maximal entropy of a black hole with $n$ initial modes is $\log(n+1)$
(counting the $n$ states plus the vacuum state), we show in Fig. 6.4 the
normalized entropy in order to be able to compare the shape of the curves as
the size of the black hole is changed. The resulting entropy curve turns out
to be strikingly similar to the one predicted by Page Page (1993) as long as
we observe evaporation for long enough (several thousand time slices). Most
importantly, the entropy that started out as a pure state reaches a maximum
(at about the time when half the black hole quanta have been converted, see
Flórez Gutiérrez (2022)) and appears to vanish as $t\to\infty$, where the
final black hole density matrix $\rho_{\rm BH}$ approaches a pure vacuum state
$|0\rangle_{d}$ in the limit $N\to\infty$, for all the input basis states
$\mathop{|n\rangle}\nolimits$ tried. It thus appears that, from the formation
to the decay of black holes, pure states are turned into pure states, and the
laws of physics remain inviolate.

Figure 6.4: Black hole entropy $S_{\rm BH}(t)$ as a function of the number of
time slices $N$ for initial states
$\mathop{|n\rangle}\nolimits_{d}\mathop{|0\rangle}\nolimits_{ab}$ with $n=5$
(orange), $n=20$ (red) and $n=50$ (pink), with $t=\Delta tN$. To normalize
entropies, logarithms are taken to the base $n+1$ so that the entropy $S_{\rm
BH}(t)\leq 1$ for any $n$. The $S$-matrix was evaluated using $r_{0}=1$ and
$\Delta t=1/15$ (modified from Brádler and Adami (2016)).

Note that using just $n=50$ initial modes already gives rise to an extremely
large Hilbert space. Using a Taylor expansion of $W^{(i)}$ with $\Delta
t=1/15$ to order up to 500 would require $2^{500}$ terms121212A Taylor
expansion of $W^{(i)}$ entails the expansion of operators of the type
$e^{A+A^{\dagger}}=\sum_{i=0}^{n}\frac{1}{n!}(A+A^{\dagger})^{i}$, where
$A=d^{\dagger}ab$. Because $A$ and $A^{\dagger}$ in general do not commute,
the number of summands in $(A+A^{\dagger})^{i}$ is not $i+1$ but instead
$2^{i}$., which is of course intractable. Brádler’s Dyck path representation
Brádler (2015) renders the calculation tractable, but it does require High-
Performance Compute Clusters. Using a smaller $\Delta t=1/25$ with
commensurately fewer time slices (to keep overall compute time constant) does
not change the curves visibly, which suggests that $\Delta t=1/15$ is
sufficiently small to allow for equilibration.

While the curves shown in Fig. 6.4 used the approximation that black hole
modes converted to Hawking and partner modes only interact once (by virtue of
the initial state (VI.9)), it is possible to relax this assumption and allow
the partner modes behind the horizon to interact with black hole modes again,
as in Fig. 6.3(b). Doing so complicates the calculation enormously so that
only small systems can be evaluated, for fewer time slices. The overall shape
of the entropy curves does not change appreciably when allowing partner modes
to interact with the black hole repeatedly. The curves become somewhat more
symmetric due to a slower rise of the entropy, and in so doing become more
similar to the symmetric curves that Page had imagined. Further, while the
calculation assumed that the black hole was initially in a pure state
$|n\rangle_{d}$, the results are unchanged if the initial state is instead in
a rotated basis $|\psi\rangle_{d}=\sum_{j=1}^{n}\epsilon_{j}|j\rangle_{d}$.

Technically speaking, the mapping from the black hole initial state to the
final state is an example of an erasure map

$\displaystyle|n\rangle_{d}|0\rangle_{ab}\stackrel{{\scriptstyle
W}}{{\longrightarrow}}|\Psi(t)\rangle_{dab}\stackrel{{\scriptstyle
t\to\infty}}{{\approx}}|0\rangle_{d}\otimes\rho_{ab}(|n\rangle)$ (VI.19)

that ultimately decouples the black hole from the Hawking and partner modes.
It turns out that this map is an explicit realization of the fully-quantum
Slepian-Wolf (FQSW) protocol Abeyesinghe  _et al._ (2009), which is the
fundamental protocol in quantum information science that quantifies how well
quantum entanglement can be transferred, stored, and distilled. The unitary
interaction (VI.7) first creates the entanglement between the black hole
(which plays the role of the reference in the FQSW protocol) and the outside
and the inside of the black hole (re-enacting the parts of Alice and Bob).
After the Page time, further dynamics erases the entanglement between the
black hole and Hawking radiation, just as in the FQSW protocol the
entanglement between Alice and the reference is erased. That the further
dynamics reverses the prior entanglement is ensured by the continued unitary
dynamics of the black hole’s interaction with the radiation field, in the same
manner as the erasure map in the FQSW protocol forces the transfer of
entanglement from Alice to Bob. It is this same unitarity that enforces the
existence of stimulated emission, which in turn preserves information in black
hole dynamics.

It is useful to consider the dynamics of black hole formation, evaporation, up
to ultimate disappearance, in terms of entropy Venn diagrams. Those diagrams
summarize how classical or quantum entropies are distributed among the
subsystems of a closed system. In particular, if we begin with a system in a
pure state (with zero entropy) those diagrams can reveal how the purity of the
system is maintained as long as the sum of all entropies remains at zero. In
Box 2 I show schematically how this is indeed achieved in the scenario I have
outlined here (see also Adami and Cerf (1999a)), and in particular suggest
that the information about the black hole’s formation is in fact encrypted in
the Hawking and partner modes (a process that is otherwise called
”scrambling”, see Hayden and Preskill (2007)). It is also clear that as a
consequence, Hawking modes can be purely thermal and yet convey information
about the black hole modes.

