Rotational superradiance with Bogoliubov dispersion

Consider now the form of the velocity field. I will assume that v is independent θ and that the density is approximately uniform, i.e. ρ ≈ const. In this case, (5a) becomes ∇ ⋅ v = 0 and (5b) implies ∇ × v = 0. The unique velocity profile for a θ independent fluid is then,

$$\begin{equation}\mathbf{v}=-\frac{D}{r}{ \overrightarrow {\mathbf{e}}}_{r}+\frac{C}{r}{ \overrightarrow {\mathbf{e}}}_{\theta },\end{equation} \tag{ 7 }$$

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 Ψ must be periodic in θ to satisfy the boundary conditions, C must be of the form,

$$\begin{equation}C=\hslash \ell /M,\end{equation} \tag{ 8 }$$

where ℓ is an integer called the winding number. This is the well-known result that circulation in a BEC is quantised [34]. The flow profile in (7) is known as the DBT in the literature. Note that vortices with winding number higher than ℓ = 1 are usually unstable [50] and in fact, this instability has been argued to be related to the presence of an ergoregion (the same mechanism responsible for superradiance) [42]. Stabilisation mechanisms, e.g. via trapping potentials, have however been demonstrated [51, 52].

In classical fluids, the angular component ${v}_{\theta }={ \overrightarrow {\mathbf{e}}}_{\theta }\cdot \mathbf{v}$ in (7) is often used as an idealisation of realistic velocity profiles [24]. 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 ${v}_{r}={ \overrightarrow {\mathbf{e}}}_{r}\cdot \mathbf{v}$, 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. [53, 54]). In order to keep N fixed, one could then resupply atoms at the same rate at the outer edge of the condensate.

Now consider fluctuations of the condensate density and phase,

$$\begin{equation}\rho \to \rho \left(1+\eta \right),\hspace{25.0pt}{\Phi}\to {\Phi}+\phi .\end{equation} \tag{ 9 }$$

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

$$\begin{equation}\begin{gathered}{D}_{t}\phi +{c}^{2}\eta -{\Lambda}{\mathbf{\nabla }}^{2}\eta =\enspace 0,\\ {D}_{t}\eta +{\mathbf{\nabla }}^{2}\phi =\enspace 0,\end{gathered}\end{equation} \tag{ 10 }$$

where D_t = ∂_t + v ⋅ ∇ is the material derivative and the constants c and Λ are given by,

$$\begin{equation}c=\sqrt{g\rho /M},\hspace{25.0pt}{\Lambda}={\hslash }^{2}/4{M}^{2}.\end{equation} \tag{ 11 }$$

For Λ = 0, the system is non-dispersive and all wavelengths will propagate at the same speed c. When Λ ≠ 0, shorter-wavelengths travel faster than c and the system becomes 'superluminally' dispersive ¹ . 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 Λ.

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

$$\begin{equation}\mathcal{S}=\int \mathrm{d}t\enspace {\mathrm{d}}^{2}\mathbf{x}\left[\frac{1}{2}\phi {D}_{t}\eta -\frac{1}{2}\eta {D}_{t}\phi -\frac{1}{2}{\eta }^{2}-\frac{1}{2}{\left(\mathbf{\nabla }\phi \right)}^{2}-\frac{1}{2}{\Lambda}{\left(\mathbf{\nabla }\eta \right)}^{2}\right],\end{equation} \tag{ 12 }$$

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

By Noether's theorem, symmetries of the action give rise to conserved currents [55]. In particular, the transformation,

$$\begin{equation}\phi \to \phi +\delta \phi ,\end{equation} \tag{ 13 }$$

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

$$\begin{equation}\delta \mathcal{L}={\partial }_{t}\left(\frac{\partial \mathcal{L}}{\partial \dot {\phi }}\delta \phi +\frac{\partial \mathcal{L}}{\partial \dot {\eta }}\delta \eta \right)+\mathbf{\nabla }\cdot \left(\frac{\partial \mathcal{L}}{\partial \mathbf{\nabla }\phi }\delta \phi +\frac{\partial \mathcal{L}}{\partial \mathbf{\nabla }\eta }\delta \eta \right).\end{equation} \tag{ 14 }$$

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

$$\begin{equation}{\partial }_{t}\rho \left[\phi \right]+\mathbf{\nabla }\cdot \mathbf{J}\left[\phi \right]=0,\end{equation} \tag{ 15 }$$

where the components of the current are given by,

$$\begin{equation}\begin{aligned}\hfill \rho \left[\phi \right]=& \enspace \frac{\partial \mathcal{L}}{\partial \dot {\phi }}\delta \phi +\frac{\partial \mathcal{L}}{\partial \dot {\eta }}\delta \eta -f,\hfill \\ \hfill \mathbf{J}\left[\phi \right]=& \enspace \frac{\partial \mathcal{L}}{\partial \mathbf{\nabla }\phi }\delta \phi +\frac{\partial \mathcal{L}}{\partial \mathbf{\nabla }\eta }\delta \eta -\mathbf{F}.\hfill \end{aligned}\end{equation} \tag{ 16 }$$

This ρ[ϕ] (which is the time component of the current) is not to be confused with the density defined earlier.

Due to the t and θ independence of v, the action will be invariant under t and θ 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,

$$\begin{equation}{J}_{E}^{r}\left[\phi \right]=\left(-\frac{1}{2}{v}_{r}\eta -{\partial }_{r}\phi \right){\partial }_{t}\phi +\left(\frac{1}{2}{v}_{r}\phi -{\Lambda}{\partial }_{r}\eta \right){\partial }_{t}\eta ,\end{equation} \tag{ 17 }$$

for the energy current and,

$$\begin{equation}{J}_{L}^{r}\left[\phi \right]=\left(-\frac{1}{2}{v}_{r}\eta -{\partial }_{r}\phi \right){\partial }_{\theta }\phi +\left(\frac{1}{2}{v}_{r}\phi -{\Lambda}{\partial }_{r}\eta \right){\partial }_{\theta }\eta ,\end{equation} \tag{ 18 }$$

for the angular momentum current.

Due to the symmetry of background, it is beneficial decompose the fields ϕ and η into their different frequency ω and azimuthal m components. In this paper, I will work with the following notation,

$$\begin{equation}\phi =\sum\limits _{\lambda }\left({\alpha }_{\lambda }{\varphi }_{\lambda }+{\alpha }_{\lambda }^{{\ast}}{\varphi }_{\lambda }^{{\ast}}\right),\hspace{25.0pt}\eta =\sum\limits _{\lambda }\left({\alpha }_{\lambda }{n}_{\lambda }+{\alpha }_{\lambda }^{{\ast}}{n}_{\lambda }^{{\ast}}\right),\end{equation} \tag{ 19 }$$

with λ denoting a particular ω, m, j triplet. The field modes are,

$$\begin{equation}{\varphi }_{\lambda }\equiv {\varphi }_{\omega mj}\left(t,\theta ,r\right)={\tilde {\varphi }}_{j}\left(\omega ,m,r\right){\mathrm{e}}^{\mathrm{i}m\theta -\mathrm{i}\omega t},\end{equation} \tag{ 20 }$$

and similarly for n_λ . The $\mathbb{C}$-fields φ and φ* are often called positive and negative frequency components respectively. The α_λ are constant amplitudes multiplying the $\mathbb{C}$-fields, which need to be taken in a symmetric combination due to the fact that ϕ and η are both real. The sum over λ is short for,

$$\begin{equation}\sum\limits _{\lambda }=\sum\limits _{m,j}\int \mathrm{d}\omega ,\end{equation} \tag{ 21 }$$

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

$$\begin{equation}\begin{gathered}-\mathrm{i}\left(\omega -\frac{mC}{{r}^{2}}\right){\tilde {\varphi }}_{j}+\frac{1}{r}{\partial }_{r}{\tilde {\varphi }}_{j}+\left(1+\frac{{\Lambda}{m}^{2}}{{r}^{2}}\right){\tilde {n}}_{j}-\frac{{\Lambda}}{r}{\partial }_{r}{\tilde {n}}_{j}-{\Lambda}{\partial }_{r}^{2}{\tilde {n}}_{j}=\enspace 0,\\ -\mathrm{i}\left(\omega -\frac{mC}{{r}^{2}}\right){\tilde {n}}_{j}+\frac{1}{r}{\partial }_{r}{\tilde {n}}_{j}-\frac{{m}^{2}}{{r}^{2}}{\tilde {\varphi }}_{j}+\frac{1}{r}{\partial }_{r}{\tilde {\varphi }}_{j}+{\partial }_{r}^{2}{\tilde {\varphi }}_{j}=\enspace 0.\end{gathered}\end{equation} \tag{ 22 }$$

