# Flying atom back-reaction and mechanically generated photons from vacuum

Alberto Mercurio<EMAIL_ADDRESS>Dipartimento di Scienze
Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Università
di Messina, I-98166 Messina, Italy  Simone De Liberato  School of Physics and
Astronomy, University of Southampton, Southampton, SO17 1BJ, United Kingdom
Franco Nori  Theoretical Quantum Physics Laboratory, Cluster for Pioneering
Research, RIKEN, Saitama 351-0198, Japan  Center for Quantum Computing, RIKEN,
Wako-shi, Saitama 351-0198, Japan  Physics Department, The University of
Michigan, Ann Arbor, Michigan 48109-1040, USA.  Salvatore Savasta
<EMAIL_ADDRESS>Dipartimento di Scienze Matematiche e Informatiche,
Scienze Fisiche e Scienze della Terra, Università di Messina, I-98166 Messina,
Italy  Roberto Stassi  Dipartimento di Scienze Matematiche e Informatiche,
Scienze Fisiche e Scienze della Terra, Università di Messina, I-98166 Messina,
Italy  Center for Quantum Computing, RIKEN, Wako-shi, Saitama 351-0198, Japan

###### Abstract

We investigate the dynamics of a two-level atom flying through a photonic
cavity when the light-matter interaction is in the ultrastrong coupling
regime. We adopt a closed full quantum description that takes into account the
quantization of the atom center-of-mass motion in addition to its internal
degree of freedom and to the quantized photonic cavity field. We find that
multiple qualitatively different dynamical regimes are achievable according to
two key figures of merit: the ratio between the kinetic energy and the bare
excitation energies, and the product of these bare energies with the time the
atom takes to fly through the cavity. According to the values of those figures
of merit, the atom can be reflected by the _dressed_ vacuum, or can convert
part of its kinetic energy into real excitations which might be emitted out of
the cavity. In the first case, the atom experiences a _quantum regenerative
braking_ mechanism, based on temporary storage of energy into virtual
excitations.

In pioneering experiments, flying atoms passing through a cavity were used to
investigate the light-matter strong coupling regime [1, 2, 3, 4]. These
experiments allowed to test fundamental aspects of measurement theory [5, 6],
and led to proof-of-concept demonstrations of basic steps in quantum
information processing [7, 8].

Flying atoms in a cavity have also been proposed to explore vacuum emission
phenomena. For example, a theoretical study [9] has shown that ground-state
atoms, if accelerated through a high-Q microwave cavity, can generate
radiation with an intensity which can exceed that of the Unruh acceleration
radiation in free space by many orders of magnitude.

Several other theoretical works predicted the generation of particles from the
vacuum upon dynamical modulation of the cavity system: varying in time the
boundary condition of an electromagnetic field (dynamical Casimir effect) [10,
11, 12], moving a qubit inside a single-mode cavity [13], having electrical
currents transit through a photonic resonator [14, 15], or an atom in systems
with broken Lorentz invariance [16].

In the ultrastrong coupling (USC) regime, in which the light-matter
interaction strength is comparable or larger than the bare frequencies of the
matter and/or light resonances, vacuum radiation has also been predicted to be
emitted when the atom-light interaction is non-adiabatically modulated [17,
18] or switched on or off [19, 20, 21].

Since its first experimental observation in doped quantum wells [22],
experimental progress in cavity quantum electrodynamics led to the achievement
of USC in many different systems and with ever-increasing coupling strengths
[23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
42]. This in turn stimulated an intense theoretical effort [41, 42, 43, 44,
45, 46, 47] and a rich phenomenology has been predicted to occur in such a
non-perturbative coupling regime. For example, a single photon can excite
multiple atoms simultaneously [48, 49], lasing emission is modified [50],
mechanical resonators can interact through virtual photons [51], electronic
ground-state configurations can be modified [52], and Higgs-like phenomena can
be observed breaking the parity symmetry of the system [53, 54]. The USC
regime could be also applied in quantum technologies [55].

Figure 1: Representation of an atom wavepacket passing through a cavity
resonator. Depending on its kinetic energy, the atom can leave the cavity
entangled with a photon mechanically generated from the vacuum.

In this article, we investigate the effects caused by an atomic two-level
system flying through a cavity, where the light-matter interaction is in the
USC regime. We consider the system initially prepared in its lowest internal
energy state (zero atomic and light excitations) with the atom flying toward
the cavity, and we calculate the quantum dynamics of the coupled system taking
into account the interaction between center-of-mass and internal degrees of
freedom of the atom. Such an interaction is due to the light-matter
interaction changing the ground state energy inside the cavity, which also
determines an effective potential barrier which can affect the center-of-mass
motion.

Simulating the time evolution of the coupled system as the atom passes through
the cavity, we find qualitatively different behaviours depending on two
figures of merit. The first is the initial kinetic energy of the atom when
compared to the height of the barrier. If the initial kinetic energy is much
lower than the barrier, the atom slows down and gets reflected, with at least
part of the kinetic energy temporarily stored in virtual excitations of the
atom-photon system. If the kinetic energy is higher than the barrier, the atom
passes instead through the cavity. The second figure of merit is instead the
product between the bare excitation frequencies of the system and the time the
atom takes to traverse the cavity. If this quantity is small, the atom
experiences a non-adiabatic modification in its interaction with the cavity
field. Then the dynamics becomes inelastic, and part of the kinetic energy is
converted into atomic and photonic real excitations entangled with the atomic
center-of-mass.

The experimental observation of the physics described in our theoretical
proposal requires to achieve the USC regime with single dipoles. While most
systems described in the introduction rely on collective coupling to reach
such a regime, there has been a sustained effort to reduce the number of
dipoles [56]. Moreover, single molecule USC has been achieved using
vibrational degrees of freedom in plasmonic picocavities [30]. Standard
electronic strong coupling has also been achieved in single molecules trapped
in plasmonic nanogaps [57], with the possibility to reach USC theoretically
predicted [58].

Results

The atom-cavity system is governed by a time-independent Hamiltonian, that
reads ($\hbar=1$)

$\hat{\mathcal{H}}^{\rm(\mathfrak{g})}=\frac{\hat{p}^{2}}{2m}+\mathcal{\hat{H}}_{\rm
R}^{(\rm\mathfrak{g})}~{},$ (1)

where $\mathfrak{g}=\\{\rm{c,d}\\}$ refers to the chosen gauge. We explicitly
indicate the gauge because, as better explained in the Methods section, gauge
invariance can cause non-trivial problems in the USC regime, and different
aspects of the physics are better understood in one gauge or the other [59,
60, 61, 62]. In the main body of the article we choose to use the Coulomb
gauge (c), while the comparisons with the dipole gauge (d) can be found in the
Methods section. The first term of the r.h.s. of Eq. (1) is the 1D kinetic
energy of the atom center-of-mass, where $\hat{p}=-id/dx$ is the linear
momentum and $m$ is the mass. The second term, $\mathcal{\hat{H}}_{\rm
R}^{(\rm c)}$, is instead a quantum Rabi Hamiltonian describing the
interaction between the atom and a single-mode cavity in the Coulomb gauge,
which is derived applying the generalized minimal coupling replacement [60],

$\displaystyle\mathcal{\hat{H}}_{\rm R}^{(\rm c)}$ $\displaystyle=$
$\displaystyle\omega_{c}\hat{a}^{\dagger}\hat{a}+\frac{1}{2}\omega_{a}\left\\{\hat{\sigma}_{z}\cos\left[2\eta(\hat{x})(\hat{a}+\hat{a}^{\dagger})\right]\right.$
(2) $\displaystyle+$
$\displaystyle\left.\hat{\sigma}_{y}\sin\left[2\eta(\hat{x})(\hat{a}+\hat{a}^{\dagger})\right]\right\\}~{}.$

Here $\omega_{c}$ and $\omega_{a}$ are, respectively, the cavity and atom
frequencies, $\hat{a}$ ($\hat{a}^{\dagger}$) is the annihilation (creation)
operator for the single-mode cavity field, and $\hat{\sigma}_{i}$ ($i=x,y,z$)
are Pauli matrices operating on the internal states of the two-level atom
$\\{\ket{g},\ket{e}\\}$.

For the sake of definiteness, we assume that the normalized space-dependent
light-matter coupling strength has a Gaussian shape centered in the cavity
center with width $\mu_{c}$ and maximal intensity $\eta_{0}$ [63, 64]

$\displaystyle\eta(x)=\eta_{0}\exp\left[-\frac{x^{2}}{2\mu_{c}^{2}}\right].$
(3)

This assumption is not essential and different field profiles could be
considered.

In the USC regime, the eigenstates of the Hamiltonian in Eq. (2) contain
virtual excitations if represented in the bare atom and photon basis, with the
$l$th state for the atom localised at position $x$ having the form

$\displaystyle\ket{l}_{x}=\sum_{n=0}^{\infty}{c_{l}^{(2n)}(x)\ket{g,2n}+c_{l}^{(2n+1)}(x)\ket{e,2n+1}},$
(4)

where the states $\ket{g,n}$ and $\ket{e,n}$ form a basis of the quantum Rabi
Hamiltonian, describing the atom in the ground or excited state with $n$
photons in the cavity. The exact form of the coefficients $c_{l}^{(i)}(x)$
depend on the coupling strength at position $x$, $\eta(x)$, and on the chosen
gauge.

Dynamics. We numerically study the full quantum dynamics of the system
applying the time evolution operator, $\hat{T}^{(c)}={\rm
exp}[-i\hat{H}^{(c)}t]$, to the initial state. In our simulations we do not
consider dissipation, which has been shown to have only a limited impact on
the population of virtual excitations [65]. At the initial time $t_{0}=0$, the
system’s state is assumed to be in the factorised form

$\displaystyle\ket{\Psi(t_{0})}=\ket{\varphi(t_{0})}\ket{g,0}.$ (5)

The center-of-mass initial wave-function, centered at $x_{0}$ and of width
$\mu_{s}$ is chosen as

$\displaystyle\bra{x}\ket{\varphi(t_{0})}=G\left(\frac{x-x_{0}}{\mu_{s}}\right)\exp(ik_{0}x),$
(6)

where $G(x)$ is a normalised symmetric Gaussian function of unit variance.
Initially, the atom is placed outside the cavity,
$\bra{\varphi(t_{0})}\hat{x}\ket{\varphi(t_{0})}=x_{0}$, with
$|x_{0}|\gg\mu_{c}$, implying $\eta(x_{0})\approx 0$, and the initial momentum
is directed towards the cavity,
$\bra{\varphi(t_{0})}\hat{p}\ket{\varphi(t_{0})}=k_{0}$, corresponding to an
initial velocity $v_{0}=k_{0}/m$.

Figure 2: Snapshots of the Gaussian wave-packets associated to the states
$\ket{g,0},\ket{e,1}$ and $\ket{g,2}$, when the initial normalized kinetic
energy is ${E}_{K}=0.02$. At $t=0\,\tau_{0}$, the atom and the cavity are both
in their ground state. At $t=0.5\,\tau_{0}$, when the atom is in the cavity,
the atom-light excitations increase, while at the same time the atom’s kinetic
energy decreases. At $t=\,\tau_{0}$, the atom leaves the cavity with conserved
initial kinetic energy. Here, $\omega_{a}=\omega_{c}$, $\eta_{0}=0.3$,
$\lambda_{0}\equiv 2\pi/k_{0}=\mu_{c}/10$.