Box 2: Quantum Entropy Venn Diagrams in Unitary Black Hole Dynamics

Entropy Venn diagrams are a useful tool to study how classical or quantum
information is distributed among the subsystems of a (larger) composite
system. Such diagrams have been used extensively to study quantum information
processing and communication Cerf and Adami (1997b, a); Adami and Cerf (1997,
1999b) as well as quantum experiments Glick and Adami (2017, 2020). In those
diagrams, a circle represents a subsystem, and the intersection of this circle
with another circle (subsystem) refers to the shared entropy between the two
subsystems. Fig. 6.5 shows a simple bi-partite Venn diagram between systems
$X$ and $Y$, labeling the conditional and shared entropies.

Figure 6.5: Entropy Venn diagram showing how the entropy of $X$, $H(X)$, is
composed of the conditional entropy $H(X|Y)$ and the shared entropy
(information) $H(X;Y)$ (similarly for $Y$).

Shared pairwise entropies (such as $H(X;Y)$ in Fig. 6.5) can never be negative
(either in classical or quantum Venn diagrams), but they can exceeed the
entropy of $X$ or $Y$ in the quantum case, and conditional entropies can be
negative in quantum physics Cerf and Adami (1997b), something that is
impossible for classical (Shannon) entropies. Shared entropies between three
or more systems can be negative both in classical and quantum physics.

The quantum entropy diagram for the black hole pure state (Fig. 6.6) shows the
black hole entangled with a reference state $R$ (as we did in the construction
of the quantum channel, when we ”purified” Alice’s density matrix in Fig.
5.1).

Figure 6.6: Entropy Venn diagram showing the initial joint state of the black
hole $BH$ purified by the reference $R$. The joint state has zero entropy,
while the inital von Neumann entropy of the black hole is $S$. As the black
hole evolves, the entropy of the reference remains at $S$, while the entropy
of the black hole (minus the entropy of the Hawking radiation) will decrease.

As the black hole evolves, it loses entropy due to pair formation of Hawking
and partner modes at the horizon (modes annihilated by $a_{k}$ and $b_{k}$). I
will use the letter $A$ to denote the Hawking modes (outside the black hole)
and $B$ to denote the partner modes (inside the black hole). They should not
be confused with the ”Alice” and ”Bob” systems defined earlier. Assuming that
the entropy of Hawking modes $H(A)=\Delta S$ (because Hawking and partner
modes are entangled, their entropy must always be the same), the entropy Venn
diagram between the black hole and radiation must be that shown in Fig.
6.7(a). Note that in this diagram the reference state R with entropy $S$ is
traced out, so that the joint state of black hole BH and the AB system must
also have entropy $S$. We can also see that the information that the Hawking
and partner modes have extracted from the black hole is encrypted: the
negativity of the shared ”triplet” information $H(BH;A;B)=-\Delta S$ is the
tell-tale sign of a symmetric Vernam cipher Vernam (1926); Shannon (1949):
each of the three systems is the cryptographic key to unlock the information
between the other two (it is easily implemented via the ”controlled NOT”
(CNOT) operation, for example, A=B.CNOT.BH). Incidentally, the CNOT operation
is precisely the one implementing the cloning operation (IV.7). This
relationship between black hole and Hawking/partner modes has previously been
described as ”scrambling” Hayden and Preskill (2007), except that in the
Hayden-Preskill protocol the decoder has access to the partner modes, which is
not possible here.

Figure 6.7: (a): Entropy Venn diagram of the black hole BH and the Hawking and
partner modes, A and B, respectively. As the reference R is traced over, the
joint entropy of BH and radiation is also $S$. The reference state’s existence
is indicated by the gray area, with conditional entropy $-S$, reminding us
that the joint system is still pure. Each of the radiation modes have marginal
entropy $\Delta S$. The entropy of the black hole given the radiation modes
(that is, not counting their contribution to the total entropy) is
$H(BH|A,B)=S-\Delta S$. (b): Entropy Venn diagram showing that the black hole
BH and the Hawking modes A are uncorrelated when tracing over reference and
partner modes. This holds true also for BH and partner modes B (when tracing
over Hawking modes).

Note further that when tracing over reference as well as partner modes, the
entropy Venn diagram between Hawking radiation and the black hole indicates
that they share zero information (Fig. 6.7b): one cannot be used to predict
the state of the other. Yet, they are still entangled via their entanglement
with the partner modes and the reference. Once $\Delta S$ has become as large
as $S$, the entire entropy of the black hole modes has been converted, so that
$H(BH|A,B)=0$ (see Fig. 6.8). The decoupling via ”entanglement erasure” has
been achieved, and the remaining state is that of a fully entangled pure GHZ
(Greene-Horne-Zeilinger) state between reference, Hawking, and partner modes.

Figure 6.8: (a): Entropy Venn diagram after complete evaporation of the black
hole. While the diagram shows the outline of the black hole BH, it actually
ceases to be a physical entity as the horizon has disappeared. (b): The
entropy Venn diagram between radiation modes and the reference after full
evaporation of the black hole.