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

Due to the linearity of the equations of motion (10), each λ 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,

$$\begin{align}\hfill {\mathcal{L}}_{\mathbb{C}}& =\frac{1}{2}\left(\frac{1}{2}{\varphi }_{\lambda }^{{\ast}}{D}_{t}{n}_{\lambda }+\frac{1}{2}{\varphi }_{\lambda }{D}_{t}{n}_{\lambda }^{{\ast}}-\frac{1}{2}{n}_{\lambda }^{{\ast}}{D}_{t}{\varphi }_{\lambda }-\frac{1}{2}{n}_{\lambda }{D}_{t}{\varphi }_{\lambda }^{{\ast}}\right.\hfill \\ \hfill & \quad \left.-{n}_{\lambda }{n}_{\lambda }^{{\ast}}-\mathbf{\nabla }{\varphi }_{\lambda }\cdot \mathbf{\nabla }{\varphi }_{\lambda }^{{\ast}}-{\Lambda}\mathbf{\nabla }{n}_{\lambda }\cdot \mathbf{\nabla }{n}_{\lambda }^{{\ast}}\right).\hfill \end{align} \tag{ 23 }$$

Applying Noether's theorem for the internal symmetry φ_λ → φ_λ e^−iα (and also for ${\varphi }_{\lambda }^{{\ast}},{n}_{\lambda },{n}_{\lambda }^{{\ast}}$) one finds the conservation of the norm current, whose components are,

$$\begin{equation}\begin{aligned}\hfill {\rho }_{\mathrm{N}}\left[\varphi \right]=& \enspace \frac{\mathrm{i}}{2}\left({\varphi }_{\lambda }{n}_{\lambda }^{{\ast}}-{n}_{\lambda }{\varphi }_{\lambda }^{{\ast}}\right),\hfill \\ \hfill {\mathbf{J}}_{\mathrm{N}}\left[\varphi \right]=& \enspace \frac{\mathrm{i}}{2}\left\{\mathbf{v}\left[{\varphi }_{\lambda }{n}_{\lambda }^{{\ast}}-{n}_{\lambda }{\varphi }_{\lambda }^{{\ast}}\right]+{\varphi }_{\lambda }\mathbf{\nabla }{\varphi }_{\lambda }^{{\ast}}-\left(\mathbf{\nabla }{\varphi }_{\lambda }\right){\varphi }_{\lambda }^{{\ast}}\right.\hfill \\ \hfill & \left.+{\Lambda}\left[{n}_{\lambda }\mathbf{\nabla }{n}_{\lambda }^{{\ast}}-\left(\mathbf{\nabla }{n}_{\lambda }\right){n}_{\lambda }^{{\ast}}\right]\right\}.\hfill \end{aligned}\end{equation} \tag{ 24 }$$

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

$$\begin{equation}\left({\varphi }_{{\lambda }_{1}},{\varphi }_{{\lambda }_{2}}\right)=\frac{\mathrm{i}}{2}\int {\mathrm{d}}^{2}\mathbf{x}\left({\varphi }_{{\lambda }_{1}}{n}_{{\lambda }_{2}}^{{\ast}}-{n}_{{\lambda }_{1}}{\varphi }_{{\lambda }_{2}}^{{\ast}}\right).\end{equation} \tag{ 25 }$$

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

$$\begin{align}\hfill W\left[{\varphi }_{{\lambda }_{1}},{\varphi }_{{\lambda }_{2}}\right]=& \enspace \frac{\mathrm{i}}{2}r\left\{{v}_{r}\left[{\varphi }_{{\lambda }_{1}}{n}_{{\lambda }_{2}}^{{\ast}}-{n}_{{\lambda }_{1}}{\varphi }_{{\lambda }_{2}}^{{\ast}}\right]+{\varphi }_{{\lambda }_{1}}{\partial }_{r}{\varphi }_{{\lambda }_{2}}^{{\ast}}-\left({\partial }_{r}{\varphi }_{{\lambda }_{1}}\right){\varphi }_{{\lambda }_{2}}^{{\ast}}\right.\hfill \\ \hfill & \left.+{\Lambda}\left[{n}_{{\lambda }_{1}}{\partial }_{r}{n}_{{\lambda }_{2}}^{{\ast}}-\left({\partial }_{r}{n}_{{\lambda }_{1}}\right){n}_{{\lambda }_{2}}^{{\ast}}\right]\right\}.\hfill \end{align} \tag{ 26 }$$

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

$$\begin{equation}{\left({\partial }_{t}S+\mathbf{v}\cdot \mathbf{\nabla }S\right)}^{2}-{\left(\mathbf{\nabla }S\right)}^{2}-{\Lambda}{\left(\mathbf{\nabla }S\right)}^{4}=0.\end{equation} \tag{ 28 }$$

Identifying the frequency and wavevector through,

$$\begin{equation}\omega =-{\partial }_{t}S,\hspace{25.0pt}\mathbf{k}=\mathbf{\nabla }S,\end{equation} \tag{ 29 }$$

with k = ||k||, the Hamilton–Jacobi equation is equivalent to the dispersion relation,

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

which determines the relationship between the local values of ω and k when v is varying. Note that for the case of v = 0, (30) is the Bogoliubov dispersion relation originally derived in [56]. Using this notation, (10) can be used to write a leading order relation between the amplitudes,

$$\begin{equation}\mathcal{B}=\mathrm{i}{\Omega}{f}^{-1}\mathcal{A},\quad f=1+{\Lambda}{k}^{2}.\end{equation} \tag{ 31 }$$

Since the dispersion relation is quadratic in ω, it has two branches,

$$\begin{equation}{\omega }_{\mathrm{D}}^{{\pm}}=\mathbf{v}\cdot \mathbf{k}{\pm}\sqrt{{k}^{2}+{\Lambda}{k}^{4}},\end{equation} \tag{ 32 }$$

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

$$\begin{equation}{\boldsymbol{v}}_{\mathrm{g}}={\mathbf{\nabla }}_{\mathbf{k}}\omega =\mathbf{v}{\pm}\mathbf{k}\frac{1+2{\Lambda}{k}^{2}}{\sqrt{{k}^{2}+{\Lambda}{k}^{4}}}.\end{equation} \tag{ 33 }$$

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,

$$\begin{equation}\mathcal{H}=-\frac{1}{2}\left(\omega -{\omega }_{\mathrm{D}}^{+}\right)\left(\omega -{\omega }_{\mathrm{D}}^{-}\right).\end{equation} \tag{ 34 }$$

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

$$\begin{equation}{\dot {x}}^{\mu }=\frac{\partial \mathcal{H}}{\partial {k}_{\mu }},\hspace{25.0pt}{\dot {k}}_{\mu }=-\frac{\partial \mathcal{H}}{\partial {x}^{\mu }}\end{equation} \tag{ 35 }$$

where x^μ = (x, t), k_μ = (k, −ω). In this section, the overdot denotes the derivative with respect to τ which parameterises the characteristics. Solving the system of equation (35) gives the coordinates and the conjugate momenta in terms of the parameter τ, i.e. x^μ = x^μ (τ) and k_μ = k_μ (τ). The phase part of φ 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,

$$\begin{equation}\mathcal{H}=0,\end{equation} \tag{ 36 }$$

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 θ independent system introduced in section 3.1. In polar coordinates, the wave vector has components,

$$\begin{equation}\mathbf{k}=\left(p,m/r\right),\hspace{25.0pt}k=\sqrt{{p}^{2}+{m}^{2}/{r}^{2}},\end{equation} \tag{ 37 }$$

where p is the radial wave vector. By Hamilton's equation (35), ω 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 τ and solve (35) for r = r(τ) and p = p(τ). 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 ² .