In the USC coupling, the local vacuum energy depends on the strength of the
light-matter coupling. This can be better understood in the dipole gauge (see
Methods), where the quantum Rabi Hamiltonian has a term equal to
$\omega_{c}\eta^{2}(\hat{x})$ [60, 66]. In the case where the atom is
considered to be fixed inside the cavity, this contribution introduces only an
energy shift, and can often be neglected. However, when considering a space-
dependent coupling strength $\eta(x)$ [67], this term corresponds to a space-
dependent effective potential which generates a force affecting the atom
dynamics. Working in the USC regime (in which we expect the maximum value of
the normalized coupling $\eta_{0}$ to be of the order of $0.1$ or more), a
meaningful figure of merit to study the mechanical motion of the atom is
therefore the initial kinetic energy normalised over the photon frequency

$\displaystyle{E}_{K}=\frac{mv_{0}^{2}}{2\omega_{c}}.$ (7)

Moreover, we expect that the atom-cavity system remains in its ground state
only when the evolution is adiabatic, which in the present system should mean
that the time the atom takes to see an increase of $\eta(x)$ from $0$ to
$\eta_{0}$, is substantially longer than the optical period of the system. In
the quasi-resonant case $\omega_{a}\approx\omega_{c}$, which we will consider
in the rest of this paper, we can then introduce the adiabatic figure of merit

$\displaystyle\Xi=\frac{v_{0}}{\omega_{c}\mu_{c}},$ (8)

and expect the evolution to be adiabatic when $\Xi\ll 1$.

Figure 3: Snapshots of the Gaussian wave packets associated to the states
$\ket{g,0},\ket{e,1}$ and $\ket{g,2}$, when the initial normalized kinetic
energy is $E_{k}=40$. At $t=0\,\tau_{0}$, the atom and the cavity are both in
their ground state. At $t=0.5\,\tau_{0}$, when the atom enters the cavity,
part of the kinetic energy is converted into atom-light excitations. At
$t=\,\tau_{0}$, the atom leaves the cavity in a superposition of light-matter
excited states. Here, $\eta_{0}=0.3$.

We start our simulation choosing two different values of the normalised
kinetic energies $E_{K}=0.02$ and $E_{K}=40$, which will allow us to explore
two completely different dynamical regimes. We will measure the time in units
of $\tau_{0}=\absolutevalue{2x_{0}/v_{0}}$, which is the time the atom would
take to fly unimpeded through the cavity to a position symmetric to the
initial one with respect to the cavity center. In all the simulations we will
fix the following parameters: $\omega_{a}=\omega_{c}$, $\eta_{0}=0.3$,
$x_{0}=-(25/6)\mu_{c}$, and $k_{0}=10\times 2\pi/\mu_{c}$. The mass $m$ is
varied to change the initial kinetic energy.

In Figure 2, we show three snapshots of the wave-packet evolution for the
initial normalised kinetic energy ${E}_{K}=0.02$, corresponding to $\Xi\approx
6.6\times 10^{-4}$. At $t=0$ the wave packet is outside the cavity, and it
moves from the left to the right with initial velocity $v_{0}$. Before the
atom enters the cavity, it remains in the initial state $\ket{g,0}$, since
$\bra{\varphi(t)}\eta(\hat{x})\ket{\varphi(t)}\approx 0$. When the atom enters
the cavity, as $\Xi\ll 1$, it sees the ground-state energy change
adiabatically, and it thus remains approximately in its local ground state
$\ket{0}_{x}$. We observe that, approaching the cavity center, the populations
of virtual excitations forming the ground state (i.e., $\ket{e,1}$ and
$\ket{g,2}$) increase, meanwhile the group velocity of the wave packet
decreases (not shown in Fig. 2). At $t\simeq\tau_{0}/2$ the atom stops and
reverses its motion. The atom then moves backwards from right to left leaving
the cavity with momentum opposite of the initial one. At $t\simeq\tau_{0}$ the
atom has left the cavity with the kinetic energy and absolute value of the
linear momentum essentially coincident with the initial ones. The reflection
happens because the gradient of the vacuum energy as a function of the atom
position is positive when the atom enters the cavity, resulting in a force
pushing back the atom when the kinetic energy is lower than the effective
potential barrier. It can be shown that excitations in Fig. 2 for
$t=0.5\tau_{0}$ are virtual. A hint comes from the fact that essentially no
excitation is present when the atom leaves the cavity ($t=\tau_{0}$). A more
direct evidence will be provided later (see Fig. 6a).

In Figure 3 we show the numerical results obtained for an initial kinetic
energy ${E}_{K}=40$, corresponding to $\Xi\approx 1.3$. Not only in this case
the kinetic energy is much larger than the barrier height, allowing the atom
to pass through the cavity, but now the atom feels a non-adiabatic change of
the local coupling strength $\eta(x)$ entering the cavity. Consequently, the
system’s state does not adiabatically follow the local ground state and the
atomic and photonic excited states become populated. In this case the
interaction of the atom with the cavity converts a fraction of the atomic
kinetic energy into real excitations: both atomic excitations and real photons
can persist after the atom leaves ($\tau=\tau_{0}$) and eventually leaks out
of the system.

Figure 4: Rate of photons emitted as a function of time and the adiabatic
parameter $\Xi$. At both small and large kinetic energies there is no creation
of photons on the output, as explained in the text.

Emitted photon rates. The rate of photon emission out of a cavity is usually
assumed proportional to the mean cavity-photon number
$\expectationvalue{\hat{a}^{\dagger}\hat{a}}$. As better detailed in the
Methods section, in the USC regime this ceases to be true and the output has
to be calculated using instead the quantity
$\langle\hat{\mathcal{X}}^{-}\hat{\mathcal{X}}^{+}\rangle$ [68, 62], where the
operator $\hat{\mathcal{X}}^{+}(\hat{\mathcal{X}}^{-})$ is related to the
positive (negative) frequencies of the electric field operator, where

$\displaystyle\hat{\mathcal{X}}^{+}=\sum_{l,k>l}i\matrixelement{l}{(\hat{a}-\hat{a}^{\dagger})}{k}\outerproduct{l}{k},$
(9)

and $\hat{\mathcal{X}}^{-}=(\hat{\mathcal{X}}^{+})^{\dagger}$, with $\ket{l}$
the system eigenstates ordered for increasing values of the energy. However,
when the atom is outside the cavity, the cavity-matter coupling is zero, and
we can retain $\expectationvalue{\hat{a}^{\dagger}\hat{a}}$. In Figure 4, the
mean number of real photons
$\langle\hat{\mathcal{X}}^{-}\hat{\mathcal{X}}^{+}\rangle$ is shown as a
function of time for different values of the adiabatic parameter $\Xi$. We can
clearly see that photons are emitted only when $\Xi>1$. As $\Xi$ increases,
the emission slowly tapers off because the time the atom spends into the
cavity grows shorter. Eventually, it becomes shorter than the Rabi oscillation
period and the atom leaves the cavity before any meaningful interaction with
the photonic field has the time to happen.

Figure 5: a Expectation value of the normalized atom momentum as a function of
time for different initial kinetic energy values, ${E}_{K}=0.02,10,40$. For
${E}_{K}=0.02$, the atom is reflected and the momentum changes sign. For
${E}_{K}=10$, the atom passes through the cavity, there is no creation of
excitations, and the momentum comes back to its initial value. For
${E}_{K}=0.02$, we can observe creation of photons and atom excitations, and
the momentum accordingly decreases. b von-Neumann entropy as a function of
time for an atom entering the cavity with initial kinetic energy ${E}_{K}=40$.
Other parameters are the same adopted in the other figures.

Entanglement. At every instant, the state of the system can be expressed as

$\ket{\Psi(t)}=\sum_{i}c_{i}(x,t)\ket{\varphi(t)}\ket{R_{i}}~{},$ (10)

where $\ket{R_{i}}$ are states which span a basis of the quantum Rabi
Hamiltonian (e.g., $\ket{g,0}$, $\ket{g,1}$, $\ket{e,0},\ldots$). Although we
chose an initially factorizable state in Eq. 5, when the dynamics is non-
adiabatic the atomic and photonic excitations generated in the process are
entangled with the atom position. In Figure 5b we plot the entanglement
calculated using the von-Neumann entropy $S=-\Tr[\rho_{\rm
R}\log_{m}(\rho_{\rm R})]$, where $\rho_{\rm R}=\Tr_{\varphi}[\rho]$ is the
reduced density matrix obtained tracing out the atom center-of-mass motion.
The coefficient $m$ is a cut-off to the dimension of the Hilbert space
describing the atom and the cavity mode. In our simulation, we choose $m=4$,
since higher lying states have vanishing populations for the parameter range
considered. In Fig. 5b we see that, after the atom leaves the cavity, the
entanglement remains constant.

Discussion

We investigated the dynamics of a flying atom prepared initially in its lowest
internal energy state (e.g., the electronic ground state) that passes through
a cavity with zero photons and interacts with it in the USC regime. We were
thus able to show how the generation of excitations out of the vacuum affects
the center of mass dynamics. We also found that any temporary reduction of the
atomic velocity due to a gradient of light-matter interaction strength results
into a temporary energy storing in virtual excitations.

When the atom enters the cavity, it experiences a force due to the gradient in
the vacuum energy felt by the atom. This leads to a quantum regenerative
braking effect in which the kinetic energy is temporarily stored in virtual
atom-light excitations inside the cavity. Different cases can take place,
according to whether the final momentum and energy of the atom are changed by
the interaction. If the kinetic energy is much lower than the difference
between the ground state energies inside and outside the cavity, the atom is
reflected. If instead its kinetic energy is large enough, the atom is
transmitted. The condition to create real excitations is connected to the
adiabatic parameter $\Xi$. If it is small enough, the atom sees and adiabatic
change in the light-matter interaction and in its ground state. In this case,
the interaction is elastic, and the atom’s final kinetic energy is the same as
the initial one. In the opposite case, the interaction is non-adiabatic and
inelastic, as part of the initial kinetic energy is converted into real
excitations.

These predictions might motivate new experimental ways to convert virtual
photons into real ones. Our theoretical proposal could be experimentally
observed in plasmonic picocavities or using single molecules trapped in
plasmonic nanogaps. Nevertheless, the achievement of the USC with single
molecule is still a challenging topic. Moreover, the experimental verification
of this effect could verify also the signatures of the counter rotating terms,
which are only present in the USC regime.

Methods

Derivation of the quantum Rabi Hamiltonian. The quantum Rabi Hamiltonian in
Eq. 2 approximates the physics of an atom coupled to a photonic resonator as a
two-level system coupled to a single photonic mode. We are thus performing two
key approximations: neglecting higher-lying matter states and photonic modes.

The first approximation leads to major problems when the coupling between
light and matter enters in the USC regime, as gauge invariance breaks in the
reduced Hilbert space and the two usually employed gauges, the Coulomb one and
the dipolar one, provide completely different results, with the latter being
the most correct one [59]. Recently, a consistent description of the quantum
Rabi model was introduced, which is able to provide gauge-invariant physical
results in any interaction regime [60]. It has been shown that this
description is closely connected to lattice gauge theories [69]. Moreover, it
was shown in Ref. [61] that adding a time dependent coupling may also lead to
gauge-dependent predictions. These ambiguities have been solved in Ref. [62].
To derive the dynamics of our system, we take into account all these recent
results regarding gauge invariance.

The neglect of higher photonic modes had also been recognised as potentially
problematic for large values of the coupling strength [70], as it can lead to
unphysical superluminal signalling. Still, the main issue for the present work
regards the calculation of the energy barrier between the vacua outside and
inside the cavity. A multimode exact calculation would have to consider that
the cavity is effective at confining the electromagntic radiation up to a
cutoff frequency $\omega_{M}$, and the energy difference felt by the atom
inside and outside the cavity should then be calculated as a difference
between a sum over the discrete modes $\omega_{c}<\omega_{M}$ in the cavity
and an integral over the continuum extending from $0$ to $\omega_{M}$ outside.
This sort of calculation is similar to the ones performed when dealing with
the static Casimir force, and the result would similarly depends on the
specific geometry considered [71]. While a multimode quantum simulation is
numerically infeasible in our case, its results would mainly boil down to a
renormalization of the potential barrier felt by the atom, which can be taken
into account in our model as a phenomenological renormalisation of the
coupling constant $\eta_{0}$ without qualitatively modify our results.

