# Amplification of electromagnetic waves by a rotating body

M.C. Braidotti  School of Physics & Astronomy, University of Glasgow, G12 8QQ
Glasgow, UK  A. Vinante  Istituto di Fotonica e Nanotecnologie - CNR and
Fondazione Bruno Kessler, I-38123 Povo, Trento, Italy  M. Cromb  School of
Physics and Astronomy, University of Southampton, SO17 1BJ, Southampton, UK.
A. Sandakumar  School of Physics and Astronomy, University of Southampton,
SO17 1BJ, Southampton, UK.  D. Faccio  School of Physics & Astronomy,
University of Glasgow, G12 8QQ Glasgow, UK  H. Ulbricht
<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Physics and
Astronomy, University of Southampton, SO17 1BJ, Southampton, UK.

###### Abstract

In 1971, Zel’dovich predicted the amplification of electromagnetic (EM) waves
scattered by a rotating metallic cylinder, gaining mechanical rotational
energy from the body.Since then, this phenomenon has been believed to be
unobservable with electromagnetic fields due to technological difficulties in
meeting the condition of amplification, that is, the cylinder must rotate
faster than the frequency of the incoming radiation. Here, we show that this
key piece of fundamental physics has been hiding in plain sight for the past
60 years in the physics of induction generators. We measure the amplification
of an electromagnetic field, generated by a toroid LC-circuit, scattered by an
aluminium cylinder spinning in the toroid gap. We show that when the
Zel’dovich condition is met, the resistance induced by the cylinder becomes
negative implying amplification of the incoming EM waves. These results reveal
the connection between the concept of induction generators and the physics of
this fundamental effect that was believed to be unobservable, and hence open
new prospects towards testing the Zel’dovich mechanism in the quantum regime,
as well as related quantum friction effects.

_Introduction_ – Electromagnetic (EM) wave amplification from rotating media
was predicted by Yakov Zel’dovich in 1971, when he showed that an incoming
beam co-rotating with a rotating absorber, such as a metallic cylinder, would
be reflected and amplified in the scattering process under the condition that
the cylinder was spinning so fast to see the incoming frequency of the
radiation to be negative Zeldovich (1971, 1972); Zeldovich  _et al._ (1986).
Zel’dovich’s prediction also showed that this mechanism would be able to
amplify any quantum fluctuations of the EM field in the vicinity of the
rotating body, therefore leading to a spontaneous emission of radiation from
the absorbing surface. The vacuum virtual photons would be converted into real
photons, draining energy from the mechanical energy of the cylinder, which
would slow down its rotation Bekenstein and Schiffer (1998); Maghrebi  _et
al._ (2012).
In Zel’dovich’s original postulation, EM waves carrying orbital angular
momentum (OAM) $\ell$ focus radially onto a spinning metallic (absorbing)
cylinder. The cylinder rotation $\Omega$ must be sufficiently high so as to
induce a rotational Doppler effect that shifts the EM frequency, $\omega$, to
negative values:

$\omega-\ell\Omega<0.$ (1)

When this condition is satisfied, the absorption coefficient changes sign and
the rotating medium loses part of its rotational energy to the outgoing waves,
which are amplified. There have been many practical proposals to verify
Zel’dovich’s predictions Bekenstein and Schiffer (1998); Faccio and Wright
(2017); Gooding  _et al._ (2020); Moussa and Alù (2022), relying e.g. on use
of a very large OAM $\ell=10,000$ or optically levitated particles spinning at
GHz rates Gooding  _et al._ (2020), restricting the size of the system to the
nano-scale Reimann  _et al._ (2018); Ahn  _et al._ (2020) or on synthetic
approaches Moussa and Alù (2022). Despite the large quantity of proposals,
technological challenges in meeting the amplification condition have prevented
its verification with EM waves, making everyone believe that this long-
foreseen effect was inaccessible in the near future. The fastest rotation
achievable by standard motors is of the order of $10$ kHz cel , and a record
of $667$ kHz is reported for a millimeter-sized magnetically levitated sphere
Schuck  _et al._ (2018). The only measurement of an analogue effect to date
has been in the acoustic regime where the speed of light is replaced by the
speed of sound that is orders of magnitude lower Cromb  _et al._ (2020). The
key step forward in this work was the implementation of a scheme where the
spinning disk is located deeply in the near-field regime, i.e. all dimensions
(size of the OAM beam, radius and thickness of the spinning disk) are
$\ll\lambda$. In spite of this achievement, the acoustic analogy does not
allow an extension to the quantum regime, leaving open the challenge to
observe the effect in the originally proposed EM domain. Furthermore,
Zel’dovich himself pointed out that with OAM the near-field approach, despite
being necessary to avoid superluminal tangential velocities, leads to a very
weak amplification. The EM field interacts with the cylinder in a region where
the field amplitude decreases like a power of $(r/\lambda)^{\ell}$, thus
reducing even further the amplification Zeldovich (1971); Zeldovich  _et al._
(1986).
In this paper, we show that this 60 year-old long-sought effect has been
concealed for all this time in the physics of induction generators. Induction
motors are constituted of two components: an external stator, composed of
circuits generating a rotating magnetic field, and a rotor, also composed of
several elementary circuit loops, usually in a squirrel cage configuration. By
replacing the internal circuits of the rotor with a solid metal cylinder as in
Zel’dovich’s original proposal, and using a gapped toroid within a LC
resonator as stator, we isolate the key physical effect and unambiguously
observe Zel’dovich amplification, which manifests itself as a negative
dissipation induced by the rotor in the LC circuit. In particular, this
overcomes two conceptual and technological difficulties related to high
rotational speeds and low amplification: 1) having a near-field interaction
allows us to use longer EM wavelengths; 2) using spin-rotational momentum, and
not OAM, allows us to maximize the geometrical interaction area, by removing
the non-wave zone intrinsic to OAM waves. Indeed, by exploiting the mechanical
equivalence between spin and OAM for the Doppler shift of EM waves Simpson
_et al._ (1997); Courtial  _et al._ (1998), we overcome the disadvantage of
the original Zel’dovich scheme, i.e. a weak or close-to-zero intensity
(decaying as $r^{|\ell|}$) in the region occupied by the rotating cylinder.