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

$$\begin{equation}{\partial }_{t}\left({f}^{-1}{\Omega}{\mathcal{A}}^{2}\right)+\mathbf{\nabla }\cdot \left({\boldsymbol{v}}_{\mathrm{g}}{f}^{-1}{\Omega}{\mathcal{A}}^{2}\right)=0,\end{equation} \tag{ 38 }$$

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 θ symmetric system of section 3.1, the amplitude is simply,

$$\begin{equation}\mathcal{A}=\vert qr{\vert }^{-\frac{1}{2}}\mathcal{N},\hspace{25.0pt}q\equiv q\left(r,{p}^{l}\right)={f}^{-1}\left(r,{p}^{l}\right){\mathcal{H}}^{\prime }\left(r,{p}^{l}\right),\end{equation} \tag{ 39 }$$

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

$$\begin{equation}{\tilde {\varphi }}_{j}=\sum\limits _{l}\vert {q}^{l}r{\vert }^{-\frac{1}{2}}{\mathcal{N}}_{j}^{l}{\text{e}}^{\text{i}\int {p}^{l}\mathrm{d}r},\end{equation} \tag{ 40 }$$

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 ${\mathcal{N}}_{j}^{l}$ will be different for each of the j independent solutions. The phase of the radial part of the wave is obtained using the definition in (29) together with the expression for the full WKB mode in (27).

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

$$\begin{equation}{\rho }_{\mathrm{N}}\left[{\tilde {\varphi }}_{j}^{l}\right]=\frac{{{\Omega}}^{l}}{{f}^{l}}\vert {\tilde {\varphi }}_{j}^{l}{\vert }^{2},\end{equation} \tag{ 41 }$$

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

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

$$\begin{align}\hfill W\left[{\tilde {\varphi }}_{j},{\tilde {\varphi }}_{{j}^{\prime }}\right]=& \enspace \frac{1}{2}\sum\limits _{l,{l}^{\prime }}\left[{v}_{r}\left(\frac{{{\Omega}}^{l}}{{f}^{l}}+\frac{{{\Omega}}^{{l}^{\prime }}}{{f}^{{l}^{\prime }}}\right)+\left({p}^{l}+{p}^{{l}^{\prime }}\right)\left(1+{\Lambda}\frac{{{\Omega}}^{l}{{\Omega}}^{{l}^{\prime }}}{{f}^{l}{f}^{{l}^{\prime }}}\right)\right]\hfill \\ \hfill & {\times}\frac{{\mathcal{N}}_{j}^{l}{{\mathcal{N}}_{{j}^{\prime }}^{{l}^{\prime }}}^{{\ast}}}{\vert {q}^{l}{q}^{{l}^{\prime }}{\vert }^{\frac{1}{2}}}{\text{e}}^{\text{i}\int \left({p}^{l}-{p}^{{l}^{\prime }}\right)\mathrm{d}r},\hfill \end{align} \tag{ 42 }$$

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

Since (26) is constant in r by definition, and the phase term will cause oscillations if l ≠ l', the factor in square brackets must vanish for these cases. This is proven in [49] 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' and one finds,

$$\begin{equation}W\left[{\tilde {\varphi }}_{j},{\tilde {\varphi }}_{{j}^{\prime }}\right]=\sum\limits _{l}\enspace \mathrm{sgn}\left({q}^{l}\right){\mathcal{N}}_{j}^{l}{{\mathcal{N}}_{{j}^{\prime }}^{l}}^{{\ast}}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.\end{equation} \tag{ 43 }$$

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 ω.

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 equation (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 its direction, i.e. the classical turning points. Denoting these locations r_tp, they are found by solving the simultaneous equations,

$$\begin{equation}{\mathcal{H}}_{\mathrm{t}\mathrm{p}}=0,\hspace{25.0pt}{\partial }_{p}{\mathcal{H}}_{\mathrm{t}\mathrm{p}}=0,\end{equation} \tag{ 44 }$$

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,

$$\begin{equation}\omega ={\omega }_{\mathrm{D}}^{{\pm}}\left({r}_{\mathrm{t}\mathrm{p}},{p}_{\mathrm{t}\mathrm{p}}\right),\hspace{25.0pt}{\partial }_{p}{\omega }_{\mathrm{D}}^{{\pm}}\left({r}_{\mathrm{t}\mathrm{p}},{p}_{\mathrm{t}\mathrm{p}}\right)=0,\end{equation} \tag{ 45 }$$

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 ${\omega }_{\mathrm{D}}^{{\pm}}$ 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. [26, 57]. 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 its asymptotic value becomes a better approximation closer to the turning point. The matrix which relates the WKB modes either side of r_tp is,

$$\begin{equation}\left(\begin{matrix}\hfill {A}^{\mathrm{R}}\hfill \\ \hfill {A}^{\mathrm{L}}\hfill \end{matrix}\right)=T\left(\begin{matrix}\hfill {A}^{{\downarrow}}\hfill \\ \hfill {A}^{{\uparrow}}\hfill \end{matrix}\right),\hspace{25.0pt}T={\mathrm{e}}^{\frac{\mathrm{i}\pi }{4}}\left(\begin{matrix}\hfill 1\hfill & \hfill -\frac{\mathrm{i}}{2}\hfill \\ \hfill -\mathrm{i}\hfill & \hfill \frac{1}{2}\hfill \end{matrix}\right),\end{equation} \tag{ 46 }$$

when the modes are real for r < r_tp and complex for r > r_tp and,

$$\begin{equation}\left(\begin{matrix}\hfill {A}^{{\uparrow}}\hfill \\ \hfill {A}^{{\downarrow}}\hfill \end{matrix}\right)=\tilde {T}\left(\begin{matrix}\hfill {A}^{\mathrm{R}}\hfill \\ \hfill {A}^{\mathrm{L}}\hfill \end{matrix}\right),\hspace{25.0pt}\tilde {T}={\mathrm{e}}^{\frac{\mathrm{i}\pi }{4}}\left(\begin{matrix}\hfill \frac{1}{2}\hfill & \hfill -\frac{\mathrm{i}}{2}\hfill \\ \hfill -i\hfill & \hfill 1\hfill \end{matrix}\right),\end{equation} \tag{ 47 }$$

when the modes are complex for r < r_tp and real for r > r_tp. Here, the propagating modes R and L are defined so that p^R > p^L. The complex modes are defined so that ↑ is the one which grows in the direction of increasing r and ↓ decays.

To relate the WKB amplitudes in the asymptotic regions of the flow, one can define an M × 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 modes,

$$\begin{equation}R\left(r\right)=\sum\limits _{l}\enspace {A}^{l}\left(r\right){\text{e}}^{\text{i}\int {p}^{l}\left(r\right)\mathrm{d}r},\end{equation} \tag{ 48 }$$

which are related to those in (40) through $R\left(r\right)=\sqrt{r}\tilde {\varphi }\left(r\right)$ (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 A, which consists of the WKB amplitudes A^l , and a row vector P, containing the WKB phases ${\text{e}}^{\text{i}\int {p}^{l}\mathrm{d}r}$. Let us 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 = P_b ⋅ A_b . Then, defining the factor,

$$\begin{equation}{\mathcal{F}}_{ab}^{l}={\left\vert \frac{{q}_{b}^{l}}{{q}_{a}^{l}}\right\vert }^{\frac{1}{2}}\enspace \mathrm{exp}\left(-\mathrm{i}{\int }_{{r}_{a}}^{{r}_{b}}{p}^{l}\mathrm{d}r\right),\end{equation} \tag{ 49 }$$

the amplitudes can be transported as,

$$\begin{equation}{\mathbf{A}}_{a}=\mathrm{diag}\left({\mathcal{F}}_{ab}^{l}\right){\mathbf{A}}_{b},\end{equation} \tag{ 50 }$$

so that the solution at r_a can be defined with respect to original phase vector through R_a = P_b ⋅ A_a .