The system Hamiltonian in the dipole gauge. By applying a Power-Zienau-Woolley
(PZW) transformation to Eq. (1) [72], we can derive the full Hamiltonian in
the dipole gauge, $\mathcal{\hat{H}}^{(\rm
d)}=\mathcal{\hat{T}}\mathcal{\hat{H}}^{(\rm c)}\mathcal{\hat{T}}^{\dagger}$,
where

$\displaystyle\mathcal{\hat{T}}=\exp\left[-i{\eta(\hat{x})}\hat{\sigma}_{x}(\hat{a}+\hat{a}^{\dagger})\right].$
(11)

The transformed Hamiltonian $\mathcal{\hat{H}}^{(\rm d)}$ has two terms: the
quantum Rabi Hamiltonian in dipole gauge $\mathcal{\hat{H}}_{\rm R}^{(\rm
d)}=\mathcal{\hat{T}}\mathcal{\hat{H}}_{\rm R}^{(\rm
c)}\mathcal{\hat{T}}^{\dagger}$, and the transformed kinetic contribution
$\mathcal{\hat{T}}\hat{p}^{2}\mathcal{\hat{T}}^{\dagger}/(2m)$. The first term
becomes

$\displaystyle\hat{\mathcal{H}}_{\rm
R}^{\rm(d)}=\omega_{c}\hat{a}^{\dagger}\hat{a}+\frac{1}{2}\omega_{a}\hat{\sigma}_{z}-i\omega_{c}\eta(\hat{x})\hat{\sigma}_{x}(\hat{a}-\hat{a}^{\dagger})+\omega_{c}\eta^{2}(\hat{x}),$

while the second one is

$\displaystyle\hat{\mathcal{T}}\hat{p}^{2}\hat{\mathcal{T}}^{\dagger}$
$\displaystyle=$
$\displaystyle\hat{p}^{2}+\hat{\sigma}_{x}\left(\hat{a}+\hat{a}^{\dagger}\right)\left[\eta^{\prime}(\hat{x})\hat{p}+\hat{p}\eta^{\prime}(\hat{x})\right]$
(13) $\displaystyle+$ $\displaystyle\eta^{\prime
2}(\hat{x})\left(\hat{a}+\hat{a}^{\dagger}\right)^{2},$

this follows from the property
$\hat{\mathcal{T}}\hat{p}\hat{\mathcal{T}}^{\dagger}=\hat{p}+\eta^{\prime}(\hat{x})\hat{\sigma}_{x}(\hat{a}+\hat{a}^{\dagger})$,
where $\eta^{\prime}(x)=d\eta(x)/dx$.

Figure 6: Comparison of the time evolution of the photon rates predicted by
the standard expectation values $\langle\hat{a}^{\dagger}\hat{a}\rangle$ in
the Coulomb (blue solid curve) and dipole gauge (orange dashed curve), and the
one calculated making use of the expectation value
$\langle\hat{\mathcal{X}}^{-}\hat{\mathcal{X}}^{+}\rangle$ (dotted green
curve) during the atom flight. a The initial normalised kinetic energy of the
atom is ${E}_{K}=1$. The dynamics is adiabatic and no photons are generated;
in fact $\langle\hat{\mathcal{X}}^{-}\hat{\mathcal{X}}^{+}\rangle$ is zero
along the atom trajectory. b The initial kinetic energy is ${E}_{K}=30$. In
this case the process is non-adiabatic and real photons are generated.

Figure 6 shows the mean photon number generated when the atom passes through
the cavity as a function of time for different initial kinetic energies. The
figure displays both the mean value of the real photons
$\langle\hat{\mathcal{X}}^{-}\hat{\mathcal{X}}^{+}\rangle$ and the mean values
of the bare photon number in the Coulomb and dipole gauge. In Figure 6a, the
atom has an initial kinetic energy comparable to the bare energies of the
system, ${E}_{K}=1$. When it enters the cavity, the evolution is _adiabatic_
and _no real photons are emitted_. The expectation value
$\langle\hat{\mathcal{X}}^{-}\hat{\mathcal{X}}^{+}\rangle$ is exactly zero
during the entire dynamics and does not depend on the chosen gauge. On the
contrary, $\langle\hat{a}^{\dagger}\hat{a}\rangle$ is different from zero
inside the cavity, and depends on the chosen gauge. A careful analysis
displayed in Ref. [62] shows that the Coulomb gauge mean value
$\langle\hat{a}^{\dagger}\hat{a}\rangle_{c}$ represents the correct mean value
for bare photons. Indeed, by changing to the dipole gauge, also the photonic
operators need to be transformed
$\hat{a}\to\hat{a}^{\prime}=\hat{\mathcal{T}}\hat{a}\hat{\mathcal{T}}^{\dagger}$,
obtaining in this case the same results using the Coulomb gauge
$\langle\hat{a}^{\prime}{}^{\dagger}\hat{a}^{\prime}\rangle_{d}=\langle\hat{a}^{\dagger}\hat{a}\rangle_{c}$.

In Figure 6b, the atom has an initial normalised kinetic energy much higher
than the bare energies of the system, ${E}_{K}=30$. When it enters the cavity,
the evolution of the system is _non-adiabatic_ and _real photons are
generated_. In fact, the expectation value
$\langle\hat{\mathcal{X}}^{-}\hat{\mathcal{X}}^{+}\rangle$ progressively
increases till the moment that the center of the atomic wave packet leaves the
cavity. Instead, the mean value $\langle\hat{a}^{\dagger}\hat{a}\rangle_{c}$,
in the Coulomb gauge, reaches its maximum when the center of the atomic wave-
packet arrives at the center of the cavity where the coupling is the largest,
and decreases afterwards.

Figure 7: Comparison between numerical and analytical calculations of the
output photon rate as a function of the initial normalised kinetic energy.

Mapping to time-dependent perturbation theory. Using a time-dependent
perturbation theory, we have analytically calculated the output photon rates.
When the initial kinetic energy is large enough, the atom velocity can be
considered constant along the trajectory, because the cavity does not affect
significantly the atom motion. With this assumption, we remove the motion
degree of freedom and consider only the simple quantum Rabi model with an
effective time-dependent coupling. In particular, the coupling follows again a
Gaussian shape (but in the time domain)
$\eta(t)=\eta_{0}\exp[-t^{2}/2\,\mu_{t}^{2}]$, with $\mu_{t}=\mu_{c}m/k_{0}$.
As a starting point, we expand the quantum Rabi Hamiltonian in the Coulomb
gauge in Eq. (2) up to the third order with respect to the photon operators.
With these approximations, the Hamiltonian becomes
$\hat{\mathcal{H}}^{\rm(c)}(t)\simeq\hat{\mathcal{H}}_{0}+\hat{\mathcal{V}}(t)$,
with
$\hat{\mathcal{H}}_{0}=\omega_{c}\hat{a}^{\dagger}\hat{a}+(\omega_{a}/2)\hat{\sigma}_{z}$,
and

$\displaystyle\hat{\mathcal{V}}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\omega_{a}\left[2\eta(t)(\hat{a}+\hat{a}^{\dagger})\hat{\sigma}_{y}-2\eta^{2}(t)(\hat{a}+\hat{a}^{\dagger})^{2}\hat{\sigma}_{z}\right.$
(14) $\displaystyle+$
$\displaystyle\left.\frac{8}{6}\eta^{3}(t)(\hat{a}+\hat{a}^{\dagger})^{3}\hat{\sigma}_{y}\right]~{}.$

Using first-order time-dependent perturbation theory, the transition
probability from the ground state $\ket{g,0}$ to the excited states $\ket{n}$
is given by

$c_{n}^{(1)}=-i\int_{-\infty}^{+\infty}\matrixelement{n}{\hat{\mathcal{V}}(t)}{g,0}e^{i(\omega_{n}-\omega_{g,0})t}dt~{}.$

For the states $\ket{e,1}$, $\ket{g,2}$ and $\ket{e,3}$, and in the case of
resonant condition $\omega_{a}=\omega_{c}$, we obtain

$\displaystyle c_{e,1}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{\eta_{0}}{\Xi}\sqrt{2\pi}e^{-2\frac{1}{\Xi^{2}}}+\frac{\eta_{0}^{3}}{\Xi}\sqrt{\frac{8\pi}{3}}e^{-\frac{2}{3}\frac{1}{\Xi^{2}}},$
$\displaystyle c_{g,2}^{(1)}$ $\displaystyle=$
$\displaystyle-i\frac{\eta_{0}^{2}}{\Xi}\sqrt{2\pi}e^{-\frac{1}{\Xi^{2}}},$
$\displaystyle c_{e,3}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{\eta_{0}^{3}}{\Xi}\sqrt{\frac{16\pi}{9}}e^{-\frac{8}{3}\frac{1}{\Xi^{2}}}~{}.$
(15)

The total photon rates can be calculated using
$\langle\hat{a}^{\dagger}\hat{a}\rangle$, because, at the end of the process,
it predicts correct results, as shown in Fig. 6. We obtain,

$\expectationvalue{\hat{a}^{\dagger}\hat{a}}_{\rm
out}\approx\absolutevalue{c_{e,1}^{(1)}}^{2}+2\absolutevalue{c_{g,2}^{(1)}}^{2}+3\absolutevalue{c_{e,3}^{(1)}}^{2}~{}.$
(16)

Therefore, we can analytically calculate the total photon rates substituting
Eqs. (Flying atom back-reaction and mechanically generated photons from
vacuum) in Eq. (16). Figure 7(a) shows the comparison between analytical and
numerical results of the output photon emission rate
$\langle\hat{a}^{\dagger}\hat{a}\rangle$ as a function of the kinetic energy.
As expected from the approximations made to obtain the analytical formula in
Eq. (16), both numerical and analytical results coincide in the limit of high
kinetic energies. Moreover, they remain in good agreement also for lower
kinetic energies. We observe that the analytical calculation overestimates the
generated photon rate, which is a direct consequence of neglecting the back-
reaction effect on the atom motion. Lowering the kinetic energy, as expected,
the generation of photons reduces and then tends to zero. Increasing the
kinetic energy increases the photon rate untill a maximum value, after that
the photon rate decreases because the atom passes trough the cavity faster
than the time required to generate excitations.

