves are parametrised by $r$, which
decreases in the direction of the arrows. Observe that as $\omega_{\star}$ is
crossed from below, the red and the green paths (and by symmetry the blue and
pink paths) meet at a saddle point in the complex plane. As this happens,
there is a discontinuous change in the character of the scattering as the
evanescent modes at infinity disconnect from the positive frequency branch and
reconnect with the negative frequency branch in the vortex core. The specific
parameters are $m=1$, $C=0.5$, $\Lambda=0.01$, with $\omega=0.41$ in A and
$\omega=0.425$ in B. These correspond to scattering Types $\mathrm{IX}^{+}$
and $\mathrm{VIII}^{+}_{\mathrm{b}}$ (defined later) for panels A and B
respectively.

####  6.1.3 Extremum equality

The equality of the $\omega$ value of the extrema of $\omega_{D}^{\pm}$
defines another important frequency $\omega_{eq}$. Using the curves
$\omega^{\pm}$, the condition for this becomes
$\omega_{eq}=\omega^{+}=\omega^{-}$ which, along with the turning point
criteria, gives,

$r_{eq}=(1-2\Lambda^{\frac{1}{2}}m)^{\frac{1}{2}},\qquad\omega_{eq}=mC/r_{eq}^{2}.$
(74)

The importance of this is that it determines the relative location of the
turning points on the upper and lower branches of the dispersion relation; for
$\omega>\omega_{eq}$, the turning point on the upper branch occurs at larger
$r$ than the one on the lower branch and vice versa for $\omega<\omega_{eq}$.
This introduces some sub-classification criteria for two of the scattering
categories defined in the next section.

####  6.1.4 Classification

Figure 8: An illustration of the trajectories of the four WKB modes through
the $(r,p)$ phase space, with $r$ on the horizontal axis and $p$ on the
vertical. As shown in Fig. 9, the different types occur in different regions
of the $(C,\omega,\Lambda)$ parameter space. In particular, superradiance
occurs in Type $\mathrm{VI}^{+}$ and $\mathrm{IX}^{+}$.

In Fig. 8, a schematic illustration is provided of the real part of $p^{l}(r)$
through the $(r,p)$ phase space for the four different modes. Again the use of
these diagrams is in identifying how many turning points there are (and which
modes they involve), as well as the asymptotic behaviour of the modes. The
turning points are labelled as follows: $r_{1}$ and $r_{2}$ are the inner and
outer turning points for the $+,-$ modes, where $r_{3}$ is the only allowed
turning point for the $t,b$ modes.

These diagrams represent the type of scattering that occurs in different
regions of the $(C,\omega)$ parameter space, which is depicted in Fig. 9. The
parameter space has a distinctively different structure depending on the sign
of $m$, as well as the size of $\Lambda$ relative to $\Lambda_{c}$. To
understand this structure, it is again useful to plot the $\omega$ value of
the extrema $\omega_{D}^{\pm}$, i.e. $\omega^{\pm}(r)=\omega_{D}^{\pm}(r,\pm
p_{ex}(r))$. The turning points (and their relative locations) can then be
deduced by looking for the intersections of these curves with a line
$\omega=\mathrm{const}$. There are four distinct cases for the shapes of
$\omega^{\pm}$, which are illustrated in Fig. 6.

Figure 9: Different types of scattering are shown to occur in different
regions of the $(C,\omega)$ parameter space (note that the bottom left corner
is the point $(0,0)$). There is a distinct change in the structure of the
parameter space depending on the size of $\Lambda$ relative to the critical
value $\Lambda_{c}$. The reason for the different scattering types can be
understood by considering Fig. 6 and observing where the turning points are
for a particular $\omega$.

A couple of noteworthy points concerning the diagrams in Fig. 8. Firstly,
there is a sub-classification for the Type $\mathrm{VII}^{+}$ and
$\mathrm{VIII}^{+}$ diagrams depending on the relative locations of
$r_{1,2,3}$. Specifically,

  * •

Type $\mathrm{VII}^{+}_{\mathrm{a}}$: $r_{3}<r_{2}$,

  * •

Type $\mathrm{VII}^{+}_{\mathrm{b}}$: $r_{2}<r_{3}$,

  * •

Type $\mathrm{VIII}^{+}_{\mathrm{a}}$: $r_{3}<r_{1}<r_{2}$,

  * •

Type $\mathrm{VIII}^{+}_{\mathrm{b}}$: $r_{1}<r_{3}<r_{2}$,

  * •

Type $\mathrm{VIII}^{+}_{\mathrm{c}}$: $r_{1}<r_{2}<r_{3}$.

Note, however, that this does not alter the computation of the scattering
coefficients since the relative location of $r_{3}$ to $r_{1,2}$ does not play
a role in this.

Secondly, the diagrams give the impression that the modes with larger
$\mathrm{Re}[p]$ appear at the top and smaller $\mathrm{Re}[p]$ are toward the
bottom. This ordering is accurate whilst ever the modes are real, but not
necessarily when they are complex. For instance, in Type $\mathrm{VIII}^{+}$
scattering it happens that
$\mathrm{Re}[p^{+}]>\mathrm{Re}[p^{t,b}]>\mathrm{Re}[p^{-}]$ as $r\to\infty$,
but $\mathrm{Re}[p^{+}]>\mathrm{Re}[p^{-}]>\mathrm{Re}[p^{t,b}]$ as $r\to 0$
(compare Fig. 5 to the Type $\mathrm{VIII}^{+}$ diagram for a clear example of
this) which is allowed since the $t$ and $b$ modes are complex as they cross
the real modes (really they move around them in the complex $p$-plane). This
is not a problem since the use of the diagrams is to determine the mode
asymptotics and identify the turning points. In fact, this overlapping of real
parts is purposefully not shown in Fig. 8 so that it is clear which modes are
interacting (i.e. share common turning points) and which are not.