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

$$\begin{equation}{\mathbf{A}}_{0}=\mathcal{M}{\mathbf{A}}_{\infty },\end{equation} \tag{ 51 }$$

where in defining these regions, it suffices (at the considered level of approximation) to find two locations r_0,∞ such that there are no turning points for r < r₀ or r > r_∞, since the energy content of the different modes beyond these points is then fixed. With this definition, one can use the solution at r_∞, i.e. R_∞ = P_∞ ⋅ A_∞, to deduce the same at r₀, i.e. R₀ = P_∞ ⋅ A₀ with A₀ given above. To construct $\mathcal{M}$, one performs a series of matrix multiplications using (46), (47) and (50) (see [57] for an explicit example of this). However, due to the way that T and $\tilde {T}$ 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,

$$\begin{equation}\left(\begin{matrix}\hfill {A}_{a}^{\mathrm{R}}\hfill \\ \hfill {A}_{a}^{\mathrm{L}}\hfill \end{matrix}\right)={\mathcal{N}}_{ab}\left(\begin{matrix}\hfill {A}_{b}^{\mathrm{R}}\hfill \\ \hfill {A}_{b}^{\mathrm{L}}\hfill \end{matrix}\right),\end{equation} \tag{ 52 }$$

with,

$$\begin{equation}\begin{aligned}\hfill {\mathcal{N}}_{ab}& =\enspace {\mathcal{F}}_{ab}^{{\downarrow}}\left[\begin{matrix}\hfill 1+\frac{1}{4}{f}_{ab}^{2}\hfill & \hfill \mathrm{i}\left(1-\frac{1}{4}{f}_{ab}^{2}\right)\hfill \\ \hfill -\mathrm{i}\left(1-\frac{1}{4}{f}_{ab}^{2}\right)\hfill & \hfill 1+\frac{1}{4}{f}_{ab}^{2}\hfill \end{matrix}\right],\hfill \\ \hfill {f}_{ab}& =\text{exp}\left(-{\int }_{{r}_{a}}^{{r}_{b}}\mathrm{I}\mathrm{m}\left[{p}^{{\downarrow}}\right]\mathrm{d}r\right),\hfill \end{aligned}\end{equation} \tag{ 53 }$$

where the ↓ mode is the complex solution of the dispersion relation which decays with increasing r between the two turning points (see [57] 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. Λ = 0) one can define r₀ = r_a and r_∞ = r_b and then (52) is equivalent to the full transfer matrix. In the four mode case (i.e. Λ ≠ 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.

Consider now the branches of the dispersion relation in (32) as a function of p at a given r (see e.g. figure 1). The intersection of a line of constant ω with the branches gives the two roots, and by (33), the gradient ${\partial }_{p}{\omega }_{\mathrm{D}}^{{\pm}}$ at these points gives the local group velocity of the modes. As r → ∞, the − is always in-going with ${\partial }_{p}{\omega }_{\mathrm{D}}^{+}\left({p}^{-}\right){< }0$, whereas the + mode is out-going with ${\partial }_{p}{\omega }_{\mathrm{D}}^{+}\left({p}^{+}\right){ >}0$. However, approaching the origin, both modes have ${\partial }_{p}{\omega }_{\mathrm{D}}^{+}\left({p}^{{\pm}}\right){< }0$ 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, ω) plane and one of the modes is sent to p → ±∞. Looking at the expression for p in (56), this occurs where the denominator is zero, which occurs for r = r_h ≡ 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.

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Note that the mode which diverges at r = r_h will be labelled with a bar inside the horizon, e.g. in figure 1 one has p⁺ and p⁻ for r > r_h but ${\bar{p}}^{+}$ and p⁻ for r < r_h. The reason for this is that whilst all the modes are contained in the two solutions of (56), the barred mode is really a different mode from the phase space perspective, as is evident from figure 2. In particular, this barred mode will map to a different mode besides the p^± ones in the dispersive case. This point will be clarified at the end of section 6.1.4.

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The different scattering possibilities can be classified using a similar scheme to that developed in [26]. In the non-dispersive case, there are three different scattering types which can be represented using phase space diagrams, see figure 2. The diagrams involve a schematic illustration of the real part of p^±(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. ω, m and C, the scattering outcome falls into one of the three categories. Which category it falls into depends on the size of ω 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 I⁰ scattering ³ .

The next possibility is that 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 II⁰ scattering.

The transition between types I⁰ and II⁰ 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 [25]) the condition ${\partial }_{r}{\omega }_{\mathrm{D}}^{+}=0$ is also satisfied. Using (34), the conditions for this location are,

$$\begin{equation}{\mathcal{H}}_{\mathrm{l}\mathrm{r}}=0,\hspace{25.0pt}{\partial }_{r}{\mathcal{H}}_{\mathrm{l}\mathrm{r}}=0,\hspace{25.0pt}{\partial }_{p}{\mathcal{H}}_{\mathrm{l}\mathrm{r}}=0,\end{equation} \tag{ 58 }$$

which yields a triplet (r_lr, p_lr, ω_lr). These conditions imply a relation between r_lr and p_lr,

$$\begin{equation}{p}_{\mathrm{l}\mathrm{r}}{r}_{\mathrm{l}\mathrm{r}}={B}_{{\pm}}\equiv mC{\pm}\sqrt{{m}^{2}{C}^{2}+{m}^{2}},\end{equation} \tag{ 59 }$$

where the + sign is for the upper branch and − sign for the lower one. Note that this relation also holds for Λ ≠ 0. The light-ring is given by,

$$\begin{equation}{r}_{\mathrm{l}\mathrm{r}}^{{\pm}}={\pm}{\left({B}_{{\pm}}^{2}+{m}^{2}\right)}^{\frac{1}{2}}/{B}_{{\pm}},\end{equation} \tag{ 60 }$$

and the light-ring frequency by,

$$\begin{equation}{\omega }_{\mathrm{l}\mathrm{r}}^{{\pm}}=mC/{r}_{\mathrm{l}\mathrm{r}}^{2}+\left(1-1/{r}_{\mathrm{l}\mathrm{r}}^{2}\right){B}_{{\pm}}.\end{equation} \tag{ 61 }$$

Since I consider ω > 0 modes, only the light-ring on the upper branch is required and I will therefore set ${r}_{\mathrm{l}\mathrm{r}}={r}_{\mathrm{l}\mathrm{r}}^{+}$ and ${\omega }_{\mathrm{l}\mathrm{r}}={\omega }_{\mathrm{l}\mathrm{r}}^{+}$ from here on. Note that r_lr is independent of m whereas ω_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 figure 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 ${\omega }_{\mathrm{D}}^{+}$ or ${\omega }_{\mathrm{D}}^{-}$, 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,

$$\begin{equation}{\omega }_{\star }=mC/{r}_{\mathrm{h}}^{2}\equiv m{{\Omega}}_{\mathrm{h}}^{\text{rot}}.\end{equation} \tag{ 62 }$$

Hence, for ω > ω_⋆, the modes will be on the upper branch just outside of r_h, whereas for ω < ω_⋆ they will be on the lower branch. Note that ω_⋆ < ω_lr for all m and C. This ω_⋆ 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 III⁰.