## References

  * Kaluzny  _et al._ [1983] Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, Observation of Self-Induced Rabi Oscillations in Two-Level Atoms Excited Inside a Resonant Cavity: The Ringing Regime of Superradiance, Phys. Rev. Lett. 51, 1175 (1983).
  * Meschede  _et al._ [1985] D. Meschede, H. Walther, and G. Müller, One-Atom Maser, Phys. Rev. Lett. 54, 551 (1985).
  * Rempe  _et al._ [1987] G. Rempe, H. Walther, and N. Klein, Observation of quantum collapse and revival in a one-atom maser, Phys. Rev. Lett. 58, 353 (1987).
  * Thompson  _et al._ [1992] R. J. Thompson, G. Rempe, and H. J. Kimble, Observation of normal-mode splitting for an atom in an optical cavity, Phys. Rev. Lett. 68, 1132 (1992).
  * Braginskii and Vyatchanin [1981] V. B. Braginskii and S. P. Vyatchanin, On the non destructive measurement of the energy of optical quanta (Russian Academy of Sciences, 1981) pp. 570–572.
  * Brune  _et al._ [2008] M. Brune, J. Bernu, C. Guerlin, S. Deléglise, C. Sayrin, S. Gleyzes, S. Kuhr, I. Dotsenko, J. M. Raimond, and S. Haroche, Process Tomography of Field Damping and Measurement of Fock State Lifetimes by Quantum Nondemolition Photon Counting in a Cavity, Phys. Rev. Lett. 101, 240402 (2008).
  * Raimond  _et al._ [2001] J. M. Raimond, M. Brune, and S. Haroche, Manipulating quantum entanglement with atoms and photons in a cavity, Rev. Mod. Phys. 73, 565 (2001).
  * Haroche [2013] S. Haroche, Nobel Lecture: Controlling photons in a box and exploring the quantum to classical boundary, Rev. Mod. Phys. 85, 1083 (2013).
  * Scully  _et al._ [2003] M. O. Scully, V. V. Kocharovsky, A. Belyanin, E. Fry, and F. Capasso, Enhancing Acceleration Radiation from Ground-State Atoms via Cavity Quantum Electrodynamics, Phys. Rev. Lett. 91, 243004 (2003).
  * Fulling and Davies [1976] S. A. Fulling and P. C. Davies, Radiation from a moving mirror in two dimensional space-time: conformal anomaly, Proc. R. Soc. London, Ser. A 348, 393 (1976).
  * Johansson  _et al._ [2009] J. R. Johansson, G. Johansson, C. M. Wilson, and F. Nori, Dynamical Casimir Effect in a Superconducting Coplanar Waveguide, Phys. Rev. Lett. 103, 147003 (2009).
  * Wilson  _et al._ [2011] C. M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J. R. Johansson, T. Duty, F. Nori, and P. Delsing, Observation of the dynamical Casimir effect in a superconducting circuit, Nature 479, 376 (2011).
  * Agustí  _et al._ [2021] A. Agustí, L. García-Álvarez, E. Solano, and C. Sabín, Qubit motion as a microscopic model for the dynamical Casimir effect, Phys. Rev. A 103, 062201 (2021).
  * Cirio  _et al._ [2016] M. Cirio, S. De Liberato, N. Lambert, and F. Nori, Ground State Electroluminescence, Phys. Rev. Lett. 116, 113601 (2016).
  * Cirio  _et al._ [2019] M. Cirio, N. Shammah, N. Lambert, S. De Liberato, and F. Nori, Multielectron ground state electroluminescence, Phys. Rev. Lett. 122, 190403 (2019).
  * Svidzinsky [2019] A. A. Svidzinsky, Excitation of a uniformly moving atom through vacuum fluctuations, Phys. Rev. Research 1, 033027 (2019).
  * Liberato  _et al._ [2007] S. D. Liberato, C. Ciuti, and I. Carusotto, Quantum Vacuum Radiation Spectra from a Semiconductor Microcavity with a Time-Modulated Vacuum Rabi Frequency, Phys. Rev. Lett. 98, 103602 (2007).
  * De Liberato  _et al._ [2009] S. De Liberato, D. Gerace, I. Carusotto, and C. Ciuti, Extracavity quantum vacuum radiation from a single qubit, Phys. Rev. A 80, 053810 (2009).
  * Garziano  _et al._ [2013] L. Garziano, A. Ridolfo, R. Stassi, O. Di Stefano, and S. Savasta, Switching on and off of ultrastrong light-matter interaction: Photon statistics of quantum vacuum radiation, Phys. Rev. A 88, 063829 (2013).
  * Stassi  _et al._ [2013] R. Stassi, A. Ridolfo, O. Di Stefano, M. J. Hartmann, and S. Savasta, Spontaneous Conversion from Virtual to Real Photons in the Ultrastrong-Coupling Regime, Phys. Rev. Lett. 110, 243601 (2013).
  * Stefano  _et al._ [2017] O. D. Stefano, R. Stassi, L. Garziano, A. F. Kockum, S. Savasta, and F. Nori, Feynman-diagrams approach to the quantum Rabi model for ultrastrong cavity QED: stimulated emission and reabsorption of virtual particles dressing a physical excitation, New Journal of Physics 19, 053010 (2017).
  * Anappara  _et al._ [2009] A. A. Anappara, S. De Liberato, A. Tredicucci, C. Ciuti, G. Biasiol, L. Sorba, and F. Beltram, Signatures of the ultrastrong light-matter coupling regime, Phys. Rev. B 79, 201303 (2009).
  * Günter  _et al._ [2009] G. Günter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. De Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, _et al._ , Sub-cycle switch-on of ultrastrong light–matter interaction, Nature 458, 178 (2009).
  * Niemczyk  _et al._ [2010] T. Niemczyk, F. Deppe, H. Huebl, E. Menzel, F. Hocke, M. Schwarz, J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, _et al._ , Circuit quantum electrodynamics in the ultrastrong-coupling regime, Nature Physics 6, 772 (2010).
  * Scalari  _et al._ [2012] G. Scalari, C. Maissen, D. Turčinková, D. Hagenmüller, S. De Liberato, C. Ciuti, C. Reichl, D. Schuh, W. Wegscheider, M. Beck, _et al._ , Ultrastrong coupling of the cyclotron transition of a 2D electron gas to a THz metamaterial, Science 335, 1323 (2012).
  * Gambino  _et al._ [2014] S. Gambino, M. Mazzeo, A. Genco, O. Di Stefano, S. Savasta, S. Patanè, D. Ballarini, F. Mangione, G. Lerario, D. Sanvitto, and G. Gigli, Exploring Light–Matter Interaction Phenomena under Ultrastrong Coupling Regime, ACS Photonics 1, 1042 (2014).
  * Goryachev  _et al._ [2014] M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev, and M. E. Tobar, High-cooperativity cavity QED with magnons at microwave frequencies, Physical Review Applied 2, 054002 (2014).
  * Pirkkalainen  _et al._ [2015] J.-M. Pirkkalainen, S. Cho, F. Massel, J. Tuorila, T. Heikkilä, P. Hakonen, and M. Sillanpää, Cavity optomechanics mediated by a quantum two-level system, Nature communications 6, 1 (2015).
  * Baust  _et al._ [2016] A. Baust, E. Hoffmann, M. Haeberlein, M. Schwarz, P. Eder, J. Goetz, F. Wulschner, E. Xie, L. Zhong, F. Quijandría, _et al._ , Ultrastrong coupling in two-resonator circuit QED, Physical Review B 93, 214501 (2016).
  * Benz  _et al._ [2016] F. Benz, M. K. Schmidt, A. Dreismann, R. Chikkaraddy, Y. Zhang, A. Demetriadou, C. Carnegie, H. Ohadi, B. De Nijs, R. Esteban, _et al._ , Single-molecule optomechanics in “picocavities”, Science 354, 726 (2016).
  * George  _et al._ [2016] J. George, T. Chervy, A. Shalabney, E. Devaux, H. Hiura, C. Genet, and T. W. Ebbesen, Multiple Rabi splittings under ultrastrong vibrational coupling, Physical review letters 117, 153601 (2016).
  * Forn-Díaz  _et al._ [2017] P. Forn-Díaz, J. J. García-Ripoll, B. Peropadre, J.-L. Orgiazzi, M. Yurtalan, R. Belyansky, C. M. Wilson, and A. Lupascu, Ultrastrong coupling of a single artificial atom to an electromagnetic continuum in the nonperturbative regime, Nature Physics 13, 39 (2017).
  * Yoshihara  _et al._ [2017a] F. Yoshihara, T. Fuse, S. Ashhab, K. Kakuyanagi, S. Saito, and K. Semba, Superconducting qubit–oscillator circuit beyond the ultrastrong-coupling regime, Nature Physics 13, 44 (2017a).
  * Yoshihara  _et al._ [2017b] F. Yoshihara, T. Fuse, S. Ashhab, K. Kakuyanagi, S. Saito, and K. Semba, Characteristic spectra of circuit quantum electrodynamics systems from the ultrastrong- to the deep-strong-coupling regime, Phys. Rev. A 95, 053824 (2017b).
  * Bayer  _et al._ [2017] A. Bayer, M. Pozimski, S. Schambeck, D. Schuh, R. Huber, D. Bougeard, and C. Lange, Terahertz light–matter interaction beyond unity coupling strength, Nano letters 17, 6340 (2017).
  * Barachati  _et al._ [2018] F. Barachati, J. Simon, Y. A. Getmanenko, S. Barlow, S. R. Marder, and S. Kéna-Cohen, Tunable third-harmonic generation from polaritons in the ultrastrong coupling regime, Acs Photonics 5, 119 (2018).
  * Yoshihara  _et al._ [2018] F. Yoshihara, T. Fuse, Z. Ao, S. Ashhab, K. Kakuyanagi, S. Saito, T. Aoki, K. Koshino, and K. Semba, Inversion of qubit energy levels in qubit-oscillator circuits in the deep-strong-coupling regime, Physical Review Letters 120, 183601 (2018).
  * Shi  _et al._ [2018] T. Shi, Y. Chang, and J. J. García-Ripoll, Ultrastrong coupling few-photon scattering theory, Physical Review Letters 120, 153602 (2018).
  * Flower  _et al._ [2019] G. Flower, M. Goryachev, J. Bourhill, and M. E. Tobar, Experimental implementations of cavity-magnon systems: from ultra strong coupling to applications in precision measurement, New Journal of Physics 21, 095004 (2019).
  * Rajabali  _et al._ [2021] S. Rajabali, E. Cortese, M. Beck, S. De Liberato, J. Faist, and G. Scalari, Polaritonic nonlocality in light–matter interaction, Nature Photonics 15, 690 (2021).
  * Kockum  _et al._ [2019] A. F. Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, Ultrastrong coupling between light and matter, Nature Reviews Physics 1, 19 (2019).
  * Forn-Díaz  _et al._ [2019] P. Forn-Díaz, L. Lamata, E. Rico, J. Kono, and E. Solano, Ultrastrong coupling regimes of light-matter interaction, Rev. Mod. Phys. 91, 025005 (2019).
  * Ma and Law [2015] K. K. W. Ma and C. K. Law, Three-photon resonance and adiabatic passage in the large-detuning Rabi model, Phys. Rev. A 92, 023842 (2015).
  * Kockum  _et al._ [2017a] A. F. Kockum, A. Miranowicz, V. Macrì, S. Savasta, and F. Nori, Deterministic quantum nonlinear optics with single atoms and virtual photons, Phys. Rev. A 95, 063849 (2017a).
  * Kockum  _et al._ [2017b] A. F. Kockum, V. Macrì, L. Garziano, S. Savasta, and F. Nori, Frequency conversion in ultrastrong cavity QED, Scientific reports 7, 1 (2017b).
  * Stassi  _et al._ [2017] R. Stassi, V. Macrì, A. F. Kockum, O. Di Stefano, A. Miranowicz, S. Savasta, and F. Nori, Quantum nonlinear optics without photons, Phys. Rev. A 96, 023818 (2017).
  * Macrì  _et al._ [2018] V. Macrì, A. Ridolfo, O. Di Stefano, A. F. Kockum, F. Nori, and S. Savasta, Nonperturbative Dynamical Casimir Effect in Optomechanical Systems: Vacuum Casimir-Rabi Splittings, Phys. Rev. X 8, 011031 (2018).
  * Garziano  _et al._ [2016] L. Garziano, V. Macrì, R. Stassi, O. Di Stefano, F. Nori, and S. Savasta, One Photon Can Simultaneously Excite Two or More Atoms, Phys. Rev. Lett. 117, 043601 (2016).
  * Macrì  _et al._ [2020] V. Macrì, F. Nori, S. Savasta, and D. Zueco, Spin squeezing by one-photon–two-atom excitation processes in atomic ensembles, Phys. Rev. A 101, 053818 (2020).
  * Bamba and Ogawa [2016] M. Bamba and T. Ogawa, Laser under ultrastrong light-matter interaction: Qualitative aspects and quantitative influences by level and mode truncations, Phys. Rev. A 93, 033811 (2016).
  * Di Stefano  _et al._ [2019a] O. Di Stefano, A. Settineri, V. Macrì, A. Ridolfo, R. Stassi, A. F. Kockum, S. Savasta, and F. Nori, Interaction of Mechanical Oscillators Mediated by the Exchange of Virtual Photon Pairs, Phys. Rev. Lett. 122, 030402 (2019a).
  * Wang and De Liberato [2021] Y. Wang and S. De Liberato, Theoretical proposals to measure resonator-induced modifications of the electronic ground state in doped quantum wells, Phys. Rev. A 104, 023109 (2021).
  * Garziano  _et al._ [2014] L. Garziano, R. Stassi, A. Ridolfo, O. Di Stefano, and S. Savasta, Vacuum-induced symmetry breaking in a superconducting quantum circuit, Phys. Rev. A 90, 043817 (2014).
  * Wang  _et al._ [2022] S.-P. Wang, A. Ridolfo, T. Li, S. Savasta, F. Nori, Y. Nakamura, and J. You, Detecting the symmetry breaking of the quantum vacuum in a light–matter coupled system, arXiv preprint arXiv:2209.05747 (2022).
  * Stassi  _et al._ [2020] R. Stassi, M. Cirio, and F. Nori, Scalable quantum computer with superconducting circuits in the ultrastrong coupling regime, npj Quantum Inf. 6, 1 (2020).
  * Ballarini and De Liberato [2019] D. Ballarini and S. De Liberato, Polaritonics: from microcavities to sub-wavelength confinement, Nanophotonics 8, 641 (2019).
  * Chikkaraddy  _et al._ [2016] R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, Single-molecule strong coupling at room temperature in plasmonic nanocavities, Nature 535, 127 (2016).
  * Kuisma  _et al._ [2022] M. Kuisma, B. Rousseaux, K. M. Czajkowski, T. P. Rossi, T. Shegai, P. Erhart, and T. J. Antosiewicz, Ultrastrong Coupling of a Single Molecule to a Plasmonic Nanocavity: A First-Principles Study, ACS Photonics 9, 1065 (2022).
  * De Bernardis  _et al._ [2018] D. De Bernardis, P. Pilar, T. Jaako, S. De Liberato, and P. Rabl, Breakdown of gauge invariance in ultrastrong-coupling cavity QED, Phys. Rev. A 98, 053819 (2018).
  * Di Stefano  _et al._ [2019b] O. Di Stefano, A. Settineri, V. Macrì, L. Garziano, R. Stassi, S. Savasta, and F. Nori, Resolution of gauge ambiguities in ultrastrong-coupling cavity quantum electrodynamics, Nature Physics 15, 803 (2019b).
  * Stokes and Nazir [2021] A. Stokes and A. Nazir, Ultrastrong time-dependent light-matter interactions are gauge relative, Phys. Rev. Research 3, 013116 (2021).
  * Settineri  _et al._ [2021] A. Settineri, O. Di Stefano, D. Zueco, S. Hughes, S. Savasta, and F. Nori, Gauge freedom, quantum measurements, and time-dependent interactions in cavity QED, Phys. Rev. Research 3, 023079 (2021).
  * Brune  _et al._ [1994] M. Brune, P. Nussenzveig, F. Schmidt-Kaler, F. Bernardot, A. Maali, J. M. Raimond, and S. Haroche, From Lamb shift to light shifts: Vacuum and subphoton cavity fields measured by atomic phase sensitive detection, Phys. Rev. Lett. 72, 3339 (1994).
  * Brune  _et al._ [1996] M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, Quantum Rabi Oscillation: A Direct Test of Field Quantization in a Cavity, Phys. Rev. Lett. 76, 1800 (1996).
  * De Liberato [2017] S. De Liberato, Virtual photons in the ground state of a dissipative system, Nat. Commun. 8, 1465 (2017).
  * Mercurio  _et al._ [2022] A. Mercurio, V. Macrì, C. Gustin, S. Hughes, S. Savasta, and F. Nori, Regimes of cavity QED under incoherent excitation: From weak to deep strong coupling, Phys. Rev. Research 4, 023048 (2022).
  * Ian  _et al._ [2012] H. Ian, Y.-x. Liu, and F. Nori, Excitation spectrum for an inhomogeneously dipole-field-coupled superconducting qubit chain, Phys. Rev. A 85, 053833 (2012).
  * Ridolfo  _et al._ [2012] A. Ridolfo, M. Leib, S. Savasta, and M. J. Hartmann, Photon blockade in the ultrastrong coupling regime, Phys. Rev. Lett. 109, 193602 (2012).
  * Savasta  _et al._ [2021] S. Savasta, O. Di Stefano, A. Settineri, D. Zueco, S. Hughes, and F. Nori, Gauge principle and gauge invariance in two-level systems, Phys. Rev. A 103, 053703 (2021).
  * Sánchez Muñoz  _et al._ [2018] C. Sánchez Muñoz, F. Nori, and S. De Liberato, Resolution of superluminal signalling in non-perturbative cavity quantum electrodynamics, Nature Communications 9, 1924 (2018).
  * Dalvit  _et al._ [2011] D. Dalvit, P. Milonni, D. Roberts, and F. Da Rosa,  _Casimir Physics_, Vol. 834 (Springer, 2011).
  * Babiker and Loudon [1983] M. Babiker and R. Loudon, Derivation of the Power-Zienau-Woolley Hamiltonian in quantum electrodynamics by gauge transformation, Proc. R. Soc. A 385, 439 (1983).