With this reshuffling of real parts allowed for complex modes, the diagrams
for Types $\mathrm{III}^{+}$ and $\mathrm{IV}^{+}$ technically represent the
same class of scattering. The reason that they have been left as separate
cases is the following: in Type $\mathrm{III}^{+}$, the $+$ and $-$ modes try
to tunnel to $\omega_{D}^{+}$ approaching the origin, but do not make it since
the branch recedes from the modes due to dispersion. In Type $\mathrm{IV}^{+}$
they try instead to tunnel to $\omega_{D}^{-}$. In this sense, Type
$\mathrm{IV}^{+}$ represents failed superradiance, since the negative energy
mode which would otherwise propagate into the vortex core is forbidden from
doing so due to the strength of dispersion.

###  6.2 Scattering coefficients

Following the procedure outlined in Section 5.3, I will now be interested in
writing down the relations between the different scattering coefficients to
show the existence of superradiance. There are four independent solutions to
the radial equation of motion, $j=1,2,3,4$, which can be defined by their
asymptotics. For each diagram in Fig. 8, one could in priciple write down an
asymptotic formula to define the modes as in (65). However, this would be a
tedious process and most of the solutions written down would contain no more
information than that which is readily apparent from looking at the diagrams.
In the following, I will therefore make some simplifying observations to avoid
having to write down each the solutions separately. It will then become
apparent that the important scattering coefficients, and the relations between
them, are exactly the same as in the non-dispersive case.

Firstly, each scattering type falls into one of four classes with different
mode asymptotics, which can be easily identified by looking at how the mode
trajectories approach $r_{0}$ and $r_{\infty}$ in the diagrams in Fig. 8.
These classes are outlined in Table 1. Note that the diagrams under asymptotic
classes A and B can be covered by a single formula in the same way that (65)
was used to represent all three diagrams in the non-dispersive case.

Class | $r_{0}~{}\\{+,-,t,b\\}$ | $r_{\infty}~{}\\{+,-,t,b\\}$ | Types
---|---|---|---
A | $\\{\mathbb{R,R,C,C}\\}$ | $\\{\mathbb{R,R,C,C}\\}$ | $\mathrm{I}^{+},\mathrm{V}^{+},\mathrm{VI}^{+}$
B | $\\{\mathbb{R,R,R,R}\\}$ | $\\{\mathbb{R,R,C,C}\\}$ | $\mathrm{II}^{+},\mathrm{VIII}^{+},\mathrm{IX}^{+}$
C | $\\{\mathbb{C,C,C,C}\\}$ | $\\{\mathbb{R,R,C,C}\\}$ | $\mathrm{III}^{+},\mathrm{IV}^{+}$
D | $\\{\mathbb{C,C,R,R}\\}$ | $\\{\mathbb{R,R,C,C}\\}$ | $\mathrm{VII}^{+}$
Table 1: The diagrams in Fig. 8 are classified according to whether each of
the WKB modes is asymptotically propagating or evanescent. $\mathbb{R}$
indicates that a particular mode is propagating, i.e. $p^{l}\in\mathbb{R}$,
where as $\mathbb{C}$ is used for evanescent modes, i.e. $p^{l}\in\mathbb{C}$.

Next, it is easy to see that in each digram, the $+,-$ modes do not interact
with the $t,b$ modes (this contrasts what happens when dispersion is
subluminal and all four modes can interact [23, 53]). Therefore, two of the
$R_{j}$ will be on the $+,-$ part of the diagram (say $R_{1,2}$) and the
remaining two will be in the $t,b$ part. There are two possibilities for the
interaction of the $t,b$ modes: either they are evanescent everywhere and do
not interact, in which case $R_{3,4}$ are the pure WKB $t,b$ modes over the
whole region and the amplitudes are unrelated; or the propagating $t,b$ modes
in the core are completely reflected, in which cases the $R_{3,4}$ are
analogous the $\mathrm{Ai}$ and $\mathrm{Bi}$ solving Airy’s equation with the
amplitudes related by a phase shift. In both of these cases, no energy is
carried by the $t,b$ modes across the system, hence, they will be of no
further interest from here on.

Lastly we have the interaction of $+,-$ modes. In asymptotics classes C and D,
there is a complete reflection of these modes at large $r$ and the two
independent solutions are again analogous to the Airy functions $\mathrm{Ai}$
and $\mathrm{Bi}$. In asymptotics classes A and B, the $+,-$ mode asymptotics
are identical to those in the non-dispersive case given in (65), this time
with the upper sign taken for Types $\mathrm{I}^{+}$, $\mathrm{II}^{+}$,
$\mathrm{V}^{+}$ and $\mathrm{VIII}^{+}$ and the lower sign for Types
$\mathrm{VI}^{+}$ and $\mathrm{IX}^{+}$. The relations between the different
amplitudes are again given by (66) and following the analysis just below (65),
one finds that superradiance occurs for Types $\mathrm{VI}^{+}$ and
$\mathrm{IX}^{+}$. In fact, there is a simpler way to see this directly from
Fig. 8, since any diagram which contains an interaction of the form,

(75)

will be superradiant. The reason for this is that this relative orientation of
the four arrows is the smoking gun for tunnelling between the upper and lower
branches of the dispersion relation. As discussed previously, a mode which
propagates into the vortex core on $\omega_{D}^{-}$ carries in a negative
energy, which by energy conservation means that the escaping mode must be
amplified.