Finally, a few words about the use of the tri-linear Hamiltonian to simulate
interactions between black hole modes and radiation. Obviously, this
interaction Hamiltonian does not follow from a fundamental theory of quantum
gravity, but instead emerges from taking the quantum optics analogy suggested
by the Bogoliubov transformation (III.1) seriously. That transformation
suggested a free-field Hamiltonian (the squeezing Hamiltonian) which, in
parametric down-conversion, can be extended so as to deal with a depletable
pump using the term (VI.2). Alternatives to using such a simple interaction
require coupling the black hole to extraneous degrees of freedom, such as for
example the dilaton (as in the CGHS model Callan, Jr and S. Giddings  _et al._
(1992)). But generally speaking (and as argued for by Strominger Strominger
(1996)), any consistent unitary theory that obeys energy conservation rules
must have a tri-linear intreraction term in the low-energy limit.

##  VII Discussion

In this review I have tried to marshal a set of arguments—some of which are
nearly as old as Hawking’s original derivation of the spontaneous emission of
radiation by black holes—to support the point of view that black hole dynamics
is unitary from formation to complete evaporation, and that information is not
lost in, swallowed by, or destroyed in, black holes. Because of the effect of
stimulated emission of radiation, which must accompany the spontaneous
emission of radiation for any black body (as Einstein’s original derivation
showed), information is always copied at the horizon with an accuracy close to
what the laws of physics allow. Stimulated emission not only ensures
information preservation, it also ensures CPT invariance, as Box 1 suggests.
The copying of information at the event horizon does not, however, violate the
quantum no-cloning theorem, which after all is a consequence of the linearity
of quantum mechanics. In fact, it is precisely the Hawking radiation that
saves the no-cloning theorem, as it is Hawking radiation that prevents perfect
cloning of quantum states (just as perfect cloning is impossible in quantum
optics due to the ”open port” in the nonlinear crystal that gives rise to
vacuum fluctuations).

How is it possible that Hawking missed this effect in his initial (and also
subsequent) publications on the quantum properties of black holes? It is clear
that he was aware of the possibility of stimulated emission, but in Hawking
(1975) he describes stimulated emission in terms of the phenomenon of
”superradiance”, which he refers to as a ”classical phenomenon” (citing Refs.
Misner (1972); Press and Teukolsky (1972); Starobinskiǐ (1973)). Superradiance
is a term used both in quantum optics and in astrophysics, and generally
refers to the emission of radiation due to excitations that owe their energy
either to inertial motion at superluminal speeds (e.g., Čerenkov radiation) or
from rotational motion. In both cases, the energy powering the radiation must
come from the kinetic energy of bulk matter. Soviet physicist Yakov Zeldovich
had calculated that a rotating cylinder would emit radiation via stimulation
Zeldovitch (1972), while Alexei Starobinskii argued that a rotating black hole
would emit radiation due to that same effect Starobinskiǐ (1973). During
Hawking’s well-documented visit to Moscow in 1973 he talked to Zeldovitch as
well as to Starobinskii, who suggested to Hawking that stimulated
(superradiant) emission must be accompanied by spontaneous emission in the
same modes but without input from bulk kinetic energy. After the visit,
Hawking engaged in the calculation we now know. His calculation, however,
treated non-rotating (Schwarzschild, rather than Kerr) black holes, and for
this reason it seems he discarded the possibility of stimulated emission out
of hand.

In hindsight, the importance of the stimulated process should have been
apparent if only Hawking had investigated the effect of particles present in
the initial vacuum (at past infinity ${\mathscr{I}}^{-}$). But Hawking
explicitly disregarded those. His reasons for doing so are not immediately
clear. As discussed earlier, to understand the effect of gray-body factors,
Hawking relied on an argument following particle trajectories backwards in
time from inside the black hole to the outside. In both Refs. Hawking (1974)
and Hawking (1976), Hawking claims that following particles inside the horizon
at future infinity back to ${\mathscr{I}}^{-}$ outside the horizon (that is,
with $v>v_{0}$ in his notation) has zero amplitude [Eq. (4.22) in Hawking
(1976) concerns precisely such particles]. It is not clear what reasons he
mustered for this assertion that there cannot be any particles at past
infinity, but we must keep in mind that Hawking also did not seem to realize
that a perfectly black hole will, when viewed from the inside, look like a
perfectly reflecting mirror (see Box 1). For such a mirror (a white hole),
those moving-backwards-in-time particles hitting the horizon would not be
transmitted to past infinity, but stimulated emission of particles from those
reflected at the white hole horizon, as discussed in section IV.2, would.

My best guess of why Hawking dismissed stimulated emission as an important
element in black hole dynamics is that he thought that stimulated emission
needs an energy source so as to lift the system out of its ground state. For
Kerr black holes, this energy is supplied by the rotational motion. For atomic
gases, inversion is created via pumping. But black holes are different: they
are ”forever in a pumped state”: their heat capacity, after all, is negative.
As a consequence, they can be stimulated to emit copies even if they lack a
charge or angular momentum.

What empirical evidence do we have today that stimulated emission must play an
important role in black hole dynamics? We have yet to detect Hawking radiation
from black holes, and are unlikely to do so because the temperature (for
average-sized black holes) is extremely low (about $6\times
10^{-8}\frac{M_{\odot}}{M}$[K] for a black hole of mass $M$, where $M_{\odot}$
is the solar mass), and the radiation is expected to be extremely faint.
However, several teams have created ”analogue black holes” by artificially
creating the causal split between space-time regions that defines black holes,
as initially suggested by Unruh Unruh (1981). One way to achieve this is to
use gravity waves in flowing fluids. As discussed for example by Leonhardt
(see in particular Fig. 8.7 in Ref. Leonhardt (2010)) it is possible to create
horizons for gravity waves by creating a fluid flow that is faster than the
maximum speed for those gravity waves. Weinfurtner et al. Weinfurtner  _et
al._ (2011) tested whether a water wave directed towards a (simulated) white
hole horizon stimulated pairs of waves with the correct amplitude ratio
(III.7) predicted by Hawking. They found this to be the case, and argued that
because stimulated emission must always be accompanied by spontaneous
emission, it should be possible in principle to observe analogue Hawking
radiation in this system also. This was achieved by another group, using
atomic Bose-Einstein condensates as the medium Muñoz de Nova  _et al._ (2019);
Kolobov  _et al._ (2021). In this experiment, two regions were created where
the flow velocity is larger (smaller) than the speed of sound in this medium,
simulating the inside (outside) of the black hole. The group observed the
formation of the black hole horizon, and the subsequent emergence of Hawking
and partner modes. Because the experiment produces not only a black hole
horizon but also a white hole horizon (an area inside of the simulated black
hole that reflects waves) the dynamics creates Čerenkov radiation that
ultimately stimulates the emission of Hawking/partner pairs.