## Acknowledgements

S.S. acknowledges the Army Research Office (ARO) (Grant No. W911NF-19-1-0065).
S.D.L. is supported by a Royal Society Research fellowship, the Philip
Leverhulme prize, and he acknowledges support from the Leverhulme Grant No.
RPG-2022-037. F.N. is supported in part by: Nippon Telegraph and Telephone
Corporation (NTT) Research, the Japan Science and Technology Agency (JST) [via
the Quantum Leap Flagship Program (Q-LEAP), and the Moonshot R&D Grant Number
JPMJMS2061], the Japan Society for the Promotion of Science (JSPS) [via the
Grants-in-Aid for Scientific Research (KAKENHI) Grant No. JP20H00134], the
Army Research Office (ARO) (Grant No. W911NF-18-1-0358), the Asian Office of
Aerospace Research and Development (AOARD) (via Grant No. FA2386-20-1-4069),
and the Foundational Questions Institute Fund (FQXi) via Grant No. FQXi-
IAF19-06.


# A New Global Embedding Approach to Study Hawking and Unruh Effects

Rabin Banerjee, Bibhas Ranjan Majhi
S. N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata-700098, India
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A new type of global embedding of curved space-times in higher dimensional
flat ones is introduced to present a unified description of Hawking and Unruh
effects. Our analysis simplifies as well as generalises the conventional
embedding approach.

##  1 Introduction

After Hawking’s famous work [1] \- the black holes radiate - known as Hawking
effect, it is now well understood that it is related to the event horizon of a
black hole. A closely related effect is the Unruh effect [2], where a similar
type of horizon is experienced by a uniformly accelerated observer on the
Minkowski space-time. A unified description of them was first put forwarded by
Deser and Levin [3, 4] which was a sequel to an earlier attempt [5]. This is
called the global embedding Minkowskian space (GEMS) approach. In this
approach, the relevant detector in curved space-time (namely Hawking detector)
and its event horizon map to the Rindler detector in the corresponding flat
higher dimensional embedding space [6, 7] and its event horizon. Then
identifying the acceleration of the Unruh detector, the Unruh temperature was
calculated. Finally, use of the Tolman relation [8] yields the Hawking
temperature. Subsequently, this unified approach to determine the Hawking
temperature using the Unruh effect was applied for several black hole space-
times [9, 10, 11, 12]. However the results were confined to four dimensions
and the calculations were done case by case, taking specific black hole
metrics. It was not clear whether the technique was applicable to complicated
examples like the Kerr-Newman metric which lacks spherical symmetry.

The motivation of this paper is to give a modified presentation of the GEMS
approach that naturally admits generalization. Higher dimensional black holes
with different metrics, including Kerr-Newman, are considered. Using this new
embedding, the local Hawking temperature (Unruh temperature) will be derived.
Then the Tolman formula leads to the Hawking temperature.

We shall first introduce a new global embedding which embeds only the
($t-r$)-sector of the curved metric into a flat space. It will be shown that
this embedding is enough to derive the Hawking result using the Deser-Levin
approach [3, 4], instead of the full embedding of the curved space-time. Hence
we might as well call this the reduced global embedding. This is actually
motivated from the fact that an $N$-dimensional black hole metric effectively
reduces to a $2$ -dimensional metric (only the ($t-r$)-sector) near the event
horizon by the dimensional reduction technique [13, 14, 15, 16] (for examples
see Appendix 1). Furthermore, this $2$-dimensional metric is enough to find
the Hawking quantities if the back scattering effect is ignored. Several
spherically symmetric static metrics will be exemplified. Also, to show the
utility of this reduced global embedding, we shall discuss the most general
solution of the Einstein gravity - Kerr-Newman space-time, whose full global
embedding is difficult to find. Since the reduced embedding involves just the
two dimensional ($t-r$)-sector, black holes in arbitrary dimensions can be
treated. In this sense our approach is valid for any higher dimensional black
hole.

The organization of the paper is as follows. In section 2 we shall find the
reduced global embedding of several black hole space-times which are
spherically symmetric. In the next section the power of this approach will be
exploited to find the Unruh/Hawking temperature for the Kerr-Newman black
hole. Finally, we shall give our concluding remarks.

##  2 Reduced global embedding

A unified picture of Hawking effect [1] and Unruh effect [2] was established
by the global embedding of a curved space-time into a higher dimensional flat
space [4]. Subsequently, this unified approach to determine the Hawking
temperature using the Unruh effect was applied for several black hole space-
times [9, 10], but usually these are spherically symmetric. For instance, no
discussion on the Kerr-Newman black hole has been given, because it is
difficult to find the full global embedding.

Since the Hawking effect is governed solely by properties of the event
horizon, it is enough to consider the near horizon theory. As already stated,
this is a two dimensional theory obtained by dimensional reduction of the full
theory. Its metric is just the ($t-r$)-sector of the original metric.

In the following sub-sections we shall find the global embedding of the near
horizon effective $2$-dimensional theory. Then the usual local Hawking
temperature will be calculated. Technicalities are considerably simplified and
our method is general enough to include different black hole metrics.