Figure 1:  (a) Experimental layout. (b) Outline of the RLC circuit used for
measurements and detection. $R_{0}$ is the ohmic coil dissipation, while
parallel dielectric losses are represented by $R_{p}$.

_The model_ – Our setup is composed of a LC toroid circuit and a spinning
cylinder within the toroid gap, as shown schematially in Fig. 1(a).
To compute the effect of the rotating cylinder on the circuit, we first
calculate the currents induced by the magnetic field on the metallic cylinder,
and then the field induced by the currents back into the circuit. Following
the calculation in Ref. BraidottiPRL2020 , we consider the spinning cylinder
axis oriented along $z$ and the magnetic flux density $\bm{B_{0}}$ produced by
the coil wrapped around the toroid to be uniform and oscillating at a
frequency $\omega$. In the lab reference frame we can write the field in the
toroid gap as

$\mathbf{B_{0}}=\beta\mathbf{b_{0}}I,$ (2)

where $I$ is the current in the coil wrapped around the toroid. The factor
$\beta$ is a geometrical factor that we measure experimentally. The vector
$\mathbf{b_{0}}=(1,0,0)^{T}e^{i\omega t}$ indicates the linear polarization of
the oscillating magnetic field. We write this as the sum of a co-rotating and
a counter-rotating vector with respect to the cylinder axis,
$\mathbf{b_{0}}=1/2\left[(1,i,0)^{T}+(1,-i,0)^{T}\right]e^{i\omega t}$. We
then move to the reference frame co-rotating with the cylinder, spinning at
frequency $\Omega$:

$\mathbf{b^{\prime}_{0}}={1\over
2}\left[(1,i,0)^{T}e^{i(\omega-\Omega)t}+(1,-i,0)^{T}e^{i(\omega+\Omega)t}\right].$
(3)

The response of the cylinder to the applied magnetic field is thus the
superposition of the response to the co-rotating and the counter-rotating
polarization components, which rotate at different Doppler shifted
frequencies, $\omega-\sigma\Omega$, where $\sigma$ indicates the wave spin
($\sigma=1$ co-rotating, $\sigma=-1$ counter-rotating). In particular, when
the condition $\Omega>\omega$ is fulfilled, the co-rotating frequency
$\omega-\Omega$ flips sign.
The response of the cylinder to each component can be found by solving
Maxwell’s equations for the specific geometry and material. Analytical
solutions can be found in the case of a spherical rotor BraidottiPRL2020  and
an infinite cylinder (see sup ). For practical purposes, the sphere solution
is found to adequately describe our experiment, as shown below. For our
conductive rotor, the response is determined by the eddy currents induced in
the conductive material of the cylinder, involving both an inductive (in-
phase) and resistive (out-of-phase) component. Induced eddy currents will
couple a magnetic flux back into the coil circuit,
$\Phi\left(\omega\right)=\alpha\left(\omega\pm\Omega\right)I\left(\omega\right)$,
where the linear susceptibility
$\alpha=\alpha^{\prime}-i\alpha^{\prime\prime}$ is evaluated in the rotating
frame. Here, $\alpha^{\prime}$ and $\alpha^{\prime\prime}$ are the in-phase
and out-of-phase components of the response function. In general, $\alpha$ is
proportional to the geometrical coupling $\beta^{2}$. Moreover, according to
linear response theory, $\alpha\left(-\omega\right)$ =
$\alpha^{*}\left(\omega\right)$, in particular
$\alpha^{\prime\prime}\left(-\omega\right)$ =
$-\alpha^{\prime\prime}\left(\omega\right)$.
From an experimental point of view it is convenient to define the resistance
$\mathcal{R}$ and the inductance $\mathcal{L}$ induced by the rotating
cylinder into the circuit:

$\displaystyle\mathcal{R}$ $\displaystyle=$ $\displaystyle
Re[V/I]=\omega\left[\alpha^{\prime\prime}\left(\omega-\Omega\right)+\alpha^{\prime\prime}\left(\omega+\Omega\right)\right]$
(4) $\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
Re[\Phi/I]=\alpha^{\prime}\left(\omega-\Omega\right)+\alpha^{\prime}\left(\omega+\Omega\right).$
(5)

where $V=i\omega\Phi$ is the induced voltage.
The in-phase component $\alpha^{\prime}$ gives the variation of inductance
$\mathcal{L}$ generated by the presence of the rotating cylinder in the gap,
while $\omega\alpha^{\prime\prime}$ can be seen as an induced resistance.
Hence, the condition $\omega-\Omega<0$ for the co-rotating component leads to
a negative resistance that in turn implies a power emission into the EM mode
as opposed to the expected (for a non-rotating or slowly rotating cylinder)
power absorbed from the EM mode. This corresponds to the amplification
predicted by Zeldovich in free space Zeldovich (1971, 1972); Zeldovich  _et
al._ (1986).
However, note that in the case of a linearly polarized oscillating field the
total resistance $\mathcal{R}$ is always composed of a co-rotating and a
counter-rotating component. We can have amplification only if the negative
resistance induced by the co-rotating term is larger than the counter-rotating
one (which is always positive). This can indeed occur due to the symmetry
breaking induced by the mechanical rotation and the counter-rotating term
could be completely eliminated in a setup with a circularly polarized field.