Earlier speculations Corley and Jacobson (1999) and experiments Steinhauer
(2014) with Bose-Einstein condensates had suggested that perhaps the
reflection of waves at the white-hole ”inner” horizon could stimulate the
emission of more modes at the black-hole horizon, which in turn after
reflection at the white-hole horizon could lead to ever-increasing
amplification of these waves: a black hole laser. In fact, this idea was
introduced rather speculatively already by Press and Teukolsky Press and
Teukolsky (1972) before the discovery of Hawking radiation. Those authors
imagined a black hole encased in a spherical mirror, and radiation amplified
superradiantly (in a Kerr black hole, of course) could reflect back onto the
mirror, creating an instability they termed the ”black hole bomb”. However, a
close analysis Wang  _et al._ (2017) of Steinhauer’s experiment Steinhauer
(2014) revealed that lasing at most played a minor role in the Bose-Einstein
cavity, most likely because the location of the white hole horizon with
respect to the black hole horizon is constantly changing, destroying the
coherence required for the lasing phenomenon.

However, these discussions open up the possibility of a black hole laser that
is not due to an ”inner” horizon, but rather emerges for the brief period of
time when a black-hole binary is inspiraling, just before the merger event.
While the period of time where the binaries are close enough so that a
significant amount of stimulated radiation could hit the partner is brief (on
the order of a fraction of a second for a typical binary), this is sufficient
for on the order of ten reflections (in the rotating frame), leading to
significant amplification131313An even more speculative notion is the idea
that a wormhole that connects two black holes could create a cavity that would
coherently amplify radiation trapped within it, giving rise to a wormhole
laser.. Could such a ”flash” of coherent black-hole laser light be detected at
the same time that we are recording the gravitational signature of such an
event? This is a difficult experimental question, since as of yet we do not
have a simulation of such a phenomenon that would allow us to tune experiments
to detect this signature. Such a simulation would allow us to distinguish the
tell-tale coherent signature from the light emitted by accretion disks (if
any). To-date, no electromagnetic signature has been observed coming from
regions where a binary black-hole merger was pinpointed to. This suggests that
possibly those mergers do not have accretion disks, and consequently there
would also be no source of radiation/matter that could initiate the black hole
laser. Should we observe an electromagnetic counterpart to a binary black-hole
merger in the future, it may be worth while to develop techniques that can
test whether that light was due to the black-hole induced simulated emission
of radiation, giving us the first direct experimental verification of
(stimulated) Hawking radiation. Funding statement This research was
unsupported.

###### Acknowledgements.

I am indebted to my collaborators in black hole information theory: Greg Ver
Steeg and Kamil Brádler. I also thank numerous friends and colleagues who
patiently listened to my arguments that black holes do not violate any laws:
Paul Davies, Nigel Goldenfeld, Nicolas Cerf, Arend Hintze, Claus Wilke, and
Richard Lenski. I dedicate this contribution to the late Jonathan P. Dowling,
who led the quantum computing group at the Jet Propulsion Laboratory where the
ideas presented in this article were first hatched.

##  Appendix A Density matrix of outgoing radiation for arbitrary absorptivity

The density matrix of outgoing radiation in region I when $m$ particles are
incident (notation $k|m$) can be calculated from the outgoing demsity matrix
$|\psi\rangle_{\rm out}\langle\psi|$ (constructed from the out-state Eq.
(III.30)) by repeated application of the disentangling theorems for SU(2) and
SU(1,1), and tracing over the degrees of freedom of region II (modes $b_{k}$
and $c_{k}$). If no antiparticles are accreting on the horizon, the
antiparticle part of the density matrix factorizes again and we can write
$\rho_{\rm I}=\rho_{k|m}\otimes\rho_{-k|0}$ where (I omit the index $k$ in the
particle numbers and the coefficients $\alpha$, $\beta$ and $\gamma$ for
succinctness in the following)

$\displaystyle\rho_{k|m}=\sum_{n=0}^{\infty}p(n|m)|n\rangle\langle n|\;,$
(A.1)

with

$\displaystyle p(n|m)$ $\displaystyle=$ $\displaystyle
Z_{m}^{2}\sum_{l=0}^{\infty}\frac{n!\,l!}{m!(l-m+n)!}\left[\sum_{i=0}^{\min(n,m)}(-1)^{i}{m\choose
i}{{l-m+n}\choose{n-i}}\Bigl{(}\frac{\gamma^{2}}{\alpha(1+\alpha)}\Bigr{)}^{i}\right]^{2}$
$\displaystyle\times$
$\displaystyle\Bigl{[}\frac{\beta^{2}(1+\alpha)^{2}}{(1+\alpha+\beta^{2})^{2}}\Bigr{]}^{n}\Bigl{[}\frac{\beta^{2}\gamma^{2}}{(1+\alpha+\beta^{2})^{2}}\Bigr{]}^{l}$

where

$\displaystyle
Z_{m}^{2}=\left(\frac{1+\alpha}{1+\alpha+\beta^{2}}\right)^{2}\left(\frac{\alpha^{2}(1+\alpha)^{2}}{\gamma^{2}}\right)^{m}\;.$
(A.3)