###  2.1 Schwarzschild metric

Near the event horizon the physics is given by just the two dimensional
($t-r$) -sector of the full Schwarzschild metric [13]:

$\displaystyle
ds^{2}=g_{tt}dt^{2}+g_{rr}dr^{2}=\Big{(}1-\frac{2m}{r}\Big{)}dt^{2}-\frac{dr^{2}}{1-\frac{2m}{r}}.$
(1)

It is interesting to see that this can be globally embedded in a flat $D=3$
space as,

$\displaystyle ds^{2}=(dz^{0})^{2}-(dz^{1})^{2}-(dz^{2})^{2}$ (2)

by the following relations among the flat and curved coordinates:

$\displaystyle
z^{0}_{out}=\kappa^{-1}\Big{(}1-\frac{2m}{r}\Big{)}^{1/2}\textrm{sinh}(\kappa
t),\,\,\,\
z^{1}_{out}=\kappa^{-1}\Big{(}1-\frac{2m}{r}\Big{)}^{1/2}\textrm{cosh}(\kappa
t),$ $\displaystyle
z^{0}_{in}=\kappa^{-1}\Big{(}\frac{2m}{r}-1\Big{)}^{1/2}\textrm{cosh}(\kappa
t),\,\,\,\
z^{1}_{in}=\kappa^{-1}\Big{(}\frac{2m}{r}-1\Big{)}^{1/2}\textrm{sinh}(\kappa
t),$ $\displaystyle z^{2}=\int
dr\Big{(}1+\frac{r_{H}r^{2}+r_{H}^{2}r+r_{H}^{3}}{r^{3}}\Big{)}^{1/2},$ (3)

where the surface gravity $\kappa=\frac{1}{4m}$ and the event horizon is
located at $r_{H}=2m$. The suffix “$in$” (“$out$”) refer to the inside
(outside) of the event horizon while variables without any suffix imply that
these are valid on both sides of the horizon. We shall follow these notations
throughout the paper. Now if a detector moves according to constant $r$
(Hawking detector) outside the horizon in the curved space, then the
corresponding Unruh detector moves on the constant $z^{2}$ plane and it will
follow the hyperbolic trajectory

$\displaystyle\Big{(}z^{1}_{out}\Big{)}^{2}-\Big{(}z^{0}_{out}\Big{)}^{2}=16m^{2}\Big{(}1-\frac{2m}{r}\Big{)}=\frac{1}{{\tilde{a}}^{2}}.$
(4)

This shows that the Unruh detector is moving in the
($z^{0}_{out},z^{1}_{out}$) flat plane with a uniform acceleration
${\tilde{a}}=\frac{1}{4m}\Big{(}1-\frac{2m}{r}\Big{)}^{-1/2}$. Then, according
to Unruh [2], the accelerated detector will see a thermal spectrum in the
Minkowski vacuum with the local Hawking temperature given by,

$\displaystyle T=\frac{\hbar{\tilde{a}}}{2\pi}=\frac{\hbar}{8\pi
m}\Big{(}1-\frac{2m}{r}\Big{)}^{-1/2}.$ (5)

So we see that with the help of the reduced global embedding the local Hawking
temperature near the horizon can easily be obtained.

Now the temperature measured by any observer away from the horizon can be
obtained by using the Tolman formula [8] which ensures constancy between the
product of temperatures and corresponding Tolman factors measured at two
different points in space-time. This formula is given by [8]:

$\displaystyle\sqrt{g_{tt}}~{}T=\sqrt{g_{0_{tt}}}~{}T_{0}$ (6)

where, in this case, the quantities on the left hand side are measured near
the horizon whereas those on the right hand side are measured away from the
horizon (say at $r_{0}$). Since away from the horizon the space-time is given
by the full metric, $g_{0_{tt}}$ must correspond to the $dt^{2}$ coefficient
of the full (four dimensional) metric.

For the case of Schwarzschild metric $g_{tt}=1-2m/r$, $g_{0_{tt}}=1-2m/r_{0}$.
Now the Hawking effect is observed at infinity ($r_{0}=\infty$), where
$g_{0_{tt}}=1$. Hence, use of the Tolman formula (6) immediately yields the
Hawking temperature:

$\displaystyle T_{0}={\sqrt{g_{tt}}}~{}T=\frac{\hbar}{8\pi m}.$ (7)

Thus, use of the reduced embedding instead of the embedding of the full metric
is sufficient to get the answer.

###  2.2 Reissner-Nordstr$\ddot{\textrm{o}}$m metric

In this case, the effective metric near the event horizon is given by [13],

$\displaystyle
ds^{2}=\Big{(}1-\frac{2m}{r}+\frac{e^{2}}{r^{2}}\Big{)}dt^{2}-\frac{dr^{2}}{1-\frac{2m}{r}+\frac{e^{2}}{r^{2}}}.$
(8)

This metric can be globally embedded into the $D=4$ dimensional flat metric
as,

$\displaystyle ds^{2}=(dz^{0})^{2}-(dz^{1})^{2}-(dz^{2})^{2}+(dz^{3})^{2}$ (9)

where the coordinate transformations are:

$\displaystyle
z^{0}_{out}=\kappa^{-1}\Big{(}1-\frac{2m}{r}+\frac{e^{2}}{r^{2}}\Big{)}^{1/2}\textrm{sinh}(\kappa
t),\,\,\,\
z^{1}_{out}=\kappa^{-1}\Big{(}1-\frac{2m}{r}+\frac{e^{2}}{r^{2}}\Big{)}^{1/2}\textrm{cosh}(\kappa
t),$ $\displaystyle
z^{0}_{in}=\kappa^{-1}\Big{(}\frac{2m}{r}-\frac{e^{2}}{r^{2}}-1\Big{)}^{1/2}\textrm{cosh}(\kappa
t),\,\,\,\
z^{1}_{in}=\kappa^{-1}\Big{(}\frac{2m}{r}-\frac{e^{2}}{r^{2}}-1\Big{)}^{1/2}\textrm{sinh}(\kappa
t),$ $\displaystyle z^{2}=\int
dr\Big{[}1+\frac{r^{2}(r_{+}+r_{-})+r_{+}^{2}(r+r_{+})}{r^{2}(r-r_{-})}\Big{]}^{1/2},$
$\displaystyle z^{3}=\int
dr\Big{[}\frac{4r_{+}^{5}r_{-}}{r^{4}(r_{+}-r_{-})^{2}}\Big{]}^{1/2}.$ (10)

Here in this case the surface gravity $\kappa=\frac{r_{+}-r_{-}}{2r_{+}^{2}}$
and $r_{\pm}=m\pm\sqrt{m^{2}-e^{2}}$. The black hole event horizon is given by
$r_{H}=r_{+}$. Note that for $e=0$, the above transformations reduce to the
Schwarzschild case (3). The Hawking detector moving in the curved space
outside the horizon, following a constant $r$ trajectory, maps to the Unruh
detector on the constant ($z^{2},z^{3}$) surface. The trajectory of the Unruh
detector is given by

$\displaystyle\Big{(}z^{1}_{out}\Big{)}^{2}-\Big{(}z^{0}_{out}\Big{)}^{2}=\Big{(}\frac{r_{+}-r_{-}}{2r_{+}^{2}}\Big{)}^{-2}\Big{(}1-\frac{2m}{r}+\frac{e^{2}}{r^{2}}\Big{)}=\frac{1}{{\tilde{a}}^{2}}.$
(11)

This, according to Unruh [2], immediately leads to the local Hawking
temperature $T=\frac{\hbar{\tilde{a}}}{2\pi}=\frac{\hbar(r_{+}-r_{-})}{4\pi
r_{+}^{2}\sqrt{1-2m/r+e^{2}/r^{2}}}$ which was also obtained from the full
global embedding [4]. Again, since in this case
$g_{0_{tt}}=1-2m/r_{0}+e^{2}/r_{0}^{2}$ which reduces to unity at
$r_{0}=\infty$ and $g_{tt}=1-2m/r+e^{2}/r^{2}$, use of Tolman formula (6)
leads to the standard Hawking temperature
$T_{0}=\sqrt{g_{tt}}~{}T=\frac{\hbar(r_{+}-r_{-})}{4\pi r_{+}^{2}}$.

###  2.3 Schwarzschild-AdS metric

Near the event horizon the relevant effective metric is [13],

$\displaystyle
ds^{2}=\Big{(}1-\frac{2m}{r}+\frac{r^{2}}{R^{2}}\Big{)}dt^{2}-\frac{dr^{2}}{\Big{(}1-\frac{2m}{r}+\frac{r^{2}}{R^{2}}\Big{)}},$
(12)

where $R$ is related to the cosmological constant $\Lambda=-1/R^{2}$. This
metric can be globally embedded in the flat space (9) with the following
coordinate transformations:

$\displaystyle
z^{0}_{out}=\kappa^{-1}\Big{(}1-\frac{2m}{r}+\frac{r^{2}}{R^{2}}\Big{)}^{1/2}\textrm{sinh}(\kappa
t),\,\,\
z^{1}_{out}=\kappa^{-1}\Big{(}1-\frac{2m}{r}+\frac{r^{2}}{R^{2}}\Big{)}^{1/2}\textrm{cosh}(\kappa
t),$ $\displaystyle
z^{0}_{in}=\kappa^{-1}\Big{(}\frac{2m}{r}-\frac{r^{2}}{R^{2}}-1\Big{)}^{1/2}\textrm{cosh}(\kappa
t),\,\,\,\
z^{1}_{in}=\kappa^{-1}\Big{(}\frac{2m}{r}-\frac{r^{2}}{R^{2}}-1\Big{)}^{1/2}\textrm{sinh}(\kappa
t),$ $\displaystyle z^{2}=\int
dr\Big{[}1+\Big{(}\frac{R^{3}+Rr_{H}^{2}}{R^{2}+3r_{H}^{2}}\Big{)}^{2}\frac{r^{2}r_{H}+rr_{H}^{2}+r_{H}^{3}}{r^{3}(r^{2}+rr_{H}+r_{H}^{2}+R^{2})}\Big{]}^{1/2},$
$\displaystyle z^{3}=\int
dr\Big{[}\frac{(R^{4}+10R^{2}r_{H}^{2}+9r_{H}^{4})(r^{2}+rr_{H}+r_{H}^{2})}{(r^{2}+rr_{H}+r_{H}^{2}+R^{2})(R^{2}+3r_{H}^{2})^{2}}\Big{]}^{1/2}$
(13)

where the surface gravity $\kappa=\frac{R^{2}+3r_{H}^{2}}{2r_{H}R^{2}}$ and
the event horizon $r_{H}$ is given by the root of the equation
$1-\frac{2m}{r_{H}}+\frac{r^{2}_{H}}{R^{2}}=0$. Note that in the
$R\rightarrow\infty$ limit these transformations reduce to those for the
Schwarzschild case (3). We observe that the Unruh detector on the
($z^{2},z^{3}$) surface (i.e. the Hawking detector moving outside the event
horizon on a constant $r$ surface) follows the hyperbolic trajectory:

$\displaystyle\Big{(}z^{1}_{out}\Big{)}^{2}-\Big{(}z^{0}_{out}\Big{)}^{2}=\Big{(}\frac{R^{2}+3r_{H}^{2}}{2r_{H}R^{2}}\Big{)}^{-2}\Big{(}1-\frac{2m}{r}+\frac{r^{2}}{R^{2}}\Big{)}=\frac{1}{{\tilde{a}}^{2}}$
(14)

leading to the local Hawking temperature
$T=\frac{\hbar{\tilde{a}}}{2\pi}=\frac{\hbar\kappa}{2\pi\Big{(}1-\frac{2m}{r}+\frac{r^{2}}{R^{2}}\Big{)}^{1/2}}$.
This result was obtained earlier [4], but with more technical complexities,
from the embedding of the full metric.

It may be pointed out that for the present case, the observer must be at a
finite distance away from the event horizon, since the space-time is
asymptotically AdS. Therefore, if the observer is far away from the horizon
($r_{0}>>r$) where $g_{0_{tt}}=1-2m/r_{0}+r_{0}^{2}/R^{2}$, then use of (6)
immediately leads to the temperature measured at $r_{0}$:

$\displaystyle
T_{0}=\frac{\hbar\kappa}{2\pi\sqrt{1-2m/r_{0}+r_{0}^{2}/R^{2}}}.$ (15)

Now, this shows that $T_{0}\rightarrow 0$ as $r_{0}\rightarrow\infty$; i.e. no
Hawking particles are present far from horizon.

##  3 Kerr-Newman metric

So far we have discussed a unified picture of Unruh and Hawking effects using
our reduced global embedding approach for spherically symmetric metrics,
reproducing standard results. However, our approach was technically simpler
since it involved the embedding of just the two dimensional near horizon
metric. Now we shall explore the real power of this new embedding.