Figure 2:  Example of the amplitude response $|V_{o}|$ (a) and phase response
$phase(V_{o})$ (b) of the LC coil-capacitor setup, measured by the lock-in
amplifier, as function of the generator frequency
$f=\omega/\left(2\pi\right)$. In this example the rotor mechanical frequency
is fixed at $F=\Omega/(2\pi)=330$ Hz and the LC resonance frequency is
$f_{0}=277$ Hz, obtained with capacitors $C=1.0$ nF, $C_{r}=22$ nF. From the
fit, shown as the red lines with modulus and phase of Eq. (6), we can extract
the parameters $L$ and $R$.

_Experimental setup_ – In our experimental setup the rotor is an aluminum
cylinder with radius $a=2$ cm, mounted on a brush-less motor max  which can be
spun up to $500$ Hz about its symmetry axis (see Fig. 1(a)). The magnetic
field is generated by a coil wound around a gapped toroidal ferrite core with
square section $4\times 4$ cm. The rotor is inserted in the $4.4$ cm gap,
slightly larger than rotor diameter. The coil is made of $2\times 10^{4}$
turns of $0.2$ mm diameter copper wire. The ohmic coil resistance at room
temperature is $R_{0}=2.03$ k$\Omega$, while the measured coil inductance is
$L_{0}\approx 263$ H. We approximate the magnetic field as quasi-uniform over
the gap. The current-to-field factor $\beta=(0.40\pm 0.03)$ T/A has been
directly calibrated with a Gaussmeter (Hirst GM05) placed in the center of the
gap.
_Measurement methods and data analysis_ – In order to perform accurate
measurements of the resistance $\mathcal{R}$ and inductance $\mathcal{L}$
induced by the rotor, the coil is placed in series to a capacitor, thus
forming a $RLC$ circuit, whose scheme is shown in Fig. 1(b). We measure the
complex transfer function from the input voltage $V_{i}$, applied to the RLC
circuit by a function generator, to the output voltage $V_{o}$, measured
across a large readout capacitor $C_{r}$. The measurement is performed by a
lock-in amplifier zur  synchronous with the input signal. The capacitor
$C_{r}$ has an impedance much lower than the lock-in input impedance
$R_{\mathrm{in}}=1$ M$\Omega$, in order to suppress the parasitic dissipation
induced by ${R_{\mathrm{in}}}$.
A typical measurement of amplitude and phase of the output voltage $V_{o}$,
measured by the lock-in amplifier, as functions of the generator frequency
$f=\omega/\left(2\pi\right)$ is shown in Fig. 2. Amplitude and phase response
of the RLC resonator are then fitted by modulus and phase of the complex
transfer function of the circuit in Fig. 1(b), which can be expressed as:

$V_{o}=\frac{Z_{r}}{Z_{r}+Z_{c}+\frac{1}{\frac{1}{R_{p}}+i\omega
C_{p}+\frac{1}{Z_{l}}}}V_{i},$ (6)

where $Z_{r}=1/\left(i\omega C_{r}\right)$, $Z_{c}=1/\left(i\omega C\right)$,
$Z_{l}=R+i\omega L=R_{0}+\mathcal{R}+i\omega\left(L_{0}+\mathcal{L}\right)$.

The resonant frequency of the circuit determined by Eq. (6) is approximately
${f_{0}=1/2\pi\sqrt{LC_{eq}}}$, where the equivalent capacitance is
${C_{eq}\approx\left(C^{-1}+C_{r}^{-1}\right)^{-1}+C_{p}}$. Thus we can tune
$f_{0}$ by varying $C$ and $C_{r}$. $C_{p}$ represents the parasitic
capacitance of the coil and $R_{p}$ the associated parallel dissipation. The
value $C_{p}=310$ pF has been estimated in a separate measurement of the bare
coil without capacitor together with the ohmic resistance $R_{0}=2.03$
k$\Omega$.

We performed measurements with 4 different sets of capacitors $C,C_{r}$, and
thus $f_{0}$. This allows us to test our model over a wide range of
parameters. For each set, we first estimate the inductance $L_{0}\simeq 263$ H
and the parasitic resistance $R_{p}\simeq 60$ M$\Omega$ by fitting the
amplitude and phase of the output voltage with the rotor removed from the gap.
Then, we insert the rotor and vary the rotor frequency $F=\Omega/2\pi$ over a
range including values lower and higher than $f_{0}$. For each $F$ we fit
again the amplitude and phase response, but now all parameters
$C,C_{r},C_{p},R_{p}$ are kept as fixed parameters, and only $R$ and $L$,
embodying the effect of the cylinder, are left as free parameters. The values
of $R$ and $L$ extracted from amplitude and phase fits are typically
consistent with each other, with relative discrepancies of the order of
$10^{-4}$ for $L$ and $10^{-2}$ for $R$.
_Results_ –

Figure 3:  Resistance and inductance induced by the rotor into the coil,
$\mathcal{R}=R-R_{0}$ (a-d) and $\mathcal{L}=L-L_{0}$ (e-h), as function of
the rotational frequency $F=\Omega/(2\pi)$ for different sets of capacitors,
and hence different resonance frequencies of the LC resonator
$f_{0}=\omega/2\pi$. Panels (a,e): $C=10$ nF, $C_{r}=47$ nF, $f_{0}=107$ Hz;
Panels (b,f): $C=3.3$ nF, $C_{r}=22$ nF, $f_{0}=175$ Hz; Panels (c,g): $C=1.0$
nF, $C_{r}=22$ nF, $f_{0}=277$ Hz; Panels (d,h): $C=1.0$ nF, $C_{r}=1.0$ nF,
$f_{0}=330$ Hz. The values obtained by the sphere model are shown as a
reference (continuous lines). The vertical lines mark $f_{0}$. We notice that
the negative values of $\mathcal{R}$ are negative by $6$ std. dev. at
$f_{0}=107$ Hz and by $75$ std. dev. at $f_{0}=330$ Hz.