There is another convenient way to understand the different scattering possibilities by plotting the evolution of the extrema of ${\omega }_{\mathrm{D}}^{{\pm}}$ with r. To do this, one solves ${\partial }_{p}{\omega }_{\mathrm{D}}^{{\pm}}=0$ 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 ω on the extrema is given by,

$$\begin{equation}{\omega }^{{\pm}}\left(r\right)\equiv {\omega }_{\mathrm{D}}^{{\pm}}\left(r,{\pm}{p}_{\mathrm{e}\mathrm{x}}\left(r\right)\right)=mC/{r}^{2}{\pm}\sqrt{\left(1-1/{r}^{2}\right){m}^{2}/{r}^{2}}.\end{equation} \tag{ 63 }$$

The turning points can be understood as the intersection of these curves with a line of ω = const. Then ω_lr is simply the extremum of ω₊ (for ω > 0) in the radial direction and ω_⋆ = ω^±(r_h). These curves are illustrated for a particular value of m and C in figure 3. Note finally that since the function V defined in (57) can be written,

$$\begin{equation}V=-\left(\omega -{\omega }^{+}\right)\left(\omega -{\omega }^{-}\right),\end{equation} \tag{ 64 }$$

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

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Since the radial equation of motion is second order in spatial derivatives, there are two independent solutions j = 1, 2. Following [47], and using the diagrams in figure 2, these are defined by the asymptotics,

$$\begin{equation}\begin{aligned}\hfill {R}_{1}\sim & \enspace \frac{1}{2\pi \vert {q}_{\infty }^{-}{\vert }^{\frac{1}{2}}}{\times}\begin{cases}{\text{e}}^{\text{i}{\int }^{{r}_{\infty }}{p}^{-}\mathrm{d}r}+\mathcal{R}{\text{e}}^{\text{i}{\int }^{{r}_{\infty }}{p}^{+}\mathrm{d}r},\hfill & r\to {r}_{\infty },\quad \hfill \\ \mathcal{T}{\text{e}}^{\text{i}{\int }^{{r}_{0}}{p}^{\mp }\mathrm{d}r},\hfill & r\to {r}_{0},\quad \hfill \end{cases}\hfill \\ \hfill {R}_{2}\sim & \enspace \frac{1}{2\pi \vert {q}_{0}^{{\pm}}{\vert }^{\frac{1}{2}}}{\times}\begin{cases}\mathcal{U}{\text{e}}^{\text{i}{\int }^{{r}_{\infty }}{p}^{+}\mathrm{d}r},\hfill & r\to {r}_{\infty },\quad \hfill \\ {\text{e}}^{\text{i}{\int }^{{r}_{0}}{p}^{{\pm}}\mathrm{d}r}+\mathcal{V}{\text{e}}^{\text{i}{\int }^{{r}_{0}}{p}^{\mp }\mathrm{d}r},\hfill & r\to {r}_{0},\quad \hfill \end{cases}\hfill \end{aligned}\end{equation} \tag{ 65 }$$

where r₀ is understood as a location just outside the horizon and the upper (lower) sign is taken for types I⁰ and II⁰ (type III⁰). Here, $\mathcal{R},\mathcal{T},\mathcal{U},\mathcal{V}$ are scattering coefficients and the factor of 2π is there so that the incident part of the mode is normalised in the inner-product (25) (see appendix 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,

$$\begin{equation}\vert {q}_{\infty }^{-}\vert {\left(2\pi \right)}^{2}\enspace W\left[{\tilde {\varphi }}_{1},{\tilde {\varphi }}_{1}\right]={q}_{\infty }^{-}+{q}_{\infty }^{+}\vert \mathcal{R}{\vert }^{2}={q}_{0}^{\mp }\vert \mathcal{T}{\vert }^{2},\end{equation} \tag{ 66a }$$

$$\begin{equation}\vert {q}_{0}^{{\pm}}\vert {\left(2\pi \right)}^{2}\enspace W\left[{\tilde {\varphi }}_{2},{\tilde {\varphi }}_{2}\right]={q}_{\infty }^{+}\vert \mathcal{U}{\vert }^{2}={q}_{0}^{{\pm}}+{q}_{0}^{\mp }\vert \mathcal{V}{\vert }^{2},\end{equation} \tag{ 66b }$$

$$\begin{equation}\vert {q}_{\infty }^{-}{q}_{0}^{{\pm}}{\vert }^{\frac{1}{2}}{\left(2\pi \right)}^{2}\enspace W\left[{\tilde {\varphi }}_{1},{\tilde {\varphi }}_{2}\right]={q}_{\infty }^{+}\mathcal{R}{\mathcal{U}}^{{\ast}}={q}_{0}^{\mp }\mathcal{T}{\mathcal{V}}^{{\ast}},\end{equation} \tag{ 66c }$$

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

Consider now the classical scattering of an in-coming wave with the vortex (i.e. the R₁ solution). Superradiance occurs when the reflected wave carries away more energy than the incident wave had coming in, i.e. ${q}_{\infty }^{+}\vert \mathcal{R}{\vert }^{2}{ >}{q}_{\infty }^{-}$, which by (66a) implies that ${q}_{0}^{\mp }{< }0$ is a necessary (and sufficient) condition for superradiance. Using $q={f}^{-1}{ \overrightarrow {\mathbf{e}}}_{r}\cdot {\boldsymbol{v}}_{\mathrm{g}}{\Omega}$ and realising that in R₁ the solution on the horizon is always in-going, superradiance will occur provided Ω₀ < 0. This is true for the solution which tunnels to the lower branch of the dispersion relation, i.e. the one in type III⁰ scattering.

The condition for superradiance assumes a more familiar form if one sets r_∞ to be true spatial infinity and takes the limit r₀ → r_h from above. Then ${q}_{\infty }^{+}=-{q}_{\infty }^{-}=\omega$ and ${q}_{0}^{\mp }=-{\tilde {\omega }}_{\mathrm{h}}$, and (66a) becomes,

$$\begin{equation}\vert \mathcal{R}{\vert }^{2}+\frac{{\tilde {\omega }}_{\mathrm{h}}}{\omega }\vert \mathcal{T}{\vert }^{2}=1,\end{equation} \tag{ 67 }$$

which is the usual relation between scattering coefficients from black hole physics. Thus amplification occurs for ${\tilde {\omega }}_{\mathrm{h}}{< }0$, as claimed in (1), which corresponds to type III⁰ scattering.

To find an expression for the reflection coefficient, I will again exploit the freedom to move the points r₀ and r_∞ (although identical results are found when these locations are fixed [26]). In type I⁰ scattering, there are no real turning points and thus, at the considered level of approximation, one has $\vert \mathcal{R}\vert =0$. For the other two cases, choose r₀ to sit just inside r₁ in figure 2, and r_∞ just outside r₂. Applying the formula in (52) and inserting the amplitudes for R₁, the reflection coefficient is given by,

$$\begin{equation}\vert \mathcal{R}\vert ={\left(\frac{1-{f}_{12}^{2}/4}{1+{f}_{12}^{2}/4}\right)}^{\mathrm{sgn}\left({\tilde {\omega }}_{\mathrm{h}}\right)},\end{equation} \tag{ 68 }$$

which as expected satisfies $\vert \mathcal{R}\vert {< }1$ for type II⁰ and $\vert \mathcal{R}\vert { >}1$ for type III⁰. This will be plotted later on in section 6.2 along with the dispersive solutions.

The scattering possibilities for Λ > 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 provide a parameter space plot later on to illustrate the parameter ranges associated with the different types of scattering. This parameter space will be 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_± from (59), the location of the light-ring (on ${\omega }_{\mathrm{D}}^{+}$) is,

$$\begin{equation}{r}_{\mathrm{l}\mathrm{r}}=\sqrt{\frac{{B}_{+}^{2}+{m}^{2}}{2{B}_{+}^{2}}}{\left(1+\sqrt{1-4{\Lambda}{B}_{+}^{2}}\right)}^{\frac{1}{2}}{\left(1-4{\Lambda}{B}_{+}^{2}\right)}^{\frac{1}{4}}.\end{equation} \tag{ 70 }$$

The light-ring momentum and frequency are then immediately given by p_lr = B₊/r_lr and ${\omega }_{\mathrm{l}\mathrm{r}}={\omega }_{\mathrm{D}}^{+}\left({r}_{\mathrm{l}\mathrm{r}},{p}_{\mathrm{l}\mathrm{r}}\right)$.

For the following discussion, it is useful to visualise the light-rings as the extrema of the ω^± curves, possibilities of which are displayed in figure 6. These curves are defined in the same way as in the non-dispersive case, i.e. the value of ω at the extrema of ${\omega }_{\mathrm{D}}^{{\pm}}$ in the p direction. I now define the following critical parameters,

$$\begin{equation}{C}_{0}=\left\vert \frac{1-4{\Lambda}{m}^{2}}{4{{\Lambda}}^{\frac{1}{2}}m}\right\vert ,\hspace{25.0pt}{{\Lambda}}_{c}=1/4{m}^{2},\end{equation} \tag{ 71 }$$

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 {ω, m} → {−ω, −m} of the dispersion relation, they will play a similarly important role in the scattering of m < 0 modes.