Finally, it is simple matter to deduce an expression for the reflection
coefficient from the diagrams. For Types $\mathrm{I}^{+}$ and
$\mathrm{II}^{+}$, $\mathcal{R}$ vanishes whereas for Types
$\mathrm{III}^{+}$, $\mathrm{IV}^{+}$ and $\mathrm{VII}^{+}$ it is simply
$|\mathcal{R}|=1$. In the remaining cases, $|\mathcal{R}|$ is obtained from
(52) as,

$|\mathcal{R}|=\left(\frac{1-f_{12}^{2}/4}{1+f_{12}^{2}/4}\right)^{\pm 1},$
(76)

with the upper sign taken for Types $\mathrm{V}^{+}$ and $\mathrm{VIII}^{+}$
and the lower sign for Types $\mathrm{VI}^{+}$ and $\mathrm{IX}^{+}$. Hence,
the form of the reflection coefficient is identical to the non-dispersive
case, and the only difference is in the location of the turning points
$r_{1,2}$ and the integral of the phase between these points.

Figure 10: Examples of the reflection coefficient for the $m=1$ mode for $C=2$
(left) and $C=5$ (right). The different curves represent different values of
$\Lambda$. The main impact of dispersion on superradiance is to reduce the
bandwidth of superradiant modes; the amount of amplification is affected much
less. An exact numerical computation of $|\mathcal{R}|$ in the non-dispersive
case (following Appendix A of [23]) is shown as a dashed curve for comparison.
Note that the discontinuous changes are an artefact of only considering
scattering due to turning points, and in reality will be smoothed over by
other (sub-dominant) sources of scattering.

The behaviour of $|\mathcal{R}|$ can be easily read off from Fig. 9 using the
knowledge gained from the diagrams in Fig. 8. For the positive $m$’s (which
are the ones which can superradiate) there are only three distinct behaviours.
For $\Lambda<\Lambda_{c}$ and $C<C_{0}$, the modes are amplified at low
$\omega$ and absorbed at high $\omega$, much like the non-dispersive case. For
$C>C_{0}$ for any $\Lambda$, the modes are amplified at small $\omega$ and
completely reflected for high $\omega$. Finally, for $\Lambda>\Lambda_{c}$ and
$C<C_{0}$, the modes are completely reflected for all $\omega$. Some examples
of this behaviour are given in Fig. 10. It is clear that for superradiant
modes, dispersion needs to be strong before any siginificant change arises to
the amount of amplication. By contrast, changes to the bandwidth of
superradiant modes become evident even for small $\Lambda$.

Note that the WKB approximation has a tendency to over-estimate
$|\mathcal{R}|$ when compared with the exact non-dispersive result. In fact,
this is to be expected since the true potential appearing in the non-
dispersive radial equation is a modification of (57) by
$m^{2}/r^{2}\to(m^{2}-1/4+5v_{r}^{2}/4)/r^{2}$ [18]. When this is used in the
approximate formula for $|\mathcal{R}|$ (which amounts to applying the WKB
approximation in the radial direction only), the exponent appearing in
$f_{12}$ becomes larger, which has the effect of reducing deviations of the
reflection coefficient away from unity. As promised, these differences quickly
become very small in the limit of large $m$.

###  6.3 Comments

Although the scattering classification for the dispersive waves was much more
involved that for non-dispersive ones, there are only a few differences when
it comes to understanding superradiance in the two cases. These have already
been alluded to earlier, but I re-emphasize them here for clarity.

When dispersion is weak, i.e. $\Lambda<\Lambda_{c}$, superradiance proceeds
for all positive frequencies below the threshold frequency $\omega_{\star}$
for any value of the rotation parameter. However, unlike the non-dispersive
case, the superradiant bandwidth does not continue to grow with $m$, and
instead levels off at a constant value of $C/2\Lambda^{\frac{1}{2}}$ in the
limit that $m\to\infty$. One would therefore expect the energy extracted by a
given $m$-mode to be less with dispersion than without it.

When dispersion is strong, i.e. $\Lambda>\Lambda_{c}$, there is another
important difference. In this regime, the $\omega\in[0,\omega_{\star}]$ modes
can only superradiate above a critical rotation parameter given by $C_{0}(m)$.
Assuming that superradiance will cause the vortex to spin down (this is
revisited in Section 8), there will be a point along time evolution of $C$
where each $m$ stops superradiating. Since, $C_{0}$ is an increasing function
of $m$ in this regime, only the small $m$ modes superradiate at low $C$. In
particular, if $\Lambda>1/4$ then the $m=1$ will not be able to superradiate,
which implies that there will be no superradiant modes in the system. The
consequence of this is that for $\Lambda<1/4$, the system will eventually
spin-down to zero rotation since there is always an angular momentum channel
into which the vortex can dissipate. However, for $\Lambda>1/4$ there is a
minimum allowed rotation given by
$C_{\mathrm{min}}=(4\Lambda-1)/4\Lambda^{\frac{1}{2}}$, and once this value is
reached, superradiance shuts off completely.