Our best experimental estimations of the resistance and inductance induced by
the rotor into the coil, $\mathcal{R}=R-R_{0}$ and $\mathcal{L}=L-L_{0}$, are
shown in Fig. S4 as functions of the rotational frequency $F$ for the 4
different resonance frequencies $f_{0}$ of the LC oscillator. The continuous
lines in the plots represent the theoretical curves for a rotating sphere. No
free parameters were used to fit the model to the measurements, showing a
substantial agreement with the experimental data. We note that the sphere-
based model appears to capture more accurately border effects compared to the
infinite cylinder model (see sup ). A residual discrepancy is however
expected. The larger discrepancy in the inductance data could be explained by
parasitic coupling to the brush-less motor, which would be stronger at lower
frequencies due to an increased penetration depth.
_Discussion_ – According to Zel’dovich’s original paper, amplification occurs
due to two principal ingredients: 1) the Doppler shifted frequency has to
become negative; 2) the imaginary (absorptive) part of the rotor material
susceptibility changes sign due to the negative Doppler frequency,
transforming losses into gain. Figure S4 shows that the experimental results
follow the Zel’dovich trend for the resistance, and hence absorption, as a
function of the rotor frequency BraidottiPRL2020 . When the rotor frequency
$\Omega/2\pi$ exceeds the LC resonant frequency $f_{0}$, the co-rotating
Doppler shifted frequency $\omega-\Omega$ and the corresponding resistance
term become negative (note that $\sigma=1$ in our case), marking the
inflection point in the resistance $\mathcal{R}$ plot. At slightly higher
frequency the negative co-rotating term exceeds the positive counterrotating
one, leading to a negative total reflected resistance $\mathcal{R}$. This is a
signature that the absorption coefficient has flipped sign and hence that
there is an electromagnetic gain induced by the mechanical rotation.
The effect is particularly evident at high resonance frequencies ($f_{0}=277$
and $335$ Hz) in agreement with the model. i.e. the imbalance between the
negative resistance (gain) induced by the co-rotating component of the field
and the positive resistance (absorption) induced by the counter-rotating
component, increases with the magnetic field frequency $f$. We conclude that
our experiment, based on the simplest possible interaction of a solid metallic
cylinder with an oscillating magnetic field, is substantially reproducing the
electromagnetic amplification mechanism predicted by Zel’dovich. Furthermore,
it is making a further step showing that this effect can be generalized to
spin angular momentum. The practical impossibility to test Zel’dovich
predictions, pointed out already in the original paper Zeldovich (1971), is
overcome here thanks to the heavy spatial confinement of the electromagnetic
field in a LC resonator, compared to free space BraidottiPRL2020 , and by
making use of spin angular momentum.
At the same time, these results show an unexpected connection between the
believed-inaccessible Zel’dovich effect and induction generators Keljik
(2009), which extract electric power from rotational motion. We identify the
Doppler shifted frequency with the ‘slip frequency’ in induction motor
terminology, and ingredient (1) is satisfied in the generator regime, when the
rotor is driven faster than the rotating magnetic field induced by the stator
excitation current. Our setup directly implements condition (2) with a
homogeneous metallic rotor. However, typical induction generators are made
with a squirrel cage rotor, i.e. circuits formed by multiple conducting bars
around an iron core. Rather than the response of the rotor being determined by
a solid material susceptibility, it is engineered through the resistances and
inductances of the squirrel cage circuits. The analogue susceptibility of a
squirrel cage enhances the amplification, for a more efficient generator than
Zel’dovich’s original proposal.

_Conclusions_ – By operating in the sub-wavelength regime and using spin
angular momentum, we have experimentally measured negative dissipation induced
by a rotating metallic cylinder, indicating the amplification of EM waves
originally predicted by Zel’dovich. This 60 year old prediction that was
believed to be untestable with current technology is found to be observable
thanks to the unforeseen link between Zel’dovich amplification and induction
motors. These findings open the way to the merging of ideas from two
previously disconnected fields. In particular, a suggestive prospect is the
realization of Zel’dovich electromagnetic amplification from a rotating body
in the quantum regime Bekenstein and Schiffer (1998); Maghrebi  _et al._
(2012), i.e. the generation of photons out of the quantum vacuum stimulated by
a mechanical rotation Pendry (1997); Zhao  _et al._ (2012); Davies (2005). In
the past this seemed an impossible task, due to the extremely low efficiency
of the effect and the need for having an EM resonator at frequency $\omega$
cooled to the ground state and combined with a rotor spinning at
$\Omega>\omega$. In contrast, induction motor schemes can be optimized through
high efficiency magneto-mechanical coupling. It remains true that realistic
values of $\omega,\Omega\sim 10^{3}$ Hz imply that the temperature required to
bring the resonator in the ground state would be prohibitively low $T\sim
10^{-9}$ K. However, ground-state cooling of a low frequency LC resonator can
be rather achieved using techniques borrowed from optomechanics, such as
feedback-cooling Vinante  _et al._ (2010). Such experiments would allow to
observe of Zel’dovich amplification in the quantum regime.