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Firstly, for Λ < Λ_c , the value of C relative to C₀ determines when there is a light-ring on the upper branch. For C < C₀, the light-ring is real and positive; for C = C₀, r_lr = 0 and for C > C₀, r_lr is complex, i.e. there is no extremum on ω⁺. 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 Λ > Λ_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 C₀ now determines whether there is an extremum on ω⁻; in particular, it is absent for C < C₀. Although we are not interested in the light-ring frequency on the lower branch (since only the ω > 0 modes are considered) its 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 m²/r² term in ${\omega }_{\mathrm{D}}^{{\pm}}$ will initially cause the two branches to separate. However, if ${\partial }_{r}{\omega }_{\mathrm{D}}^{-}=0$ is nowhere satisfied, the lower branch will continue moving towards lower values of ω approaching r = 0, thereby making it impossible for positive frequency modes to reach this branch. Hence, superradiance (which relies on tunnelling to the ${\omega }_{\mathrm{D}}^{-}$ branch) is impossible in this regime. However, above C₀ superradiance will still occur. The implication of this is that above Λ_c , each m > 0 mode has a minimum possible rotation below which it cannot superradiate. In the language of black hole physics, this is the same as saying that when dispersion is strong, co-rotating modes satisfying $2{{\Lambda}}^{\frac{1}{2}}m{ >}C+\sqrt{{C}^{2}+1}$ do not exhibit an ergosphere, which is the mechanism that allows for black hole superradiance.

Note finally that all of the properties discussed above can be deduced using the example ω^± curves given in figure 6.

6.1.2. Threshold frequency

As seen in the non-dispersive case, the onset of superradiance is signaled by the tunnelling of the + and − modes from ${\omega }_{\mathrm{D}}^{+}$ at large r to ${\omega }_{\mathrm{D}}^{-}$ at small r. Let us start by assuming that there exists a threshold frequency ω_⋆ that governs when this occurs. For Λ = 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 two pairs of propagating modes to exist simultaneously on the upper and lower branches of the dispersion relation in the core (as demonstrated by figure 4). Hence, in the dispersive case, one must make sure that it is indeed the + and − modes which appear on ${\omega }_{\mathrm{D}}^{-}$ and not the t and b modes. For example, figure 5 shows a situation where the t and b modes are on ${\omega }_{\mathrm{D}}^{-}$ approaching the origin. If instead the + and − modes tunnel to ${\omega }_{\mathrm{D}}^{-}$, the mode trajectories in the evanescent region should connect to opposite branches of the dispersion relation at small r compared to this example. What is needed is a criterion which distinguishes these two scenarios.

To find the necessary condition, consider the following argument. Using only the dispersion relation, one can find an example where p^± tunnel to ${\omega }_{\mathrm{D}}^{-}$ and one where they do not simply by testing different values of ω, m, C, Λ. The trajectories of the four p^l (r) through the complex p-plane in these examples would look like those shown in figure 7. The difference between these two cases is that p^± 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 ω = ω_⋆, the modes undergo a head on collision at the point p_⋆ (and also at the point given by its complex conjugate). The task at hand then is to find the expression for these points.

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

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

Using the fact that p_⋆ is a saddle point, one has the condition that all x, y derivatives of U, W must vanish, but since $\mathcal{H}\left(p\right)$ 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,

$$\begin{equation}{r}_{\star }={\left(1+2{{\Lambda}}^{\frac{1}{2}}m\right)}^{\frac{1}{2}},\hspace{25.0pt}{\omega }_{\star }=mC/{r}_{\star }^{2},\hspace{25.0pt}{p}_{\star }=\mathrm{i}{\left({m}^{2}+\left({r}_{\star }^{2}-1\right)/2{\Lambda}\right)}^{\frac{1}{2}}/{r}_{\star }.\end{equation} \tag{ 73 }$$

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

To summarize, the quantities in (73) give the location of a complex saddle point in phase space and the frequency for a mode to be on this saddle point. For ω > ω_⋆, the + and − modes coming from infinity will recoil from this saddle point toward increasing real p and connect back onto the upper branch of the dispersion relation, whereas for ω < ω_⋆ they recoil toward decreasing p and connect to the lower branch.

6.1.3. Extremum equality

The equality of the ω value of the extrema of ${\omega }_{\mathrm{D}}^{{\pm}}$ defines another important frequency ω_eq. Using the curves ω^±, the condition for this becomes ω_eq = ω⁺ = ω⁻ which, along with the turning point criteria, gives,

$$\begin{equation}{r}_{\mathrm{e}\mathrm{q}}={\left(1-2{{\Lambda}}^{\frac{1}{2}}m\right)}^{\frac{1}{2}},\hspace{25.0pt}{\omega }_{\mathrm{e}\mathrm{q}}=mC/{r}_{\mathrm{e}\mathrm{q}}^{2}.\end{equation} \tag{ 74 }$$

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 ω > ω_eq, the turning point on the upper branch occurs at larger r than the one on the lower branch and vice versa for ω < ω_eq. This introduces some sub-classification criteria for two of the scattering categories defined in the next section.

It is important to note that naively looking at the top left panel of figure 6 gives the impression that ω < ω_eq is the superradiance condition since, tracking the modes from large r inward in this plot, one would assume that the + and − modes appear on ${\omega }_{\mathrm{D}}^{-}$ if the red curve is crossed before the black curve. This is not the case since it is actually the t and b modes that appear on ${\omega }_{\mathrm{D}}^{-}$ when crossing the red curve for ω > ω_⋆. As explained in figure 7, the superradiance criterion for Λ > 0 has to do with mode behaviour at complex values of p and, as such, is not evident from inspecting the ω^± curves.

6.1.4. Classification

In figure 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: r₁ and r₂ are the inner and outer turning points for the +, − modes, whereas r₃ is the only allowed turning point for the t, b modes.

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These diagrams represent the type of scattering that occurs in different regions of the (C, ω) parameter space, which is depicted in figure 9. The parameter space has a distinctively different structure depending on the sign of m, as well as the size of Λ relative to Λ_c . To understand this structure, one can again consult figure 6 and look for the intersection of an ω = const. line with the ω^± curves to see where the turning points arise. Which modes are involved at the turning points is determined by the size of ω relative to the important frequencies displayed. This information produces precisely the scattering schematics of figure 8 and where they occur in figure 9.

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A couple of noteworthy points concerning the diagrams in figure 8. Firstly, there is a sub-classification for the type VII⁺ and VIII⁺ diagrams depending on the relative locations of r_1,2,3. Specifically,

• Type ${\mathrm{V}\mathrm{I}\mathrm{I}}_{\text{a}}^{+}$: r₃ < r₂,
• Type ${\mathrm{V}\mathrm{I}\mathrm{I}}_{\text{b}}^{+}$: r₂ < r₃,
• Type ${\mathrm{V}\mathrm{I}\mathrm{I}\mathrm{I}}_{\text{a}}^{+}$: r₃ < r₁ < r₂,
• Type ${\mathrm{V}\mathrm{I}\mathrm{I}\mathrm{I}}_{\text{b}}^{+}$: r₁ < r₃ < r₂,
• Type ${\mathrm{V}\mathrm{I}\mathrm{I}\mathrm{I}}_{\text{c}}^{+}$: r₁ < r₂ < r₃.

Note that this does not alter the computation of the scattering coefficients since the relative location of r₃ to r_1,2 does not play a role in scattering. However, the distinction between these cases would be important, for example, if one wanted to predict the spatial form of the wave, since the presence of four propagating modes would lead to noticeable interference.

Secondly, the diagrams give the impression that the modes with larger Re[p] appear at the top and smaller 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 VIII⁺ scattering it happens that Re[p⁺] > Re[p^t,b ] > Re[p⁻] as r → ∞, but Re[p⁺] > Re[p⁻] > Re[p^t,b ] as r → 0 (compare figure 5 to the type 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 figure 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 III⁺ and IV⁺ technically represent the same class of scattering. The reason that they have been left as separate cases is the following: in type III⁺, the + and − modes try to tunnel to ${\omega }_{\mathrm{D}}^{+}$ approaching the origin, but do not make it since the branch recedes from the modes due to dispersion. In type IV⁺ they try instead to tunnel to ${\omega }_{\mathrm{D}}^{-}$. In this sense, type 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.