In the next section, I will show that the vortex will shed energy and angular
momentum spontaneously as a result of superradiance acting upon it’s vacuum
fluctuations.

##  7 Spontaneous emission

Due to quantum fluctuations in the system, there will be a steady flux of
energy and angular momentum out of the vortex, even when the fields $\phi$ and
$\eta$ are in their vacuum states. To compute this quantum emission, I will
work in the fully dispersive case. This is principally done so that both
dispersive and non-dispersive cases can be handled simultaneously (the non-
dispersive result is found as the $\Lambda\to 0$ limit of the dispersive one).
Physically, however, the dispersive case is the one relevant to real
experiments, where the effects of dispersion are never completely negligible.

###  7.1 Quantisation

The quantisation procedure given here follows that of [43]. The conjugate
momentum to $\phi$ is obtained from (12) as,

$\pi=\frac{\partial\mathcal{L}}{\partial\dot{\phi}}=-\eta.$ (77)

Canonical quantisation then proceeds by replacing complex conjugation by
Hermitian conjugation, promoting the fields $\phi$ and $\eta$ to operators and
imposing the equal time commutation relations,

$\begin{split}[\hat{\phi}(x),\hat{\phi}(y)]=&\
[\hat{\pi}(x),\hat{\pi}(y)]=0,\\\ [\hat{\phi}(x),\hat{\pi}(y)]=&\
i\delta^{(2)}(x-y).\end{split}$ (78)

In (19), the amplitudes $\alpha^{*}_{\lambda}$ and $\alpha_{\lambda}$ and are
replaced by creation and annihilation operators, $\hat{a}^{\dagger}_{\lambda}$
and $\hat{a}_{\lambda}$ respectively, which obey,

$[\hat{a}_{\lambda_{1}},\hat{a}^{\dagger}_{\lambda_{2}}]=\delta_{\lambda_{1}\lambda_{2}}.$
(79)

The normal mode expansion of the fields then reads,

$\hat{\phi}=\sum_{\lambda}(\hat{a}_{\lambda}\varphi_{\lambda}+\hat{a}^{\dagger}_{\lambda}\varphi^{*}_{\lambda}),\qquad\hat{\eta}=\sum_{\lambda}(\hat{a}_{\lambda}n_{\lambda}+\hat{a}^{\dagger}_{\lambda}n^{*}_{\lambda}).$
(80)

The vacuum is defined as the state which is annihilated by all
$\hat{a}_{\lambda}$, i.e. $\hat{a}_{\lambda}|0\rangle=0$.

The goal is to compute the vacuum expectation value (VEV) of the energy and
angular momentum current far away from the vortex. To do this, an expression
is needed for the energy and angular momentum current operators. Starting from
the expressions in (17) and (18), define the following operators,

$\widehat{\textfrak{S}}_{E}^{r}=\tfrac{1}{2}\big{(}\tfrac{1}{2}v_{r}\\{\hat{\phi},\partial_{t}\hat{\eta}\\}-\tfrac{1}{2}v_{r}\\{\hat{\eta},\partial_{t}\hat{\phi}\\}-\\{\partial_{r}\hat{\phi},\partial_{t}\hat{\phi}\\}-\Lambda\\{\partial_{r}\hat{\eta},\partial_{t}\hat{\eta}\\}\big{)},$
(81)

and,

$\widehat{\textfrak{S}}_{L}^{r}=\tfrac{1}{2}\big{(}\tfrac{1}{2}v_{r}\\{\hat{\phi},\partial_{\theta}\hat{\eta}\\}-\tfrac{1}{2}v_{r}\\{\hat{\eta},\partial_{\theta}\hat{\phi}\\}-\\{\partial_{r}\hat{\phi},\partial_{\theta}\hat{\phi}\\}-\Lambda\\{\partial_{r}\hat{\eta},\partial_{\theta}\hat{\eta}\\}\big{)},$
(82)

where $\\{~{},~{}\\}$ is the anti-commutator. This ordering of terms is chosen
to make the operator symmetric in the fields whilst still recovering (17) and
(18) in the classical limit.

Inserting the expansions in (80), the VEVs of the different anti-commutators
appearing in Eq. (81) are,

$\begin{split}\langle 0|\\{\hat{\phi},\partial_{t}\hat{\eta}\\}|0\rangle=&\
-i\sum_{\lambda}\nolimits\omega\left[n_{\lambda}\varphi_{\lambda}^{*}-\varphi_{\lambda}n^{*}_{\lambda}\right],\\\
\langle 0|\\{\hat{\eta},\partial_{t}\hat{\phi}\\}|0\rangle=&\
-i\sum_{\lambda}\nolimits\omega\left[\varphi_{\lambda}n^{*}_{\lambda}-n_{\lambda}\varphi_{\lambda}^{*}\right],\\\
\langle 0|\\{\partial_{r}\hat{\phi},\partial_{t}\hat{\phi}\\}|0\rangle=&\
-i\sum_{\lambda}\nolimits\omega\left[\varphi_{\lambda}\partial_{r}\varphi^{*}_{\lambda}-(\partial_{r}\varphi_{\lambda})\varphi_{\lambda}^{*}\right],\\\
\langle 0|\\{\partial_{r}\hat{\eta},\partial_{t}\hat{\eta}\\}|0\rangle=&\
-i\sum_{\lambda}\nolimits\omega\left[n_{\lambda}\partial_{r}n^{*}_{\lambda}-(\partial_{r}n_{\lambda})n_{\lambda}^{*}\right].\end{split}$
(83)