It is possible to rewrite this expression141414Converting the double sums in
(A) into the single sum in (A.4) is highly non-trivial and took two years to
discover after the identity was confirmed numerically by G. Ver Steeg. for
$p(n|m)$, the probability to detect $n$ outgoing particles at
${\mathscr{I}}^{+}$ if $m$ were incident on the static black hole, by using a
resummation technique described by Panangaden and Wald Panangaden and Wald
(1977) to read

$\displaystyle
p(n|m)=R_{nm}\sum_{i=0}^{\min(n,m)}(-1)^{i}{{m}\choose{i}}{{m+n-i}\choose{n-i}}\left(1-\frac{\gamma^{2}}{\alpha^{2}\beta^{2}}\right)^{i}$
(A.4)

with

$\displaystyle
R_{nm}=\frac{1}{1+\beta^{2}}\left(\frac{\beta^{2}}{1+\beta^{2}}\right)^{m+n}\left(\frac{\alpha^{2}}{\beta^{2}}\right)^{m}\;,$
(A.5)

Expression (A.4) agrees precisely with the conditional probability derived by
Bekenstein and Meisels Bekenstein and Meisels (1977) using maximum entropy
methods, and by Panangaden and Wald Panangaden and Wald (1977) in quantum
field theory (but using a very different method than the one described here).
While the expression given by Bekenstein and Meisels looks quite different
from (A.4) as they write (with $x=\omega/T_{\rm BH}$ as before)

$\displaystyle p(n|m)$ $\displaystyle=$
$\displaystyle\frac{(e^{x}-1)e^{mx}\,\Gamma^{n+m}}{(e^{x}-1+\Gamma)^{n+m+1}}\sum_{i=0}^{\min(n,m)}\frac{(-1)^{i}(m+n-i)!}{i!(m-i)!(n-i)!}$
(A.6) $\displaystyle\times$
$\displaystyle\biggl{[}1-2\frac{1-\Gamma}{\Gamma^{2}}(\cosh(x)-1)\biggr{]}^{i}\;,$

we can nevertheless see it agrees with (A.4) (and therefore also with (A)) by
noting, for example, that

$\displaystyle
2\frac{1-\Gamma}{\Gamma^{2}}(\cosh(x)-1)=\frac{\gamma^{2}}{\alpha^{2}\beta^{2}}\;.$
(A.7)

and

$\displaystyle\frac{(e^{x}-1)e^{mx}\,\Gamma^{n+m}}{(e^{x}-1+\Gamma)^{n+m+1}}=R_{nm}\;.$
(A.8)

The expression (A.6) has the advantage that it manifestly observes the
detailed balance condititions, something that is not immediately apparent from
Eqs. (A) and (A.4).

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# RF regulation with superconducting cavities and beam operation using a
frequency shifted cavity

S. Pfeiffer<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany  V. Ayvazyan  Deutsches Elektronen-
Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  J. Branlard  Deutsches
Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  T. Buettner
Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany
S. Choroba  Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607
Hamburg, Germany  B. Faatz  Shanghai Advanced Research Institute, Chinese
Academy of Sciences, Haike Road 99, Shanghai 201210, China  K. Honkavaara
Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany
V. Katalev  Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607
Hamburg, Germany  H. Schlarb  Deutsches Elektronen-Synchrotron DESY, Notkestr.
85, 22607 Hamburg, Germany  C. Schmidt  Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany  S. Schreiber  Deutsches Elektronen-
Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  A. Sulimov  Deutsches
Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  E. Vogel
Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany
H. Weise  Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg,
Germany

###### Abstract

The free-electron laser FLASH at DESY and the European XFEL are operated with
superconducting radio frequency cavities and supply beam to several user
experiments. The switching time between experiments is limited to dozens of
microseconds. This contribution will show a regulation with a frequency
shifted superconducting cavity to manipulate and change the accelerating
properties of electron bunches with 250 kHz.The main challenge of the concept
presented in this contribution can be summarized in this statement: finding a
way to modulate the energy of individual bunches in a single-source multiple-
cavity scheme for a potential CW upgrade of the EuXFEL.

Free-electron laser, RF gun, Control system, Frequency control, NRF cavity

##  I Introduction

The Free-electron LASer in Hamburg (FLASH) at Deutsches Elektronen-SYnchrotron
(DESY) started user operation in 2005 as the first free-electron laser for XUV
and soft X-ray radiation, [1]. It is operated in the self-amplified
spontaneous emission (SASE) mode and currently covers a wavelength range from
4 nm to 90 nm with photon pulse durations between 10 fs and 200 fs depending
on the undulator section, FLASH1 and FLASH2. FLASH1 is currently equipped with
fixed gap undulator, and offers five beamlines for users in the ’Albert
Einstein’ experimental hall. The variable gap undulators of the ’Kai Siegbahn’
experimental hall (FLASH2) enhances significantly the capacities for XUV and
soft X-ray FEL users at DESY since 2014 [2]. A schematic layout of FLASH is
depicted in Fig. 1.

The European X-ray Free Electron Laser (EuXFEL), in user operation since
September 2017, is the largest SRF accelerator and offers currently the
world’s highest peak brilliance in the soft and hard X-rays radiations [3, 4].