Finally, it is interesting to look at how the scattering outcomes in the non-dispersive case transform into their dispersive counterparts as the parameter Λ is increased from zero. Looking back at figure 2, the effect of switching on Λ is to take the two divergent ends of the out-going mode trajectory at r = r_h and attach both ends to the line r = 0 in those plots. In doing so, one obtains the mapping type I⁰ → II⁺, II⁰ → VIII⁺ and III⁰ → IX⁺. In the first two (non-superradiant) cases, the p⁺ mode outside the horizon is extended all the way into the vortex core, whereas the ${\bar{p}}^{+}$ mode is converted into the p^t,b modes. The same happens in the last (superradiant) case except with the p⁻ mode extended and the ${\bar{p}}^{-}$ mode converted.

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 figure 8, one could in principle 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 r₀ and r_∞ in the diagrams in figure 8. These classes are outlined in table 1. Note that the p^± modes in 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.

Table 1. The diagrams in figure 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. ${p}^{l}\in \mathbb{C}$.

Class	r0 {+, −, t, b}	r∞ {+, −, t, b}	Types
A	$\left\{\mathbb{R,R,C,C}\right\}$	$\left\{\mathbb{R,R,C,C}\right\}$	I+, V+, VI+
B	$\left\{\mathbb{R,R,R,R}\right\}$	$\left\{\mathbb{R,R,C,C}\right\}$	II+, VIII+, IX+
C	$\left\{\mathbb{C,C,C,C}\right\}$	$\left\{\mathbb{R,R,C,C}\right\}$	III+, IV+
D	$\left\{\mathbb{C,C,R,R}\right\}$	$\left\{\mathbb{R,R,C,C}\right\}$	VII+

Next, it is easy to see that in each diagram, the +, − modes do not interact with the t, b modes (this contrasts what happens when dispersion is subluminal and all four modes can interact [26, 57]). 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 Ai and 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 Ai and 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 I⁺, II⁺, V⁺ and VIII⁺ and the lower sign for types VI⁺ and 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 VI⁺ and IX⁺. In fact, there is a simpler way to see this directly from figure 8, since any diagram which contains an interaction of the form,

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 ${\omega }_{\mathrm{D}}^{-}$ 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 I⁺ and II⁺, $\mathcal{R}$ vanishes whereas for types III⁺, IV⁺ and VII⁺ it is simply $\vert \mathcal{R}\vert =1$. In the remaining cases, $\vert \mathcal{R}\vert$ is obtained from (52) as,

$$\begin{equation}\vert \mathcal{R}\vert ={\left(\frac{1-{f}_{12}^{2}/4}{1+{f}_{12}^{2}/4}\right)}^{{\pm}1},\end{equation} \tag{ 76 }$$

with the upper sign taken for types V⁺ and VIII⁺ and the lower sign for types VI⁺ and 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.

The behaviour of $\vert \mathcal{R}\vert$ can be easily read off from figure 9 using the knowledge gained from the diagrams in figure 8. For the positive m's (which are the ones which can superradiate) there are only three distinct behaviours. For Λ < Λ_c and C < C₀, the modes are amplified at low ω and absorbed at high ω, much like the non-dispersive case. For C > C₀ for any Λ, the modes are amplified at small ω and completely reflected for high ω. Finally, for Λ > Λ_c and C < C₀, the modes are completely reflected for all ω. Some examples of this behaviour are given in figure 10. It is clear that for superradiant modes, dispersion needs to be strong before any significant change arises to the amount of amplication. By contrast, changes to the bandwidth of superradiant modes become evident even for small Λ.

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Note that the WKB approximation has a tendency to over-estimate $\vert \mathcal{R}\vert$ 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 ${m}^{2}/{r}^{2}\to \left({m}^{2}-1/4+5{v}_{r}^{2}/4\right)/{r}^{2}$ [18]. When this is used in the approximate formula for $\vert \mathcal{R}\vert$ (which amounts to applying the WKB approximation in the radial direction only), the exponent appearing in f₁₂ 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.

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. Λ < Λ_c , superradiance proceeds for all positive frequencies below the threshold frequency ω_⋆ 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 → ∞. 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. Λ > Λ_c , there is another important difference. In this regime, the ω ∈ [0, ω_⋆] modes can only superradiate above a critical rotation parameter given by C₀(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 C₀ is an increasing function of m in this regime, only the small m modes superradiate at low C. In particular, if Λ > 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 Λ < 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 Λ > 1/4 there is a minimum allowed rotation given by ${C}_{\mathrm{min}}=\left(4{\Lambda}-1\right)/4{{\Lambda}}^{\frac{1}{2}}$, 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 its vacuum fluctuations.

The quantisation procedure given here follows that of [47]. The conjugate momentum to ϕ is obtained from (12) as,

$$\begin{equation}\pi =\frac{\partial \mathcal{L}}{\partial \dot {\phi }}=-\eta .\end{equation} \tag{ 77 }$$

Canonical quantisation then proceeds by replacing complex conjugation by Hermitian conjugation, promoting the fields ϕ and η to operators and imposing the equal time commutation relations,

$$\begin{equation}\begin{aligned}\hfill & \left[\hat{\phi }\left(x\right),\hat{\phi }\left(y\right)\right]=\left[\hat{\pi }\left(x\right),\hat{\pi }\left(y\right)\right]=0,\hfill \\ \hfill & \left[\hat{\phi }\left(x\right),\hat{\pi }\left(y\right)\right]=\mathrm{i}{\delta }^{\left(2\right)}\left(x-y\right).\hfill \end{aligned}\end{equation} \tag{ 78 }$$

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

$$\begin{equation}\left[{\hat{a}}_{{\lambda }_{1}},{\hat{a}}_{{\lambda }_{2}}^{{\dagger}}\right]={\delta }_{{\lambda }_{1}{\lambda }_{2}}.\end{equation} \tag{ 79 }$$

The normal mode expansion of the fields then reads,

$$\begin{equation}\hat{\phi }=\sum\limits _{\lambda }\left({\hat{a}}_{\lambda }{\varphi }_{\lambda }+{\hat{a}}_{\lambda }^{{\dagger}}{\varphi }_{\lambda }^{{\ast}}\right),\hspace{25.0pt}\hat{\eta }=\sum\limits _{\lambda }\left({\hat{a}}_{\lambda }{n}_{\lambda }+{\hat{a}}_{\lambda }^{{\dagger}}{n}_{\lambda }^{{\ast}}\right).\end{equation} \tag{ 80 }$$

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

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,

$$\begin{equation}{\hat{\mathfrak{S}}}_{E}^{r}=\frac{1}{2}\left(\frac{1}{2}{v}_{r}\left\{\hat{\phi },{\partial }_{t}\hat{\eta }\right\}-\frac{1}{2}{v}_{r}\left\{\hat{\eta },{\partial }_{t}\hat{\phi }\right\}-\left\{{\partial }_{r}\hat{\phi },{\partial }_{t}\hat{\phi }\right\}-{\Lambda}\left\{{\partial }_{r}\hat{\eta },{\partial }_{t}\hat{\eta }\right\}\right),\end{equation} \tag{ 81 }$$

and,

$$\begin{equation}{\hat{\mathfrak{S}}}_{L}^{r}=\frac{1}{2}\left(\frac{1}{2}{v}_{r}\left\{\hat{\phi },{\partial }_{\theta }\hat{\eta }\right\}-\frac{1}{2}{v}_{r}\left\{\hat{\eta },{\partial }_{\theta }\hat{\phi }\right\}-\left\{{\partial }_{r}\hat{\phi },{\partial }_{\theta }\hat{\phi }\right\}-{\Lambda}\left\{{\partial }_{r}\hat{\eta },{\partial }_{\theta }\hat{\eta }\right\}\right),\end{equation} \tag{ 82 }$$