###### Acknowledgements.

_Acknowledgments_ – We thank Damon Grimsey for expert technical support with
the setup. The authors acknowledge financial support from EPSRC (UK Grant No.
EP/P006078/2) and the European Union’s Horizon 2020 research and innovation
programme, grant agreement No. 820392. A. V. and H.U. acknowledge financial
support from the QuantERA grant LEMAQUME, funded by the QuantERA II ERA-NET
Cofund in Quantum Technologies implemented within the EU Horizon 2020
Programme, from the UK funding agency EPSRC (grants EP/W007444/1, EP/V035975/1
and EP/V000624/1), the Leverhulme Trust (RPG-2022-57), the EU Horizon 2020
FET-Open project TeQ (766900) and the EU Horizon Europe EIC Pathfinder project
QuCoM (GA no.10032223). Data is available at:
DOI:10.5525/gla.researchdata.1457

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Amplification of electromagnetic waves by a rotating body: Supplementary
material

M. C. Braidotti1, A. Vinante2, M. Cromb3, A. Sandakumar3, D. Faccio1,∗, H.
Ulbricht3,∗

_1 School of Physics and Astronomy, University of Glasgow, G12 8QQ, Glasgow,
UK.
2Istituto di Fotonica e Nanotecnologie - CNR and Fondazione Bruno Kessler,
I-38123 Povo, Trento, Italy.
3Department of Physics and Astronomy, University of Southampton, SO17 1BJ,
Southampton, UK.
∗_

##  I The sphere model

We model the system composed by the LC toroid circuit and the spinning
cylinder within the toroid gap as shown in Fig. 1 of the main manuscript.
Amplification occurs when the resistance $\mathcal{R}$ induced by the cylinder
in the circuit becomes negative. To compute the effect of the cylinder on the
circuit we approximate the cylinder as a sphere with same radius $R$. This is
justified by the fact that the gap is quasi-cubical, so the effective length
of the cylinder inside the gap is close to $R$. Then, we compute the magnetic
dipole moment $\bm{m_{0}}$ induced by the magnetic field in the conductive
sphere. This will allow us to calculate the magnetic flux back-reflected into
the circuit, and hence the induced resistance. To do this we follow the
derivation in BraidottiPRL2020 , the difference lays in the magnetic field
$\mathbf{B_{0}}$ which is oscillating rather than rotating.

Let’s consider the spinning axis oriented along $z$ and the magnetic flux
density $\bm{B_{0}}$ produced by the coil wrapped around the toroid to be
uniform and oscillating at a frequency $\omega$.
In the lab reference frame we can write the field in the toroid gap as:

$\mathbf{B_{0}}=\beta\mathbf{b_{0}}I,$ (S1)

where $I$ is the current in the coil wrapped around the toroid. The
geometrical factor $\beta=(0.40\pm 0.03$) T/A has been measured
experimentally. The vector $\mathbf{b_{0}}=(1,0,0)^{T}e^{i\omega t}$
represents the linear polarization of the field oscillating at frequency
$\omega$. To our purposes it is convenient to write it as the sum of two
rotating vectors, co-rotating and counter-rotating with respect to the
cylinder rotation, such that:

$\mathbf{b_{0}}={1\over 2}\left[(1,i,0)^{T}+(1,-i,0)^{T}\right]e^{i\omega t}.$
(S2)

In order to see the amplification effect we are looking for, calculations are
carried out in the reference frame co-rotating with the rotor. In this frame,
we write the vector $\mathbf{b_{0}}$ as

$\mathbf{b_{0}}={1\over
2}\left[(1,i,0)^{T}e^{i(\omega-\Omega)t}+(1,-i,0)^{T}e^{i(\omega+\Omega)t}\right]=\mathbf{b}_{co}+\mathbf{b}_{counter},$
(S3)

where $\omega$ and $\Omega$ are the frequency of the field $\bm{B_{0}}$ and
the rotation frequency of the rotor in the lab frame, respectively.
Our purpose is to find the form of the vector potential $\bm{\mathcal{A}}$,
which allows us to compute the magnetic dipole moment, coupled to the magnetic
flux $\phi=\beta|\bm{m_{0}}|$ back-reflected into the toroid circuit.

To this aim, we solve Maxwell’s equations, which describe the interaction of
the magnetic field $\bm{B}$ generated by the circuit and the conductive sphere
and we find the equation for the vector potential BraidottiPRL2020 :

$-\Delta\mathbf{\mathcal{A}}+\mu_{0}\mu_{r}\sigma\frac{\partial\mathbf{\mathcal{A}}}{\partial
t}=0.$ (S4)

In order to solve this last equation for $\bm{\mathcal{A}}$, we divide the
space in two regions: one inside the cylinder with radius $R$ and one outside
it. At the interface of the two areas, we consider the following three
boundary conditions for a spherical body:

$\displaystyle B_{\perp}^{in}$ $\displaystyle=$ $\displaystyle
B_{\perp}^{out}$ (S5) $\displaystyle H_{\parallel}^{in}$ $\displaystyle=$
$\displaystyle H_{\parallel}^{out}$ (S6)
$\displaystyle\mathbf{\mathcal{A}}\underset{r\rightarrow\infty}{\longrightarrow}$
$\displaystyle\mathbf{\mathcal{A}_{\infty}}$
$\displaystyle=\frac{1}{2}\mathbf{B_{0}}\times\mathbf{r}$ (S7)

where $\mathbf{B}=\mu_{0}\mu_{r}\mathbf{H}$ and $\bm{r}$ is the position
vector in the co-rotating reference frame centred at the center of the sphere.
The magnetic field outside the sphere
$\mathbf{B^{out}}=\mathbf{B_{0}}+\mathbf{B^{refl}}$ is composed of two
components: the incident magnetic field generated by the circuit
$\mathbf{B_{0}}$ plus the magnetic field reflected from the cylinder
$\mathbf{B^{refl}}$. Let us remind that $\mathbf{B_{0}}$ is composed by a co-
rotating and a counter-rotating component (see eq. (S3)). The magnetic field
inside the cylinder $\mathbf{B^{in}}$ is given by the component of the
incident field transmitted in the scattering.
Considering the last boundary condition, the complex vector potential
$\mathbf{\mathcal{A}}$ can be generally written as

$\mathbf{\mathcal{A}}=\frac{1}{2}\left[\left(F_{co}(r)\mathbf{B_{co}}+F_{counter}(r)\mathbf{B_{counter}}\right)\times\mathbf{r}\right]=\mathbf{\mathcal{A}_{co}}+\mathbf{\mathcal{A}_{counter}},$
(S8)

being $\mathbf{B_{0}}=\mathbf{B_{co}}+\mathbf{B_{counter}}$. The function
$F(r)$, such that $F(r)\to\infty=1$, can be computed following
BraidottiPRL2020  as