The utility of the reduced embedding approach comes to the fore for the Kerr-
Newman black hole which is not spherically symmetric. The embedding for the
full metric, as far as we are aware, is not done in the literature.

The effective $2$-dimensional metric near the event horizon is given by [15,
16],

$\displaystyle
ds^{2}=\frac{\Delta}{r^{2}+a^{2}}dt^{2}-\frac{r^{2}+a^{2}}{\Delta}dr^{2},$
(16)

where

$\displaystyle\Delta=r^{2}-2mr+a^{2}+e^{2}=(r-r_{+})(r-r_{-});\,\,\,\
a=\frac{J}{m};$ $\displaystyle r_{\pm}=m\pm\sqrt{m^{2}-a^{2}-e^{2}}.$ (17)

The event horizon is located at $r=r_{+}$. This metric can be embedded in the
following $D=5$-dimensional flat space:

$\displaystyle
ds^{2}=\Big{(}dz^{0}\Big{)}^{2}-\Big{(}dz^{1}\Big{)}^{2}-\Big{(}dz^{2}\Big{)}^{2}+\Big{(}dz^{3}\Big{)}^{2}+\Big{(}dz^{4}\Big{)}^{2},$
(18)

where the coordinate transformations are

$\displaystyle
z^{0}_{out}=\kappa^{-1}\Big{(}1-\frac{2mr}{r^{2}+a^{2}}+\frac{e^{2}}{r^{2}+a^{2}}\Big{)}^{1/2}\textrm{sinh}(\kappa
t),\,\,\,\
z^{1}_{out}=\kappa^{-1}\Big{(}1-\frac{2mr}{r^{2}+a^{2}}+\frac{e^{2}}{r^{2}+a^{2}}\Big{)}^{1/2}\textrm{cosh}(\kappa
t),$ $\displaystyle
z^{0}_{in}=\kappa^{-1}\Big{(}\frac{2mr}{r^{2}+a^{2}}-\frac{e^{2}}{r^{2}+a^{2}}-1\Big{)}^{1/2}\textrm{cosh}(\kappa
t),\,\,\,\
z^{1}_{in}=\kappa^{-1}\Big{(}\frac{2mr}{r^{2}+a^{2}}-\frac{e^{2}}{r^{2}+a^{2}}-1\Big{)}^{1/2}\textrm{sinh}(\kappa
t),$ $\displaystyle z^{2}=\int
dr\Big{[}1+\frac{(r^{2}+a^{2})(r_{+}+r_{-})+r_{+}^{2}(r+r_{+})}{(r^{2}+a^{2})(r-r_{-})}\Big{]}^{1/2},$
$\displaystyle z^{3}=\int
dr\Big{[}\frac{4r_{+}^{5}r_{-}}{(r^{2}+a^{2})^{2}(r_{+}-r_{-})^{2}}\Big{]}^{1/2},$
$\displaystyle z^{4}=\int
dra\Big{[}\frac{r_{+}+r_{-}}{(a^{2}+r_{-}^{2})(r_{-}-r)}+\frac{4(a^{2}+r_{+}^{2})(a^{2}-r_{+}r_{-}+(r_{+}+r_{-})r)}{(r_{+}-r_{-})^{2}(a^{2}+r^{2})^{3}}$
$\displaystyle+\frac{4r_{+}r_{-}(a^{2}+2r_{+}^{2})}{(r_{+}-r_{-})^{2}(a^{2}+r^{2})^{2}}+\frac{rr_{-}-a^{2}+r_{+}(r+r_{-})}{(a^{2}+r_{-}^{2})(a^{2}+r^{2})}\Big{]}^{1/2}.$
(19)

Here the surface gravity $\kappa=\frac{r_{+}-r_{-}}{2(r_{+}^{2}+a^{2})}$. For
$e=0,a=0$, as expected, the above transformations reduce to the Schwarzschild
case (3) while only for $a=0$ these reduce to the Reissner-
Nordstr$\ddot{\textrm{o}}$m case (10).

As before, the trajectory adopted by the Unruh detector on the constant
($z^{2},z^{3},z^{4}$) surface corresponding to the Hawking detector on the
constant $r$ surface is given by the hyperbolic form,

$\displaystyle\Big{(}z^{1}_{out}\Big{)}^{2}-\Big{(}z^{0}_{out}\Big{)}^{2}=\kappa^{-2}\Big{(}1-\frac{2mr}{r^{2}+a^{2}}+\frac{e^{2}}{r^{2}+a^{2}}\Big{)}=\frac{1}{{\tilde{a}}^{2}}.$
(20)

Hence the local Hawking temperature is

$\displaystyle
T=\frac{\hbar{\tilde{a}}}{2\pi}=\frac{\hbar\kappa}{2\pi\sqrt{\Big{(}1-\frac{2mr}{r^{2}+a^{2}}+\frac{e^{2}}{r^{2}+a^{2}}\Big{)}}}.$
(21)

Finally, since $g_{tt}=1-\frac{2mr}{r^{2}+a^{2}}+\frac{e^{2}}{r^{2}+a^{2}}$
(corresponding to the near horizon reduced two dimensional metric) and
$g_{0_{tt}}=\frac{r_{0}^{2}-2mr_{0}+a^{2}+e^{2}-a^{2}{\textrm{sin}}^{2}\theta}{r_{0}^{2}+a^{2}{\textrm{cos}}^{2}\theta}$
(corresponding to the full four dimensional metric), use of the Tolman
relation (6) leads to the Hawking temperature

$\displaystyle
T_{0}=\frac{\sqrt{g_{tt}}}{\sqrt{(g_{0}{{}_{tt}})_{r_{0}\rightarrow\infty}}}~{}T=\frac{\hbar\kappa}{2\pi}=\frac{\hbar(r_{+}-r_{-})}{4\pi(r_{+}^{2}+a^{2})},$
(22)

which is the well known result [15].

##  4 Conclusion

We provide a new approach to the study of Hawking/Unruh effects including
their unification, initiated in [3, 4, 5], popularly known as global embedding
Minkowskian space-time (GEMS). Contrary to the usual formulation [3, 4, 5, 9,
10, 11], the full embedding was avoided. Rather, we required the embedding of
just the two dimensional ($t-r$)-sector of the theory. This was a consequence
of the fact that the effective near horizon theory is basically two
dimensional. Only near horizon theory is significant since Hawking/Unruh
effects are governed solely by properties of the event horizon.

This two dimensional embedding ensued remarkable technical simplifications
whereby the treatment of more general black holes (e.g. those lacking
spherical symmetry like the Kerr-Newman) was feasible. Also, black holes in
any dimensions were automatically considered since the embedding just required
the ($t-r$)-sector.

## References

  * [1] S. W. Hawking, Nature 248, 30 (1974).
S. W. Hawking, Commun. Math. Phys. 43, 199 (1975) [Erratum-ibid. 46, 206
(1976)].

  * [2] W. G. Unruh, Phys. Rev. D 14, 870 (1976).
  * [3] S. Deser and O. Levin, Class. Quant. Grav. 14, L163 (1997) [arXiv:gr-qc/9706018].
S. Deser and O. Levin, Class. Quant. Grav. 15, L85 (1998) [arXiv:hep-
th/9806223].

  * [4] S. Deser and O. Levin, Phys. Rev. D 59, 064004 (1999) [arXiv:hep-th/9809159].
  * [5] H. Narnhofer, I. Peter and W. E. Thirring, Int. J. Mod. Phys. B 10 (1996) 1507.
  * [6] H. F. Goenner, “General Relativity and Gravitation, edited by A. Held (Plenum, Newyork,1980), p.441.
  * [7] J. Rosen, Rev. Mod. Phys. 37 (1965) 204.
  * [8] R. C. Tolman, “Relativity, Thermodynamics and Cosmology”, New York: Dover Publication, (1987), p.318.
  * [9] Y. W. Kim, Y. J. Park and K. S. Soh, Phys. Rev. D 62, 104020 (2000) [arXiv:gr-qc/0001045].
S. T. Hong, Y. W. Kim and Y. J. Park, Phys. Rev. D 62, 024024 (2000)
[arXiv:gr-qc/0003097].
S. T. Hong, W. T. Kim, Y. W. Kim and Y. J. Park, Phys. Rev. D 62, 064021
(2000) [arXiv:gr-qc/0006025].
S. T. Hong, W. T. Kim, J. J. Oh and Y. J. Park, Phys. Rev. D 63, 127502 (2001)
[arXiv:hep-th/0103036].
H. Z. Chen, Y. Tian, Y. H. Gao and X. C. Song, JHEP 0410, 011 (2004)
[arXiv:gr-qc/0409107].
N. L. Santos, O. J. C. Dias and J. P. S. Lemos, Phys. Rev. D 70, 124033 (2004)
[arXiv:hep-th/0412076].
H. Z. Chen and Y. Tian, Phys. Rev. D 71, 104008 (2005).

  * [10] Y. Tian, JHEP 0506, 045 (2005) [arXiv:gr-qc/0504040].
  * [11] E. J. Brynjolfsson and L. Thorlacius, JHEP 0809, 066 (2008) [arXiv:0805.1876 [hep-th]]. Y. W. Kim, J. Choi and Y. J. Park, arXiv:0909.3176 [gr-qc].
  * [12] S. T. Hong, Gen. Rel. Grav. 36, 1919 (2004) [arXiv:gr-qc/0310118].
  * [13] S. P. Robinson and F. Wilczek, Phys. Rev. Lett. 95, 011303 (2005) [arXiv:gr-qc/0502074].
  * [14] S. Carlip, Phys. Rev. Lett. 82, 2828 (1999) [arXiv:hep-th/9812013].
  * [15] S. Iso, H. Umetsu and F. Wilczek, Phys. Rev. D 74, 044017 (2006) [arXiv:hep-th/0606018].
  * [16] K. Umetsu, arXiv:0907.1420 [hep-th].


Method | SF | CI | C$\rightarrow$P | C$\rightarrow$R | C$\rightarrow$S | P$\rightarrow$C | P$\rightarrow$R | P$\rightarrow$S | R$\rightarrow$C | R$\rightarrow$P | R$\rightarrow$S | S$\rightarrow$C | S$\rightarrow$P | S$\rightarrow$R | Avg.
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
ResNet-50 [48] | ✗ | ✗ | 57.15 | 76.22 | 56.00 | 60.76 | 84.16 | 65.12 | 64.31 | 69.00 | 59.71 | 57.81 | 56.28 | 74.27 | 65.07
DANN [9] | ✗ | ✗ | 63.90 | 82.80 | 68.10 | 64.80 | 78.10 | 60.80 | 74.80 | 73.30 | 71.80 | 74.50 | 66.00 | 81.60 | 71.71
MDD [50] | ✗ | ✗ | 68.84 | 87.87 | 73.62 | 67.64 | 86.59 | 69.48 | 79.38 | 74.91 | 74.42 | 78.35 | 69.84 | 85.36 | 76.36
MCC [53] | ✗ | ✗ | 52.75 | 83.54 | 62.46 | 66.60 | 85.26 | 60.64 | 69.94 | 61.78 | 56.73 | 67.24 | 54.93 | 77.41 | 66.61
ToAlign [66] | ✗ | ✗ | 65.10 | 87.78 | 73.32 | 71.67 | 85.98 | 73.98 | 77.27 | 75.78 | 74.47 | 76.97 | 67.91 | 84.76 | 76.25
COAL [35] | ✗ | ✓ | 68.52 | 85.73 | 70.95 | 72.06 | 87.98 | 68.04 | 75.20 | 73.76 | 62.48 | 77.81 | 68.43 | 85.21 | 74.68
PCT [38] | ✗ | ✓ | 71.34 | 89.86 | 75.14 | 78.17 | 89.43 | 77.00 | 79.73 | 77.16 | 75.45 | 81.26 | 72.29 | 87.59 | 79.53
SHOT [18] | ✓ | ✗ | 73.83 | 89.67 | 73.83 | 77.81 | 89.95 | 75.03 | 78.25 | 75.32 | 71.77 | 77.31 | 70.18 | 87.73 | 78.39
BAIT [55] | ✓ | ✗ | 75.83 | 88.73 | 77.05 | 77.82 | 87.33 | 74.63 | 76.55 | 76.11 | 73.68 | 82.07 | 71.53 | 88.48 | 79.15
NRC [31] | ✓ | ✗ | 75.32 | 91.54 | 76.47 | 80.49 | 91.29 | 75.87 | 82.55 | 75.60 | 75.13 | 82.05 | 77.57 | 91.96 | 81.32
CPGA [27] | ✓ | ✗ | 67.73 | 85.01 | 65.70 | 69.96 | 86.14 | 68.91 | 71.96 | 74.19 | 63.81 | 74.77 | 69.12 | 84.38 | 73.47
ISFDA [22] | ✓ | ✓ | 75.31 | 90.10 | 76.03 | 80.83 | 90.39 | 72.67 | 81.29 | 76.32 | 72.36 | 79.58 | 71.37 | 87.98 | 79.52
T-CPGA (Ours) | ✓ | ✓ | 80.02 | 93.94 | 84.41 | 89.31 | 93.45 | 84.60 | 89.24 | 80.81 | 85.43 | 89.36 | 80.36 | 93.50 | 87.04