The anti-commutators in Eq. (82) are given by similar relations except there,
the $\omega$ sitting inside of the sum is replaced by $m$. The flux of energy
and angular momentum out of the vortex are the quantities of interest, hence
one must integrate the VEV of (81) and (82) around a ring far away from the
origin. Noticing that the four terms in (83) are simply those appearing in
(26), one finds,

$\int^{2\pi}_{0}d\theta~{}r\langle
0|\widehat{\textfrak{S}}_{E}^{r}|0\rangle=2\pi\sum_{\lambda}\omega
W[\varphi_{\lambda},\varphi_{\lambda}],$ (84)

and similarly for the angular momentum current, where the integral has been
evaluated by noticing that $W[\varphi_{\lambda},\varphi_{\lambda}]$ is
independent of $\theta$.

To perform the normal mode sum, one can make a few simplifying observations.
Firstly, notice that $W[\tilde{\varphi}_{j},\tilde{\varphi}_{j}]$ vanishes for
solutions which are everywhere evanescent and those which are completely
reflected. This means that none of the diagrams in asymptotics classes C or D
will contribute. For the same reason, the $R_{3,4}$ solutions in asymptotics
classes A and B will also not contribute. Thus, only the $R_{1,2}$ solutions,
given in (65), need to be taken into consideration. Taking the expressions for
$W[\tilde{\varphi}_{1},\tilde{\varphi}_{1}]$ and
$W[\tilde{\varphi}_{2},\tilde{\varphi}_{2}]$ evaluated at $r_{\infty}$
directly from (66), one arrives at,

$\begin{split}2\pi\sum_{j}W[\tilde{\varphi}_{j},\tilde{\varphi}_{j}]=&\
\frac{1}{2\pi}\left[\frac{q_{\infty}^{-}+q_{\infty}^{+}|\mathcal{R}|^{2}}{|q_{\infty}^{-}|}+\frac{q^{+}_{\infty}|\mathcal{U}|^{2}}{|q^{\pm}_{0}|}\right],\\\
=&\
\frac{|\mathcal{R}|^{2}-1}{2\pi}\left[1+\mathrm{sgn}(q^{\pm}_{0}/q^{-}_{\infty})\right],\end{split}$
(85)

where in the second line, I have used
$q^{+}_{\infty}q^{-}_{\infty}|\mathcal{U}|^{2}=q^{\pm}_{0}(q_{\infty}^{-}+q_{\infty}^{+}|\mathcal{R}|^{2})$,
which is obtained by combining the three relations in (66), and also the fact
that the reflection coefficient is evaluated just outside $r_{2}$ so
$q_{\infty}^{+}=-q_{\infty}^{-}>0$. Recalling that the upper sign should be
taken for scattering Types $\mathrm{I}^{0}$, $\mathrm{II}^{0}$,
$\mathrm{I}^{+}$, $\mathrm{II}^{+}$, $\mathrm{V}^{+}$ and $\mathrm{VIII}^{+}$
and the lower sign for Types $\mathrm{III}^{0}$, $\mathrm{VI}^{+}$ and
$\mathrm{IX}^{+}$, one finds $\mathrm{sgn}(q^{\pm}_{0}/q^{-}_{\infty})=\mp 1$.
Thus, the term in the square brackets vanishes for all non-superradiant
scattering scenarios.

Finally, equating this energy flux with an energy loss from the system (and
the same for the angular momentum flux) one finds,

$\begin{pmatrix}\dot{E}\\\
\dot{L}\end{pmatrix}=-\frac{1}{\pi}\sum_{m}\int_{SR}d\omega~{}\begin{pmatrix}\omega\\\
m\end{pmatrix}\left(|\mathcal{R}|^{2}-1\right),$ (86)

in agreement with [43]. Here $SR$ indicates that the $\omega$ integral is
performed only over superradiant frequencies, and the expression for
$|\mathcal{R}|$ is given by that in (76) with the lower sign.

Figure 11: The values of $\dot{E}$ and $\dot{L}$ as a function of $C$ using
the WKB approximation (solid and dashed lines respectively) and an exact
result obtained from numerical simulation of the equations of motion (dots and
crosses respectively). The relative error between the WKB and exact result is
a decreasing function of $C$, indicating that the approximation improves for
large $C$.