Figure 1: Schematic layout of the FLASH facility, [5]. The electron gun is on
the left, the experimental halls are on the right (components are not to
scale). Behind the last accelerating module, the beam is switched between
FLASH1, which is the undulator line in operation since 2005, and FLASH2 in
operation since 2014.

These two unique accelerators are based on TESLA superconducting radio
frequency (SRF) cavities, housed into 8-cavity cryomodules, the building brick
of these accelerators. FLASH has 7 such 1.3 GHz cryomodules for a final energy
of 1.25 GeV, while EuXFEL has 97 for a final energy of 17.5 GeV.

Both FLASH and EuXFEL accelerators operate in burst mode where the SRF
cavities are filled at a 10-Hz repetition rate, for a duration of 1.4 msec per
RF pulse, see Fig. 2. The electron bunches are accelerated over these short
energy bursts (ranging from $600\,\mu s$ to $800\,\mu s$), with a 1-4.5 MHz
bunch repetition rate depending on user requirements and machine capability.
This pulsed mode of operation allows for very high gradients inside the SRF
resonators (20-30 MV/m) while keeping the dynamic heat load within the
capacity of the cryogenic plant and the liquid helium flow within the
cryomodule 2-phase pipe capability. While some user experiments are only
possible at the extreme high brilliance FLASH and EuXFEL can offer, other
experiments would benefit from a more relaxed bunch repetition rate (kHz
versus MHz) but provided continuously instead of being bunched. This calls for
a different mode of RF operation, namely continuous wave (CW). The feasibility
of a CW upgrade of EuXFEL is currently considered and several R&D projects are
being carried out at DESY, [6, 7, 8]. Since its early days as a demonstrator
of SRF accelerator-based FEL, FLASH continues to offer the opportunity to
develop new techniques for SRF accelerators, which can be applied at FLASH and
at XFEL. This contribution presents such an example of R&D originally
presented in 2016 [9]. In a synchrotron, the bunch-to-bunch energy spread of
the synchrotron frequency can be increased by pure amplitude modulation [10]
or with a detuned cavity [11]. Pure amplitude modulation requires a broadband
cavity, while operating a detuned cavity requires sufficient motor tuning
margin without changing the infrastructure. The latter were tested at FLASH
with its two beamlines in mid-2018 as a proof of concept for possible use in a
CW EuXFEL. Such an approach have been taken into account for pure amplitude
modulation maintaining the beam energy [12].

Figure 2: The RF is sent to the cavity during the fill time, while the
electron bunches are accelerated during the next pulse section where the field
should be kept constant to provide the same acceleration to all bunches. This
section is hence referred to as flat top. The following section is the RF
decay.

Both facilities share a common RF architecture, where a single RF source (a
multi beam klystron) is used to generate a pulsed high power RF wave, split
and distributed via waveguides into several cavities (up to 16 at FLASH and up
to 32 at EuXFEL). The RF regulation scheme has been developed accordingly,
introducing the so-called vector-sum (VS) control where the RF fields measured
inside all cavities of an RF station are added up into a single complex vector
used as input to the feedback controller, details of RF field sampling,
processing and regulation are given in [13]. The controlled output is
amplified at the single klystron and passively distributed to the multiple
cavities. This RF architecture choice (driven by cost savings and technology
availability at the design time) has an impact on the CW upgrade scenario. The
waveguide distribution associated with each cryomodule and the concept of a
single source providing power for multiple cavities are to be preserved.

In pulsed mode, the RF energy can be modulated from pulse to pulse, [14], or
within a single RF pulse, by acting on the accelerating gradient in amplitude
and phase during the beam acceleration phase of the RF pulse, [15]. The latter
is routinely done at FLASH and XFEL where operators define several beam
regions (typically 1-3) where different amplitude and phase parameters are
set, providing different compression parameters for bunches as a function of
their position within a single bunch train. The bunches of different beam
regions can be adjusted by RF settings to have e.g. a different chirp after
the bunch compression section. This concept allows operators to distribute
subsections of a given bunch train into different beam lines with various beam
properties. While this concept is well suited for pulsed-RF operation, it
cannot be applied in CW. As an alternative, it was proposed to use some
isolated cavities to provide not only energy but also amplitude and phase
modulation at the level of individual bunches [9]. This concept is explained
and experimentally demonstrated in this contribution.
The paper is organized as follows: The next section shortly introduces the
digital LLRF system with multi beamline support. Section 3 discusses the
operation of a detuned cavity with the normal LLRF operation scheme. Adding
and tuning beam as experiment is shown in Sec. 4 with measurements at FLASH
for various conditions. The last section concludes the paper and gives a short
outlook to a possible CW upgrade of EuXFEL.

##  II LLRF system and multi-beam operation

The digital LLRF system, implemented on an FPGA based on MicroTCA.4 standard,
is used to maintain and regulate the operating point of an RF station on
microsecond level, [16]. The LLRF regulation is depicted in Fig. 3.

Figure 3: LLRF system overview.