$\displaystyle
F(r)=1+D(c_{EC}R)\left(\frac{R}{r}\right)^{3}\qquad\mbox{for}\quad r\leq R$
(S9) $\displaystyle
F(r)=\left[1+D(c_{EC}R)\right]\frac{f(c_{EC}r)}{f(c_{EC}R)}\qquad\mbox{for}\quad
r>R.$ (S10)

where $c_{EC}=\sqrt{-i\mu_{0}\mu_{r}\sigma\omega_{\pm}}$, with
$\omega_{\pm}=\omega\pm\Omega$ depending whether we are considering the co-
rotating component or the counter-rotating one. The function $D$ is

$\displaystyle
D(c_{EC}R)=\frac{(2\mu_{r}+1)g(c_{EC}R)-1}{(\mu_{r}-1)g(c_{EC}R)+1}$ (S11)
$\displaystyle
g(c_{EC}R)=f(c_{EC}R)\frac{c_{EC}R}{\sin(c_{EC}R)}=\frac{1-(c_{EC}R)\cot(c_{EC}R)}{(c_{EC}R)^{2}}.$
(S12)

These last equations define the explicit form of the vector potential
$\bm{\mathcal{A}}=\mathbf{\mathcal{A}_{co}}+\mathbf{\mathcal{A}_{counter}}$ in
the whole space.
It is now possible to compute the form of the magnetic dipole moment
$\mathbf{m_{0}}$ on the cylinder in the reference frame co-rotating with the
cylinder, which will be composed by two components: one co-rotating with the
cylinder and another one counter-rotating. From the relation between
$\mathbf{m_{0}}$ and the reflected component of the vector potential
$\mathbf{\mathcal{A}}$

$\mathbf{\mathcal{A}}=\frac{\mu_{0}}{4\pi}\frac{\mathbf{m_{0}}\times\mathbf{r}}{r^{3}}$
(S13)

we find

$\displaystyle\mathbf{m_{0}}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\mu_{0}}IR^{3}\beta[F_{co}(R)-1]\begin{pmatrix}1\\\
i\\\ 0\end{pmatrix}e^{i(\omega-\Omega)t}+$ (S14) $\displaystyle+$
$\displaystyle\frac{\pi}{\mu_{0}}IR^{3}\beta[F_{co}(R)-1]\begin{pmatrix}1\\\
-i\\\ 0\end{pmatrix}e^{i(\omega+\Omega)t}=$ (S15) $\displaystyle=$
$\displaystyle\left(\chi_{co}\mathbf{B_{co}}+\chi_{counter}\mathbf{B_{counter}}\right).$
(S16)

where $\chi=\chi^{\prime}-i\chi^{\prime\prime}$ is the complex response
function of the sphere in the presence of the field in the co-rotating
reference frame. It is worth noting that in order to compute the amount of
reflected flux into the coil, we subtracted a factor $1$ from $F(R)$
corresponding to the contribution of the incident field $\mathbf{B_{0}}$.
Looking at the amplification from the point of view of the LC circuit, the
rotating magnetic moment will couple with a magnetic flux $\phi$ back-
reflected into the circuit and with peak value $\phi=\beta|\mathbf{m_{0}}|$.
We can thus write a relation $\phi=(\alpha_{co}+\alpha_{counter})I$, with
$\alpha=\alpha^{\prime}-i\alpha^{\prime\prime}$: the flux response function is
given by $\alpha=\beta^{2}\chi$. The voltage given by the flux is then:

$V=i\omega\phi=i\omega(\alpha_{co}+\alpha_{counter})I$ (S17)

and we can define the resistance $\mathcal{R}$ and the inductance
$\mathcal{L}$ generated by the presence of the rotating cylinder as:

$\displaystyle\mathcal{R}$ $\displaystyle=$ $\displaystyle
Re[V/I]=\omega(\alpha^{\prime\prime}_{co}+\alpha^{\prime\prime}_{counter})$
(S18) $\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
Re[\phi/I]=(\alpha^{\prime}_{co}+\alpha^{\prime}_{counter}).$ (S19)

The factor $\alpha^{\prime}$ gives the inductance $\mathcal{L}$ generated by
the presence of the rotating cylinder in the gap, while
$\omega\alpha^{\prime\prime}$ can be seen as a resistance $\mathcal{R}$
induced by the rotating cylinder into the circuit. Hence amplification
corresponds to a negative resistance that leads to power emission into the
electromagnetic (EM) mode as opposed to the expected (for a non-rotating or
slowly rotating cylinder) power absorbed from the EM mode. We observe that the
resistance $\mathcal{R}$ is composed by a co-rotating and a counter-rotating
components. We have amplification only if the co-rotating one is larger than
the counter-rotating one. This is possible since the response function
$\alpha$ is not symmetric with respect to $\omega$.