###  7.2 Non-dispersive results

To evaluate $\dot{E}$ and $\dot{L}$, the reflection coefficients are computed
using the WKB formula in Eq. (76), as well as from exact simulation of the
wave equation (see e.g. Appendix A of [23]). Fig. 11 demonstrates that the
rates of energy and angular momentum loss are dramatically increased for
larger $C$ values (in dimensionful units, this means for large $C/D$ ratios).
It is also shown that the agreement between the exact and WKB results
increases with $C$. The reason for this is dominant mode in the sum in (86) is
the $m\sim|C/D|$ mode as shown in Fig. 12. Since the WKB approximation
improves for large $m$, the values of $\dot{E}$ and $\dot{L}$ will become more
accurate for large $C$. This contrasts the black hole case where the lowest
angular momentum mode always dominates the sum [43]. However, this is to be
expected since in a Kerr black hole $a/M<1$, where $a$ and $M$ are the
rotation and mass parameters of the spacetime. The difference with a fluid
mechanical vortex is that the ratio $C/D$ is (in principle) not bound from
above. In the large $C$ limit, the rates are approximately,

$\dot{E}\approx-0.05~{}C^{5},\qquad\dot{L}\approx-0.07~{}C^{4},$ (87)

where the coefficients and exponents have been obtained from a numerical fit
over the range $C\in[0,100]$.

Figure 12: The integral appearing in $\dot{E}$ for different values of $m$ and
$C$. The solid line respresents the WKB reflection coefficients, whereas the
crosses are from numerical simulation. Although (86) is a sum over integer
values of $m$, the WKB result is extended to non-integer values here to
clearly indicate the location of the peak, which appears at roughly $m\sim C$.
Since the WKB result becomes closer to the exact one for higher $m$, the
agreement between $\dot{E}_{\mathrm{num}}$ and $\dot{E}_{\mathrm{wkb}}$ will
improve for large $C$ where the higher $m$ modes dominate the sum. A similar
plot can be obtained for the components of $\dot{L}$.

###  7.3 Dispersive results

When $\Lambda>0$, the dependence of $\dot{E}$ and $\dot{L}$ on $C$ is
qualitatively similar to the non-dispersive case. Fig. 13 demonstrates that as
the parameter $\Lambda$ increases, the values of $\dot{E}$ and $\dot{L}$
decrease. As explained at the end of Section 6.3, there are two principle
reasons for this; 1) the superradiant bandwidth becomes narrower as $\Lambda$
increases and 2) modes with $m$ larger than $m_{\mathrm{max}}$ (obtained by
inverting the expression for $C_{0}$ in (71)) will not superradiate. For fixed
$C$, $m_{\mathrm{max}}$ decreases with increasing $\Lambda$ until eventually
there are no superradiant modes left in the system, and $\dot{E}$ and
$\dot{L}$ go to zero. Inverting this statement, for $\Lambda$ fixed (and
larger than $1/4$) there is a value $C_{\mathrm{min}}$ below which
superradiance will not occur for any $m$. $C_{\mathrm{min}}$ goes to zero at
$\Lambda=1/4$, which means that for $\Lambda<1/4$, superradiance will occur at
least for some $m$ modes for all $C$.

Figure 13: Dispersion decreases the rate at which superradiance extracts
energy and angular momentum from the vortex. Non-dispersive rates are
indicated as horizontal lines on the left of the plots for comparison. The
critical $\Lambda$ value of $\Lambda=1/4$ is shown as a dashed black line.

##  8 Discussion

###  8.1 Effect of quantised $\ell$

In contrast to previous sections, from here on the dimensions on various
quantities will be restored. In analogy with the circulation parameter defined
in (8), let the drain parameter be given by,

$D=\hbar d/M,$ (88)

where (unlike $\ell$) $d$ is not constrained to be an integer. There are two
reasons to write $D$ this way. The first is that the ratio $C/D=\ell/d$, which
permits an easy comparison of the drain strength to the winding number. The
second is that $d=r_{h}/\xi$ gives the ratio of the “would be” horizon (i.e.
$D/c$) to the healing length $\xi=\hbar/\sqrt{M\mu}$, which characterises the
scale over which the condensate heals back to it’s bulk value around
boundaries. In particular $\xi$ gives a characteristic length on which the
density varies in the vortex core.

The dimensionless dispersion parameter discussed throughout this work is now
really $\Lambda/D^{2}=1/4d^{2}$. Therefore, the relative strength of
dispersion is controlled by how fast the system is draining. Importantly, it
was noted above that superradiance shuts off at a critical $C$ value when
$\Lambda/D^{2}>1/4$. In terms of $\ell$ and $d$, this really means that when
the system is weakly draining, i.e. $d<1$, superradiance can only spin down
the system whilst $\ell>\ell_{\mathrm{min}}=(1-d^{2})/2$. However, since $l$
can only take on integer values, the minimum rotation for a spinning vortex
will be $\ell=1$, which is always greater than $\ell_{\mathrm{min}}$. Thus,
the vortex will still be able to superradiate whilst ever it is spinning,
irrespective of the value of $d$.

Although the rates in (86) will force the system to evolve, it remains unclear
exactly how this evolution might occur. From $\mathcal{S}_{\mathrm{GPE}}$, the
conserved current for rotations in $\theta$ gives the angular momentum of the
system,

$L=\hbar\ell N.$ (89)

This quantity must decrease as a result of $\dot{L}$ in (86). There are two
ways this may occur; either by a reduction in the number of atoms in the
condensate or a reduction in $\ell$. If it is the former, then the rotation
$C$ remains fixed during the evolution, whereas if it is the latter then the
density decreases. To determine which of these is the dominant effect, one
would need to solve the full backreaction equations. Furthermore, since $\ell$
is quantised, it can only presumably only decrease in discrete lumps, but this
may itself induced problems since it would require an instantaneous global
change in the phase of the condensate. One possibility is that evolution may
proceed through the emission of a quantised vortex. The precise detail of this
evolution certainly warrants further investigation.