The baseband drive signal in I and Q (in-phase and quadrature) is up-converted
to operating frequency by a vector modulator before it is amplified from Watt
to Megawatt level by a klystron providing RF to up to 32 SRF cavities at a
time. The generated and stored RF field in a cavity is coupled out with a
probe pickup antenna and used for signal detection. This signal is down-
converted first to an intermediate frequency (IF) and finally into baseband.
The sum of all probe signals is compared to a set-point and its error is
processed with digital filters and a feedback controller before being added
this to the nominal drive signal. This closes the feedback loop. To relax the
feedback controller, the error of each pulse is used to optimize the digital
drive signal from pulse to pulse to minimize repetitive errors. Several beam
regions can be defined by splitting the digital drive signal into parts to
deliver beam to more than one experiment at FLASH. A kicker-septum system has
been installed behind the FLASH accelerator to split the electron bunch train
between two undulator lines. The typical rise time for the kicker is in the
order of 50 $\mu s$, [17]. This defines the lower bound on the transition time
between the two beam regions. This transition time is also used independently
of beam regions, to change RF related beam properties such as compression or
energy. [18]. The amplitude and phase change rate between two beam regions is
also limited by the small bandwidth of the SRF cavities, [16]. The operation
at FLASH and EuXFEL allows to have different flattops for different SASE
regions. By this an independent beam tuning of e.g. FLASH1 and FLASH2 can be
done.

##  III Frequency shifted cavity

Modulating the accelerating amplitude and phase of a single bunch on a
microsecond time scale can be done with different methods. Installing and
operating a normal conducting cavity with a high bandwidth in standing or
traveling wave mode configuration is one option. Another option that doesn’t
require changing the infrastructure is to operate one or more SRF cavities at
a different operating frequency. A cavity detuned by a subharmonic frequency
of the laser repetition rate will provide energy modulation to the fraction of
the bunches corresponding to the subharmonic order. In the case of FLASH a 250
kHz cavity detuning for a laser repetition rate of 500 kHz will modulate every
other bunch. There are several aspects to take into consideration when
choosing the cavity detuning: the detuning should be large enough to be
interesting from a user perspective (i.e. laser repetition rate). The
additional contribution to the vector-sum field regulation by the detuned
cavity operation should be minimized (i.e., several 100 kHz) and it should be
within the tuning range of the motorized resonator frequency tuner ($<$ 500
kHz) for typical TESLA resonators. In the remainder of this paper, we will
refer to the frequency-shifted cavity as the “slip” cavity. The name of “slip”
cavity comes from the fact that the field inside the detuned cavity rotates at
a frequency slower than the other cavities in a phasor diagram representation,
hence providing frequency slippage.

### Slip cavity basics

Bunch by bunch SASE fine tuning using a slip cavity is illustrated in Fig. 4.

Figure 4: Amplitude (black) of the 1.3 GHz resonance frequency and the
superposed contribution of a detuned frequency (grey), both mapped into
baseband representation. The bunches with repetition rate twice the detuned
cavity frequency are shown in blue and red.

Manipulating every bunch by the operation of such a slip cavity changes the RF
property of every bunch in amplitude and phase. If for instance the second
part of the bunch train (visualized by red charge components) is to be
optimized, the first part (illustrated by blue bunches) will change the RF
field as well. To avoid this, the second part is to be corrected by the
remaining (in this case 7) cavities which are still operated in vector-sum
configuration, shown in Fig. 5.

Figure 5: RF field for cavities operated in vector-sum and slip cavity (C7) in
complex plane. The slip cavity is changing phase by $\pi$ for every other
bunch, depicted as $\phi$ seen by first bunch $b_{1}$ and second bunch $b_{2}$
or for the given slip cavity operating frequency for every odd and even bunch,
respectively.

### Slip cavity operation with VS

Consider a slip cavity operating with some amplitude and a zero phase shift
between the slip cavity RF field and the vector-sum at the time of the first
bunches. Every odd bunch is accelerated and every even bunch decelerated due
to the energy modulation of the slip cavity.
If the slip cavity is operated with a 90 degree phase difference from the
operation of the VS cavities, a change in slip cavity amplitude would lead to
a phase change for every bunch while passing all 8 cavities, see Fig. 5. E.g.
every odd bunch sees a decrease in phase (blue arrow) and every even bunch
(red arrow) sees an increase in phase. The previous nominal operating point is
depicted by the black arrow as sum of 7 cavities, each depicted as dashed
arrow. Hereby, part of the bunches increase and the other part decrease their
compression. To correct for this, the RF field of the blue arrow is to be
corrected such that the blue and black arrows match again. This allows to tune
only every even bunch without interfering with the odd bunches.

### Driving the slip cavity

The LLRF system is driving 8 cavities with one klystron. Cavities 1 to 6 and
cavity 8 are added to the vector-sum, while cavity 7 is operated with a
frequency detuning of -250 kHz. As seen in Fig. 6 and 7 the forward signal to
all cavities is the same and differs only by its amplitude and phase from non-
uniform power distribution and forward phasing. The latter can be corrected
using phase shifters to compensate for waveguide length differences.

Figure 6: Cavity 1 forward, reflected and probe signal. The mean value is
shown as a solid line, while the bright dots represent several consecutive RF
pulses.

The reflected signal differs in pulse shape since cavity 1 is filled with
energy and the reflected signal reduces with external coupling factor defining
the cavity bandwidth. This is different for cavity 7 operated at -250 kHz.
Here, the entire forward signal is reflected since the 1.3 GHz baseband
frequency does not match the operating frequency of the cavity. Nevertheless,
the cavity is filled with RF with frequency difference of -250 kHz, visible as
decay of the oscillations on the reflected signal. Hence the cavity is filled
with this frequency and the amplitude is increasing as for cavity 1. The fast
phase change is related to frequency offset and given as
$d\phi/dt=\Delta\omega=\omega_{0}-\omega_{0,C_{7}}$, with
$\omega_{0}=2\pi\cdot 1.3$ GHz as operating frequency and $\omega_{0,C_{7}}$
as operating frequency of cavity 7.

Figure 7: Cavity 7 forward, reflected and the demodulated (without -250 kHz
frequency offset) and scaled probe signal.