##  II The cylinder model

Let us consider a non rotating cylinder with infinite length in a transverse
oscillating magnetic field, along $x$. To compute the effect of the cylinder
we write the oscillating magnetic field back-reflected from the cylinder as a
vector $\mathbf{B}=(B_{r},B_{\phi},B_{z})$ in cylindrical coordinates.
Considering the geometry of our system the $B_{z}$ component is null since the
external magnetic field is transverse to the cylinder and no induced $\hat{z}$
component is anticipated. This leaves us with a 2D problem Perry1978 . Solving
Maxwell equations for this problem gives the components of the back-reflected
field from the cylinder:

$\displaystyle B_{r}$ $\displaystyle=$ $\displaystyle B_{0}\frac{R^{\prime
2}}{r^{2}}\cos(\phi)e^{-i\omega t}$ (S20) $\displaystyle B_{\phi}$
$\displaystyle=$ $\displaystyle-B_{0}\Bigl{(}-\frac{R^{\prime
2}}{r^{2}}\Bigr{)}\sin(\phi)e^{-i\omega t},$ (S21)

where $R^{\prime 2}$ is given by

$R^{\prime
2}(\omega)=R^{2}\frac{(\mu_{r}+1)J_{1}(\sqrt{iR_{m}})-\sqrt{iR_{m}}J_{0}(\sqrt{iR_{m}})}{(\mu_{r}-1)J_{1}(\sqrt{iR_{m}})-\sqrt{iR_{m}}J_{0}(\sqrt{iR_{m}})}$
(S22)

where $R_{m}(\omega)=\left[R/\delta(\omega)\right]^{2}$ is the magnetic
Reynolds number and $\delta(\omega)=1/\sqrt{\mu_{0}\mu_{r}\sigma\omega}$ is
the penetration depth, depending on the frequency of oscillation of the
magnetic field $\omega$ Perry1978 . Let us consider the case of rotating
cylinder. Equations (S20) needs to account the rotation of the cylinder at
frequency $\Omega$. In the reference frame co-rotating with the cylinder, the
co-rotating and a counter-rotating components of the oscillating magnetic
field rotate at the frequencies $\omega_{-}=\omega-\Omega$ and
$\omega_{+}=\omega+\Omega$ respectively. To include the rotation we observe
that the field in eq. (S20) corresponds to a linearly polarized field
expressed in polar coordinates Matsuo2011  111The linearly polarized vector in
$(x,y)$ coordinates  $\begin{pmatrix}1\\\ 0\end{pmatrix}$ (S23) corresponds to
the vector  $\begin{pmatrix}\cos(\phi)\\\ -\sin(\phi)\end{pmatrix}$ (S24) in
polar coordinatesMatsuo2011 .. In polar coordinates, right and left rotating
vectors can be written as

$|\mbox{right}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}\cos(\phi)-i\sin(\phi)\\\
-\sin(\phi)-i\cos(\phi)\end{pmatrix}\qquad\mbox{and}\qquad|\mbox{left}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}\cos(\phi)+i\sin(\phi)\\\
-\sin(\phi)+i\cos(\phi)\end{pmatrix}.$ (S25)

We can hence express the field in (S20) as a sum of co-rotating (right) and
counter-rotating (left) components. Importantly, in this frame the frequency
of the co-rotating field is different from the frequency of the counter-
rotating component, being $\omega_{-}=\omega-\Omega$ and
$\omega_{+}=\omega+\Omega$ respectively. The back-reflected field for our
system is

$\displaystyle B_{r}$ $\displaystyle=$ $\displaystyle
B_{0}\Bigl{[}\frac{R^{\prime
2}(\omega_{-})}{2r^{2}}\Bigl{(}\cos(\phi)-i\sin(\phi)\Bigr{)}e^{-i\omega_{-}t}+\frac{R^{\prime
2}(\omega_{+})}{2r^{2}}\Bigl{(}\cos(\phi)+i\sin(\phi)\Bigr{)}e^{-i\omega_{+}t}\Bigr{]}$
(S26) $\displaystyle B_{\phi}$ $\displaystyle=$ $\displaystyle-
B_{0}\Bigl{[}\frac{R^{\prime
2}(\omega_{-})}{2r^{2}}\Bigl{(}-\sin(\phi)-i\cos(\phi)\Bigr{)}e^{-i\omega_{-}t}+\frac{R^{\prime
2}(\omega_{+})}{2r^{2}}\Bigl{(}-\sin(\phi)+i\cos(\phi)\Bigr{)}e^{-i\omega_{+}t}\Bigr{]}.$
(S27)

Similarly to the sphere case we now compute the equivalent magnetic moment
induced in the cylinder. To this end, we write the $\bm{B}$ field generated by
a linear distribution of magnetic dipole moment with density per unit length
$\bm{\lambda}$ along the cylinder axis, that is:

$\mathbf{B}(r,\phi)=\int_{\infty}^{\infty}\frac{\mu_{0}}{4\pi}\Bigl{(}\frac{3\mathbf{r}(\bm{\lambda}\cdot\mathbf{r})}{r^{5}}-\frac{\bm{\lambda}}{r^{3}}\Bigr{)}dz,$
(S28)

where $\mathbf{r}$ is the position vector in cylindrical coordinates. The
solution of this equation is:

$\mathbf{B}(r,\phi)=\Bigl{(}\frac{\mu_{0}}{2\pi}\frac{\lambda_{r}}{r^{2}},-\frac{\mu_{0}}{2\pi}\frac{\lambda_{\phi}}{r^{2}}\Bigr{)}^{T}.$
(S29)

Comparing eqs. (S29) and (S26) we find the expression for the magnetic moment
density $\bm{\lambda}=(\lambda_{r},\lambda_{\phi})^{T}$ being

$\displaystyle\lambda_{r}$ $\displaystyle=$ $\displaystyle\frac{2\pi
r^{2}}{\mu_{0}}\frac{1}{2r^{2}}B_{0}\Bigl{[}R^{\prime
2}(\omega_{-})\Bigl{(}\cos(\phi)-i\sin(\phi)\Bigr{)}e^{-i\omega_{-}t}+R^{\prime
2}(\omega_{+})\Bigl{(}\cos(\phi)+i\sin(\phi)\Bigr{)}e^{-i\omega_{+}t}\Bigr{]}=$
(S30) $\displaystyle\lambda_{\phi}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{\mu_{0}}B_{0}\frac{1}{2}\Bigl{[}R^{\prime
2}(\omega_{-})\Bigl{(}-\sin(\phi)-i\cos(\phi)\Bigr{)}e^{-i\omega_{-}t}+R^{\prime
2}(\omega_{+})\Bigl{(}-\sin(\phi)+i\cos(\phi)\Bigr{)}e^{-i\omega_{+}t}\Bigr{]}.$
(S31)