###  8.2 Gravity waves

It was argued above that the suppression of superradiance due to strong
dispersion (weak drain) would not be observable in a BEC due to the quantised
nature of circulation. Might there be another system where this suppression is
observable?

The equations of motion in (10) are equivalent to those governing capillary-
gravity waves in the shallow water regime $hk\ll 1$, with $h$ the height of
the fluid. In that case, the wave speed is $c=\sqrt{gh}$, with $g$ now the
acceleration due to gravity, and the dispersive parameter is $\Lambda=\sigma
h/\rho_{f}$, with $\sigma$ the surface tension of the fluid and $\rho_{f}$
it’s density. In the experiment of [2], $D$ and $h$ were of the order of
$10^{-3}$ and $10^{-2}$ respectively. Taking the values of $\sigma$ and
$\rho_{f}$ for water, the dimensionless dispersive parameter is
$\Lambda/D^{2}\sim\mathcal{O}(1)$. At this strength of dispersion, the $m=1,2$
modes would cease to superradiate around $C/D\sim\mathcal{O}(1)$. However,
since $C/D$ was closer to 15 for that experiment, one would still expect to
see superradiance for these modes. This is consistent with observation.

Note that the present estimates are for shallow water waves, whereas the
experiments in [2] were performed closer to the deep water regime. It was
shown in [23] that superradiance can still occur in deep water, and a next
natural step would be to add in the capillary modification to the dispersion
relation. Since this makes the dispersion superluminal at high $k$, one would
expect some of the features of scattering in the present system to carry over.
In particular, one might expect superradiance to be suppressed for low
circulation. Importantly, since the fluid is classical, there is no longer a
minimum value for $C$, raising the possibility of observing this suppression
of superradiance in a water tank experiment akin to that of [2].

##  9 Conclusion

In this work, a general framework for studying superradiance in dispersive
systems (which follows on from [23]) has been presented. This framework was
then applied to study how superradiance occurs under Bogoliubov type
dispersion. Through a detailed analysis of the possible scattering outcomes,
it was found that the superradiance condition in (1) gets modified to,

$\omega<m\Omega_{h}^{\mathrm{rot}}\left(1+m\xi/r_{h}\right)^{-1},$ (90)

where $r_{h}$ and $\Omega_{h}^{\mathrm{rot}}$ are defined as in the non-
dispersive case, although $r_{h}$ is no longer a true horizon. This condition
essentially determines when the in-going mode ceases to tunnel back to the
$\Omega>0$ branch of the dispersion relation at small $r$, and instead
attempts to tunnel to the $\Omega<0$ branch. However, (90) is no guarantee
that the mode will have reached this branch by $r=0$, which means as a
consequence that (in this particular regime) incident waves are completely
reflected by the vortex and superradiance suppressed. In turned out that this
suppression of superradiance was outside of the parameter range for
superfluids owing to the effects of quantised circulation and an analogous
classical system, namely shallow water capillary-gravity waves, was instead
suggested as an alternative where this suppression may be observable.

It was then argued that the main influence of dispersion on the superradiance
spectrum would be to reduce the bandwidth in $\omega$ over which superradiance
occurs, whereas the amount of amplification for a given superradiant mode
would not change significantly. The physical reason for this is that the modes
involved in the scattering are the same ones present in the non-dispersive
case, which live on the part of the dispersion relation where the effects of
$\Lambda\neq 0$ are smallest. In particular, the location of the turning
points (which determine size of the reflection coefficient) are only slightly
modified. A further difference with the non-dispersive system is that just
above the superradiance threshold, the combination of large rotations and
dispersion meant that incident $m>0$ modes would not be absorbed by the vortex
and instead completely reflected.

Finally, the spontaneous emission of the vortex due to it’s superradiant
vacuum fluctuations was studied. As expected from classical considerations, it
was found that the rate of energy and angular momentum loss is decreased by
dispersion, since there are less $\omega$ and $m$ modes to superradiate.
Another interesting finding was that (in the non-dispersive case where all
$m>0$ superradiate) the mode which extracts most from the system is the
$m\sim|C/D|$ mode. This highlights an interesting property of superradiance
which is not present in black hole physics, where the equivalent ratio ($a/M$)
is always less than unity of a consequence of cosmic censorship.

For simplicity, I have only treated the constant density approximation here,
due to the possibility of obtaining concise formulae for the various important
parameters (in particular $\omega_{\star}$). Since $\rho(r)$ varies over a
(minimum) scale of $\sim\xi$, the constant density approximation is applicable
roughly when $r_{h}\gg\xi$, or equivalently $\Lambda/D^{2}\ll 1/4$. This has
implications for the regions of parameter space in Fig. 9 where the scattering
coefficients provided herein are a good approximation. Despite this, the
effects of varying $\rho$ could easily be incorporated into the formalism,
simply by inserting a coefficient of $\rho(r)/\rho(\infty)$ in front of the
$k^{2}$ term in (32), and the predictions improved.

Many thanks to Cisco Gooding, August Geelmuyden, Sebastian Erne and Silke
Weinfurtner for discussions involving various aspects of this project.
Financial support was provided by Silke Weinfurtner’s Leverhulme Research
Leadership Award (RL-2019-020).