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 equation (81) are,

$$\begin{equation}\begin{aligned}\hfill \langle 0\vert \left\{\hat{\phi },{\partial }_{t}\hat{\eta }\right\}\vert 0\rangle & =-\mathrm{i}\sum\limits _{\lambda }\enspace \omega \left[{n}_{\lambda }{\varphi }_{\lambda }^{{\ast}}-{\varphi }_{\lambda }{n}_{\lambda }^{{\ast}}\right],\hfill \\ \hfill \langle 0\vert \left\{\hat{\eta },{\partial }_{t}\hat{\phi }\right\}\vert 0\rangle & =-\mathrm{i}\sum\limits _{\lambda }\enspace \omega \left[{\varphi }_{\lambda }{n}_{\lambda }^{{\ast}}-{n}_{\lambda }{\varphi }_{\lambda }^{{\ast}}\right],\hfill \\ \hfill \langle 0\vert \left\{{\partial }_{r}\hat{\phi },{\partial }_{t}\hat{\phi }\right\}\vert 0\rangle & =-\mathrm{i}\sum\limits _{\lambda }\enspace \omega \left[{\varphi }_{\lambda }{\partial }_{r}{\varphi }_{\lambda }^{{\ast}}-\left({\partial }_{r}{\varphi }_{\lambda }\right){\varphi }_{\lambda }^{{\ast}}\right],\hfill \\ \hfill \langle 0\vert \left\{{\partial }_{r}\hat{\eta },{\partial }_{t}\hat{\eta }\right\}\vert 0\rangle & =-\mathrm{i}\sum\limits _{\lambda }\enspace \omega \left[{n}_{\lambda }{\partial }_{r}{n}_{\lambda }^{{\ast}}-\left({\partial }_{r}{n}_{\lambda }\right){n}_{\lambda }^{{\ast}}\right].\hfill \end{aligned}\end{equation} \tag{ 83 }$$

The anti-commutators in equation (82) are given by similar relations except there, the ω 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 VEVs 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,

$$\begin{equation}{\int }_{0}^{2\pi }\mathrm{d}\theta \enspace r\langle 0\vert {\hat{\mathfrak{S}}}_{E}^{r}\vert 0\rangle =2\pi \sum\limits _{\lambda }\enspace \omega W\left[{\varphi }_{\lambda },{\varphi }_{\lambda }\right],\end{equation} \tag{ 84 }$$

and similarly for the angular momentum current, where the integral has been evaluated by noticing that W[φ_λ , φ_λ ] is independent of θ.

To perform the normal mode sum, one can make a few simplifying observations. Firstly, notice that $W\left[{\tilde {\varphi }}_{j},{\tilde {\varphi }}_{j}\right]$ 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 $W\left[{\tilde {\varphi }}_{1},{\tilde {\varphi }}_{1}\right]$ and $W\left[{\tilde {\varphi }}_{2},{\tilde {\varphi }}_{2}\right]$ evaluated at r_∞ directly from (66), one arrives at,

$$\begin{align}\hfill 2\pi \sum\limits _{j}\enspace W\left[{\tilde {\varphi }}_{j},{\tilde {\varphi }}_{j}\right]=& \enspace \frac{1}{2\pi }\left[\frac{{q}_{\infty }^{-}+{q}_{\infty }^{+}\vert \mathcal{R}{\vert }^{2}}{\vert {q}_{\infty }^{-}\vert }+\frac{{q}_{\infty }^{+}\vert \mathcal{U}{\vert }^{2}}{\vert {q}_{0}^{{\pm}}\vert }\right],\hfill \\ \hfill =& \enspace \frac{\vert \mathcal{R}{\vert }^{2}-1}{2\pi }\left[1+\mathrm{sgn}\left({q}_{0}^{{\pm}}/{q}_{\infty }^{-}\right)\right],\hfill \end{align} \tag{ 85 }$$

where in the second line, I have used ${q}_{\infty }^{+}{q}_{\infty }^{-}\vert \mathcal{U}{\vert }^{2}={q}_{0}^{{\pm}}\left({q}_{\infty }^{-}+{q}_{\infty }^{+}\vert \mathcal{R}{\vert }^{2}\right)$, which is obtained by combining the three relations in (66), and also the fact that the reflection coefficient is evaluated just outside r₂ so ${q}_{\infty }^{+}=-{q}_{\infty }^{-}{ >}0$. Recalling that the upper sign should be taken for scattering types I⁰, II⁰, I⁺, II⁺, V⁺ and VIII⁺ and the lower sign for types III⁰, VI⁺ and IX⁺, one finds $\mathrm{sgn}\left({q}_{0}^{{\pm}}/{q}_{\infty }^{-}\right)=\mp 1$. 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,

$$\begin{equation}\left(\begin{matrix}\hfill \dot {E}\hfill \\ \hfill \dot {L}\hfill \end{matrix}\right)=-\frac{1}{\pi }\sum\limits _{m}{\int }_{SR}\mathrm{d}\omega \enspace \left(\begin{matrix}\hfill \omega \hfill \\ \hfill m\hfill \end{matrix}\right)\left(\vert \mathcal{R}{\vert }^{2}-1\right),\end{equation} \tag{ 86 }$$

in agreement with [47]. Here SR indicates that the ω integral is performed only over superradiant frequencies, and the expression for $\vert \mathcal{R}\vert$ is given by that in (76) with the lower sign. ⁴

To evaluate $\dot {E}$ and $\dot {L}$, the reflection coefficients are computed using the WKB formula in equation (76), as well as from exact simulation of the wave equation (see e.g. appendix A of [26]). Figure 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 ∼ |C/D| mode as shown in figure 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 [47]. 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,

$$\begin{equation}\dot {E}\approx -0.05{C}^{5},\hspace{25.0pt}\dot {L}\approx -0.07{C}^{4},\end{equation} \tag{ 87 }$$

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

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When Λ > 0, the dependence of $\dot {E}$ and $\dot {L}$ on C is qualitatively similar to the non-dispersive case. Figure 13 demonstrates that as the parameter Λ 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 Λ increases and (2) modes with m larger than m_max (obtained by inverting the expression for C₀ in (71)) will not superradiate. For fixed C, m_max decreases with increasing Λ until eventually there are no superradiant modes left in the system, and $\dot {E}$ and $\dot {L}$ go to zero. Inverting this statement, for Λ fixed (and larger than 1/4) there is a value C_min below which superradiance will not occur for any m. C_min goes to zero at Λ = 1/4, which means that for Λ < 1/4, superradiance will occur at least for some m modes for all C.

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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,

$$\begin{equation}D=\hslash d/M,\end{equation} \tag{ 88 }$$

where (unlike ℓ) 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 = ℓ/d, which permits an easy comparison of the drain strength to the winding number. The second is that d = r_h/ξ gives the ratio of the 'would be' horizon (i.e. D/c) to the healing length $\xi =\hslash /\sqrt{M\mu }\equiv 2{{\Lambda}}^{\frac{1}{2}}/c$, which characterises the scale over which the condensate heals back to its bulk value around boundaries. In particular ξ gives a characteristic length on which the density varies in the vortex core.

The dimensionless dispersion parameter discussed throughout this work is now really Λ/D² = 1/4d². 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 Λ/D² > 1/4. In terms of ℓ and d, this really means that when the system is weakly draining, i.e. d < 1, superradiance can only spin down the system whilst ℓ > ℓ_min = (1 − d²)/2. However, since ℓ can only take on integer values, the minimum rotation for a spinning vortex will be ℓ = 1, which is always greater than ℓ_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 ${\mathcal{S}}_{\text{GPE}}$, the conserved current for rotations in θ gives the angular momentum of the system,

$$\begin{equation}L=\hslash \ell N.\end{equation} \tag{ 89 }$$

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 ℓ. If it is the former, then the rotation C remains fixed during the evolution and one would expect the density to change, whereas if it is the latter then C will vary. To determine which of these is the dominant effect, one would need to solve the full backreaction equations. Furthermore, since ℓ 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.

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 ≪ 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 Λ = σh/ρ_f, with σ the surface tension of the fluid and ρ_f its density. In the experiment of [2], D and h were of the order of 10⁻³ and 10⁻² respectively. Taking the values of σ and ρ_f for water, the dimensionless dispersive parameter is ${\Lambda}/{D}^{2}\sim \mathcal{O}\left(1\right)$. At this strength of dispersion, the m = 1, 2 modes would cease to superradiate around $C/D\sim \mathcal{O}\left(1\right)$. 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 [26] 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].