##  IV Beam operation at FLASH

The following conditions were used for the beam operation study at FLASH.
Cavity 7 of the first accelerating structure (ACC1) was detuned by -250 kHz
and taken out of the vector-sum feedback control. Therefore its contribution
is not added to the remaining 7 cavities, used for VS regulation. The previous
operating amplitude for VS can also be achieved with only 7 cavities in VS.
The VS regulation is operated with an iterative learning control (ILC)
algorithm, reducing repetitive errors from pulse to pulse, and feedback.
Learning for the ILC is switched off while corrections are kept when operating
the system with the slip cavity. This ensures that the systems does not learn
from the additional drive signal. The feedback is kept active to ensure
minimal error between setpoint and VS during operation. The slip cavity can be
controlled in amplitude and phase with feedforward only, see Fig. 8.

Figure 8: Graphical user interface to adjust the parameters for ACC1 - Cavity
7 during operation. The main operation parameters are amplitude and phase
setpoints, detuning frequency etc. can be adjusted on left side. The middle
plot shows the drive amplitude and phase for slip cavity only (black) and as
modulated superposition with the ILC correction table. The demodulated slip
cavity RF field probe signal is shown on the right.

The 250 kHz modulated drive signal is added to the VS drive signal in baseband
before up-conversion. To add the drive contribution of the slip cavity, we
make use of the ILC table. Learning is enabled first without the slip cavity;
the learned correction is then frozen and the slip cavity drive signal is
added in I and Q to the total ILC correction table. ILC learning is disabled
while driving the slip cavity to avoid overwriting the tables.

The time period with the operational conditions during the study time is
summarized in Tab. 1. Figure 9 shows the main result of slip cavity operation
for about 20 minutes under different conditions.

Table 1: Operational conditions # | Time | Las. 1 | Las. 2 | Slip | SASE | comment
---|---|---|---|---|---|---
| [s] | q=0.2nC | q=0.4nC | cav. | [$\mu$J] |
I. | 0-100 | - | 500kHz | OFF | 82 | Reference
II. | 100-300 | 250kHz | 250kHz | OFF | 40 | Laser change
III. | 300-560 | 250kHz | 250kHz | ON | 70 | Tuning
IV. | 560-1000 | 250kHz | 250kHz | ON | 75 | Operation
V. | 1000-1200 | 250kHz | 250kHz | OFF | 40 | OFF

### I. Setup of beam reference

We started with zero gradient in slip cavity and restored transmission through
accelerator with 7 cavities in VS with one laser operated at 500 kHz. This
data set serves as reference for nominal beam operation at FLASH. This
operation is maintained by using a slow feedback system looking on the first
bunch and correcting the VS amplitude and phase with beam property changes
such as arrival time and compression related to operation of the slip cavity,
[19]. This allows independent tuning of the second beam part.

Figure 9: Main result for operation in the different time periods given in
seconds with 2 different lasers and 2 different bunch charges. RF and beam
signals starting from 25-Sep-2018 01:35:29. The colors corresponds to first,
respectively second bunch.

### II. Change to 2 laser operation

Laser 1 is switched on after 100 s producing interleaved bunches (the even
bunches). The charge for these bunches is about half the charge of Laser 2.
Switching on Laser 1 reduces the SASE intensity from 80 $\mu J$ to 40 $\mu J$
since this ion-current detector shows the mean pulse energy [20]. With now
twice the number of bunches, assuming the same SASE intensity for the odd
bunches (bunch properties, i.e. BCM and arrival time remain same) and expected
no SASE intensity for the even bunches, the shown intensity is reduced by
half. This reduction by a factor of two and the argumentation are based on
experience at FLASH. It could not be measured directly at bunch resolution
during the experiment.

### III. Tuning and IV. Operation with slip cavity

The slip cavity has been switched on after 300 s. The slow RF feedback keeps
the beam properties for odd bunches constant acting on the amplitude and the
phase of the first accelerating structure (ACC1). It is used to compensate the
additional phase shift and energy gain induced by the slip cavity. After some
adaptation steps and further manual tuning of the slip cavity, the final mean
SASE level of about 75 $\mu J$ is reached. Because the beam parameters of the
odd bunches did not change and only the even bunches have a significantly
different compression signal in BC3, we conclude that the even bunches changed
from almost no lasing to a SASE level which is comparable to the odd bunches,
ignoring small changes in other beam parameters.

### V. Switching off the slip cavity

The slip cavity is switched off after 1000 s while the beam generated with
laser 1 is still present, recovering the SASE intensity level of 40 $\mu J$
after convergence of the slow RF feedbacks. This experiment shows the
usability of a slip cavity which can be used to change beam properties on
microsecond level.

##  V Conclusion/Outlook

The contribution shows the usability of beam manipulation using a cavity
operated detuned with a multiple of the bunch repetition rate for possible CW
operation scheme. Different user experiments can be served and beam properties
can be varied on upper kHz repetition rate. This is hardly possible with
bandwidth limitation of SRF cavities and no infrastructure change. We show
that the operation with only 1 LLRF system is possible, however, certain
algorithms like ILC need to separate the operation in frequency domain such
that any superposition of VS correction by ILC and additional frequency
modulation do not start influencing each other. Hereby it would be beneficial
to add a second LLRF system for detuned cavity with its own high power drive
chain. This would also open the possibility for fast feedback and slow
correction of the detuned cavity in amplitude and phase.

##  VI Acknowledgment

The authors acknowledge support from DESY (Hamburg, Germany), a member of the
Helmholtz Association HGF. The work was carried out when B. Faatz was still
employed as a scientist at DESY.

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