## Appendix A

In this appendix, the normalisation of incident WKB modes (which defines the
different independent solutions, e.g. (65), of the radial equation of motion)
is performed explicitly. The inner product (25) evaluated for two WKB modes
is,

$\begin{split}(\varphi_{\lambda_{1}},\varphi_{\lambda_{2}})=&\ \frac{1}{2}\int
d^{2}\mathbf{x}~{}\frac{\mathcal{N}_{\lambda_{1}}\mathcal{N}^{*}_{\lambda_{2}}}{r}\left|\frac{f_{\lambda_{1}}f_{\lambda_{2}}}{\mathcal{H}^{\prime}_{\lambda_{1}}\mathcal{H}^{\prime}_{\lambda_{2}}}\right|^{\frac{1}{2}}\left(\frac{\Omega_{\lambda_{1}}}{f_{\lambda_{1}}}+\frac{\Omega_{\lambda_{2}}}{f_{\lambda_{2}}}\right)e^{i\int(p_{\lambda_{1}}-p_{\lambda_{2}})dr+i(m_{1}-m_{2})-i(\omega_{1}-\omega_{2})t},\\\
=&\
\tfrac{1}{2}(2\pi)\delta_{m_{1}m_{2}}e^{-i(\omega_{1}-\omega_{2})t}\mathcal{N}_{\lambda_{1}}\mathcal{N}^{*}_{\lambda_{2}}\int
dr\left|\frac{f_{\lambda_{1}}f_{\lambda_{2}}}{\mathcal{H}^{\prime}_{\lambda_{1}}\mathcal{H}^{\prime}_{\lambda_{2}}}\right|^{\frac{1}{2}}\left(\frac{\Omega_{\lambda_{1}}}{f_{\lambda_{1}}}+\frac{\Omega_{\lambda_{2}}}{f_{\lambda_{2}}}\right)e^{i\int(p_{\lambda_{1}}-p_{\lambda_{2}})dr},\end{split}$
(91)

where the $\mathcal{N}_{\lambda}$ are constants to be found from
normalisation, and to get to the second line the $\theta$ integral has been
evaluated. At this point, it would be possible to evaluate the $r$ integral if
the exponent were a difference of constants multiplied by $r$ and the
prefactor under the integral were also constant. This is indeed the case when
the modes are normalised in the non-dispersive case at spatial infinity.
However, it is not the case (in general) when the modes are dispersive, or are
being normalised in a region where the background flow is non-vanishing.

To proceed, recognise that in the WKB approximation, $p_{\lambda_{1}}$ and
$p_{\lambda_{2}}$ are assumed large. Hence if $p_{\lambda_{1}}\neq
p_{\lambda_{2}}$, the phase term will oscillate rapidly leading to
cancellation in the integral. The dominant contribution comes from near
$p_{\lambda_{1}}=p_{\lambda_{2}}$. This cannot be satisfied unless one works
with the same type of WKB mode, but this defines the specific $j$ solution of
the radial equation of motion. Hence the expression above picks up a factor of
$\delta_{j_{1}j_{2}}$. Next, expand the momentum,

$p(\omega_{1})=p(\omega_{2})+\partial_{\omega}p(\omega_{2})(\omega_{1}-\omega_{2}),$
(92)

and similarly for $\Omega_{1}$, $f_{1}$ and $\mathcal{H}^{\prime}_{1}$. Note,
I have dropped the subscript $\lambda_{1,2}$ since it is now assumed we are
working with the same $m,j$-mode and hence, the different quantities only
differ due to their dependence on $\omega$. At leading order in
$(\omega_{1}-\omega_{2})$, the integral becomes,

$2\int
dr\left|\frac{f(\omega_{2})}{\mathcal{H}^{\prime}(\omega_{2})}\right|\frac{\Omega(\omega_{2})}{f(\omega_{2})}e^{i(\omega_{1}-\omega_{2})\int\partial_{\omega}p(\omega_{2})dr}.$
(93)

By noting $f>0$ for propagating modes, the $f$ terms cancel. Next, define a
new coordinate $X=\int\partial_{\omega}p(\omega_{2})dr$, which satisfies
$dr=\vec{\mathbf{e}}_{r}\cdot\bm{v}_{g}~{}dX$. The integral now reads,

$\mathrm{sgn}(\Omega)~{}2\int
dXe^{i(\omega_{1}-\omega_{2})X}=\mathrm{sgn}(\Omega)~{}4\pi\delta(\omega_{1}-\omega_{2}),$
(94)

where to get to the left hand side, I have used
$\mathcal{H}^{\prime}=\vec{\mathbf{e}}_{r}\cdot\bm{v}_{g}\Omega$. Plugging
this back into the expression above, one obtains,

$(\varphi_{\lambda_{1}},\varphi_{\lambda_{2}})=\mathrm{sgn}(\Omega)(2\pi)^{2}\mathcal{N}_{\lambda_{1}}\mathcal{N}^{*}_{\lambda_{2}}\delta_{m_{1}m_{2}}\delta_{j_{1}j_{2}}\delta(\omega_{1}-\omega_{2}).$
(95)

Thus, for these modes to be normalised, the amplitude must be,

$\mathcal{N}_{\lambda}=(2\pi)^{-1},$ (96)

and the normalisation condition can be written in the compact notation,

$(\varphi_{\lambda_{1}},\varphi_{\lambda_{2}})=\mathrm{sgn}(\Omega)\delta_{\lambda_{1}\lambda_{2}}.$
(97)

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