Now, for the actual finite case we take the solution for the density of
magnetic moment of the infinite case and integrate over the finite length of
the gap $L$, where the $\bm{B}$ field is non negligible. This gives the total
effective magnetic moment induced in the finite cylinder, neglecting border
effects:

Being
$\mathbf{m}=\chi_{co}\mathbf{B_{co}}+\chi_{counter}\mathbf{B_{counter}}$, we
find

$\displaystyle\chi_{co}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{\mu_{0}}R^{\prime 2}(\omega_{-})L$ (S32)
$\displaystyle\chi_{counter}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{\mu_{0}}R^{\prime 2}(\omega_{+})L.$ (S33)

The resistance $\mathcal{R}$ and inductance $\mathcal{L}$ generated by the
presence of the rotating cylinder can be written as

$\displaystyle\mathcal{R}$ $\displaystyle=$
$\displaystyle\omega\beta^{2}(\chi^{\prime\prime}_{co}+\chi^{\prime\prime}_{counter})$
(S34) $\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\beta^{2}(\chi^{\prime}_{co}+\chi^{\prime}_{counter}).$ (S35)

Figure S4:  Comparison of the resistance and inductance calculated through the
sphere (blue) and cylinder (orange) models. $\mathcal{R}=R-R_{0}$ (a,b,c,d)
and $\mathcal{L}=L-L_{0}$ (e,f,g,h) computed as function of the rotational
frequency $F=\Omega/(2\pi)$ for different resonance frequencies
$f=\omega/(2\pi)~{}=~{}\left[105\textrm{(a,e)},175\textrm{(b,f)},275\textrm{(c,g)},335\textrm{(d,h)}\right]$
Hz of the LC oscillator.

##  III Squirrel-cage induction generator vs solid rotor

Induction machines are a type of rotating machine where a stator circuit - the
non-rotating part, usually embedded in a hollow cylinder - has alternating
currents oscillating at a single frequency $\omega_{ac}$ \- the frequency of
the power supply, which are used to generate a rotating field. The stator can
have multiple magnetic poles $p$, hence the frequency of rotation of the
stator magnetic field is $\omega_{s}=\omega_{ac}/p$.

Figure S5:  Sketch of copper squirrel cage rotor.

Secondary circuits are mounted on the rotor, placed inside the hollow stator
cylinder, and rotate at frequency $\omega_{m}$ \- the mechanical rotation. The
mechanism is based on the principle of induction: the rotor circuits are
short-circuited and their currents are induced by the stator rotating field.

In most induction machines, a squirrel cage rotor is used. This is made of a
solid cylinder of iron or steel carved by conductor bars, usually made of
copper or aluminium, that are short-circuited by a conductive ring around the
cylinder bases (see an sketch of the conductive bars in the typical squirrel
cage circuit in Figure S5). The thickness and the shape of these conductive
bars determines the inductive and resistive properties of the rotor circuits.
In our analysis we consider the induction system described in Woodson1968  and
we write the voltage $V_{s}$ induced by the stator on the rotor as

$V_{s}=\frac{\omega_{s}^{2}M^{2}(R_{rot}/s)}{(R_{rot}/s)^{2}+\omega_{s}^{2}L_{rot}^{2}}I_{s}+i\frac{\omega_{s}^{3}M^{2}L_{rot}}{(R_{rot}/s)^{2}+\omega_{s}^{2}L_{rot}^{2}}I_{s},$
(S36)

where $s=(\omega_{s}-\omega_{m})/\omega_{s}$ is the so called slip, that can
be related to the Zel’dovich condition considering the correspondence
$p\rightarrow\ell$ ($\ell=1$ in our case), $\omega_{m}\rightarrow\Omega$ and
$\omega_{ac}\rightarrow\omega$. $M$ is the mutual inductance, while
$R_{rot},L_{rot}$ are the resistance and inductance of the rotor’s circuits.
It is possible to find the effective susceptibility $\chi_{\mathrm{eff}}$ of
the squirrel cage induction generator by recalling that
$\alpha=\beta^{2}\chi_{\mathrm{eff}}$. The geometrical factor $\beta$ for the
squirrel cage circuit is just the mutual inductance $M$. After some
manipulation, we find:

$\chi_{\mathrm{eff}}=\frac{\alpha}{M^{2}}=\frac{L_{rot}(\omega-\Omega)^{2}}{R_{rot}^{2}+L_{rot}^{2}(\omega-\Omega)^{2}}-i\frac{R_{rot}(\omega-\Omega)}{R_{rot}^{2}+L_{rot}^{2}(\omega-\Omega)^{2}},$
(S37)

where the first term on the right hand side is the inductive component
($\chi_{\mathrm{eff}}^{\prime}$) and the second is the resistive component
($\chi_{\mathrm{eff}}^{\prime\prime}$), which indeed flips sign if the slip
frequency $(\omega-\Omega)<0$, i.e. the Zel’dovich condition is fulfilled.

## References

  * (1) M.C. Braidotti, A. Vinante, G. Gasbarri, D. Faccio, and H. Ulbricht.  Zel’dovich amplification in a superconducting circuit.  Phys. Rev. Lett., 125:140801, 2020.
  * (2) Shigeki Matsuo.  Matrix calculus for axially symmetric polarized beam.  Opt. Express, 19(13):12815–12824, 2011.
  * (3) Michel P. Perry and Thomas B. Jones.  Eddy current induction in a solid conducting cylinder with a trasverse magentic field.  IEEE Transactions on Magnetics, 14(4):227, 1978.
  * (4) H. H. Woodson and J.R. Melcher.  Electromechanical Dynamics.  John Wiley and Sons, New York, NY, 1968.