# Supplemental Materials for “Three-Dirac-fermion approach to unexpected
universal gapless surface states in van der Waals magnetic topological
insulators”

Dinghui Wang1, Huaiqiang Wang1∗, Dingyu Xing1,2, and Haijun Zhang1,2∗ 1
National Laboratory of Solid State Microstructures, School of Physics, Nanjing
University, Nanjing 210093, China
2 Collaborative Innovation Center of Advanced Microstructures, Nanjing
University, Nanjing 210093, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Contents

  1. I Emergent PT symmetry in coupled two-Dirac-fermion model
  2. II Ferromagnetic case by three-Dirac-fermion model
  3. III Surface LDOS and Fermi contours by three-Dirac-fermion model
  4. IV Results from different vdW functionals
  5. V Results for other vdW magnetic topological insulators
    1. V.1 MnBi4Te7
      1. V.1.1 SL termination case
      2. V.1.2 QL termination case
    2. V.2 MnBi6Te10, MnSb2Te4 and MnSb4Te7
  6. VI Wavefunction profile of topological surface states
  7. VII First-principles calculation details

##  I Emergent PT symmetry in coupled two-Dirac-fermion model

Here, we show the existence of the combined symmetry of inversion $P$ and
time-reversal $T$ in a coupled two-Dirac-fermion system, where the two Dirac
fermions have opposite T-breaking mass terms (induced by opposite magnetic
moments) and opposite helicities. For the three-Dirac-fermion model with
$h_{1}=h_{2}=-h_{3}=h$ in the main text, when $\Delta_{12}=0$, $D_{1}$ is
isolated from $D_{2}$ and $D_{3}$, and the coupled $D_{2}$ and $D_{3}$
subsystem can be exactly described by the above model. In the ordered bases of
($|D_{2\uparrow}\rangle$,$|D_{2\downarrow}\rangle$,$|D_{3\uparrow}\rangle$,$|D_{3\downarrow}\rangle$),
the four-band model Hamiltonian is given by

$H(\mathbf{k})=\left(\begin{array}[]{cccc}h&ivk^{-}&\Delta_{23}&0\\\
-ivk^{+}&-h&0&\Delta_{23}\\\ \Delta_{23}&0&-h&-ivk^{-}\\\
0&\Delta_{23}&ivk^{+}&-h\end{array}\right)=-\tau_{z}(k_{x}\sigma_{y}-k_{y}\sigma_{x}-h\sigma_{z})+\Delta_{23}\tau_{x},$
(S1)

where $k^{\pm}=k_{x}\pm ik_{y}$, $v$ is the Fermi velocity, $h$ represents the
Zeeman coupling strength, and the Pauli matrices $\tau_{i}$ and $\sigma_{i}$
act in the orbital and spin subspaces. The inversion and time-reversal
operators are given by $\tau_{x}$ and $i\sigma_{y}K$ ($K$ mean complex
conjugate), respectively. The combined $PT$ operator is then simply obtained
as $i\tau_{x}\sigma_{y}K$. It is straightforward to show that the above
Hamiltonian satisfies the $PT$ symmetry as

$PTH(\mathbf{k})(PT)^{-1}=H(\mathbf{k}).$ (S2)

Since the antiunitary $PT$ operator satisfies $(PT)^{2}=-1$, Kramers
degeneracy is ensured at every momentum, leading to doubly degenerate band
structures with vanishing total Berry curvatures and thus zero Chern number
for the occupied bands.

Fig. S 1: Bands at the $\Gamma$ point of the coupled three-Dirac model with FM
moments and (a) $\Delta_{12}<h$, (b) $\Delta_{12}>h$ in the expansion process
of the van der Waals gap between surface and layers, where
$\Delta_{23}/\Delta_{12}$ gradually decreases to zero in the surface-detached
limit. The gap closing-and-reopening process appears only when
$\Delta_{12}<h$, which is accompanied by a unit change of the Chern number
($C$). In the numerical calculation, $h/\Delta_{12}$ is chosen to be $1.2$ and
$0.8$ in (a) and (b), respectively. (c) Surface gap evolution from first-
principles calculation of a (5+1)-SL (red lines) and (5+3)-SL (blue lines) FM
MnBi2Te4 with out-of-plane FM moments.

##  II Ferromagnetic case by three-Dirac-fermion model

Here, we present the detailed analysis of the ferromagnetic (FM) counterpart
of three-Dirac-fermion model, where the gapless transition point appears only
when $\Delta_{12}<h$ is satisfied. For the coupled three-Dirac-fermion model
with FM moments, i.e., $h_{1}=h_{2}=h_{3}=h$ ($h>0$ is assumed), the
Hamiltonian is given by

$H=\left(\begin{array}[]{cccccc}h&-ivk^{-}&\Delta_{12}&0&0&0\\\
ivk^{+}&-h&0&\Delta_{12}&0&0\\\ \Delta_{12}&0&h&ivk^{-}&\Delta_{23}&0\\\
0&\Delta_{12}&-ivk^{+}&-h&0&\Delta_{23}\\\ 0&0&\Delta_{23}&0&h&-ivk^{-}\\\
0&0&0&\Delta_{23}&ivk^{+}&-h\end{array}\right).$ (S3)

The energy spectrum can be easily obtained as

$E(\mathbf{k})=\pm\sqrt{h^{2}+k^{2}},\pm\sqrt{k^{2}+(\sqrt{\Delta_{12}^{2}+\Delta_{23}^{2}}-h)^{2}},\pm\sqrt{k^{2}+(\sqrt{\Delta_{12}^{2}+\Delta_{23}^{2}}+h)^{2}}.$
(1)

Obviously, the energy gap closes at the $\Gamma$ point when
$\sqrt{\Delta_{12}^{2}+\Delta_{23}^{2}}=h$ is satisfied.

In the detaching process of the surface system with the van der Waals gap
distance between surface and bulk layers changing from $d_{0}$ to $\infty$, as
shown in Fig. 2(a) in the main text, $\Delta_{12}$ remains unchanged, while
$\Delta_{23}/\Delta_{12}$ gradually decreases from a large value to zero. It
follows that, when $\Delta_{12}<h$, there exists a gap closing-and-reopening
process at a critical value of $\Delta_{23}/\Delta_{12}$, as shown in Fig.
S1(a). This is accompanied by a change of the total Chern number from 1/2 to
3/2. In contrast, when $\Delta_{12}>h$, the gap always keeps open, as shown in
Fig. S1(b), and the Chern number keeps unchanged as 1/2. Note that the
condition for the mergence of gap closing in the FM case is opposite to that
in the antiferromagnetic (AFM) case in the main text.

To confirm the above analysis based on the three-Dirac-fermion model, we have
carried out first-principles calculations of (5+1)-SL and (5+3)-SL FM MnBi2Te4
with out-of-plane magnetic moments, where the topmost 1 SL and 3SL are
gradually detached from the the 5SL bulk. The evolutions of the surface gap
size in the expansion process of the vdW gap distance between the detached
layer and the bulk are shown in Fig. S1(c), where a gapless point emerges for
the (5+3)-SL case with $\Delta_{12}<h$, while the gap remains open for the
(5+1)-SL case with $\Delta_{12}>h$. These results are consistent with the
theoretical analysis of the FM case by the three-Dirac fermion model.

##  III Surface LDOS and Fermi contours by three-Dirac-fermion model

In this section, to further show the functionality of the three-Dirac model,
we will use it to calculate the surface local density of states (LDOS) and
Fermi contours in the vdW gap expansion process for both the nonmagnetic case
and AFM case exhibiting a gapless transition point, corresponding to the
(7+1)-SL nonmagnetic CaBi2Te4 and AFM MnBi2Te4 films in the main text.

Fig. S 2: Surface LDOS (first column) and corresponding Fermi contours in
ascending order of the three energy cuts. The upper three rows correspond to
the three stages in the interlayer vdW gap expansion process of a layered AFM
topological insulator, which presents the gapless transition point in the
expansion process. (First row) the unexpanded case with a representative value
of $\Delta_{23}/\Delta_{12}=2$. (Second row) at the critical gap expansion
value of $\Delta_{23}/\Delta_{12}\approx 0.95$. (Third row) the surface-
detached limit with $\Delta_{23}/\Delta_{12}=0$. The last row describes the
nonmagnetic case with a representative $\Delta_{23}/\Delta_{12}=2$. In the
numerical calculations, the Zeeman field strength is set as
$h/\Delta_{12}=0.3$, and the coefficient $\lambda$ of the third-order terms in
Eq. (S4) is chosen as $\lambda_{1}=-\lambda_{2}=\lambda_{3}$ with
$\lambda/\Delta_{12}=0.2$.

To reveal the trigonal or hexagonal structures of the surface states, we need
to include symmetry-allowed higher-order corrections up to third-order terms
of $k$ in the Hamiltonian of Eq. (2) in the main text. A nonmagnetic
Bi2Te3-type surface state satisfies the following three symmetries, namely,
the time-reversal symmetry $T$, three-fold rotation symmetry $C_{3z}$, and
mirror symmetry $M_{x}$. In the presence of the out-of-plane magnetic order,
both $T$ and $M_{x}$ are broken, whereas the $C_{3z}$ symmetry and combined
time-reversal and mirror symmetries $M_{x}T$ are preserved. Through the method
of invariant in the $k\cdot p$ theory, all symmetry allowed terms up to third
order in $k$ can be obtained (see, e.g., Ref. Naselli et al. (2022)). For
simplicity, we consider the following symmetry allowed third-order correction
for each of the three Dirac fermions

$H^{\prime}_{D_{i}}=\lambda_{i}(k_{+}^{3}+k_{-}^{3})\sigma_{z},$ (S4)

where $k_{\pm}=k_{x}\pm ik_{y}$. It is worth mentioning that this third-order
correction term for Dirac surface states of TIs has also been derived in Ref.
Liu et al. (2010). The full Hamiltonian $H^{\prime}$ of the coupled three-
Dirac-fermion system is then obtained by combining the Hamiltonian in Eq. (2)
of the main text with the above terms.

To calculate the LDOS and Fermi contours of the gradually detached surface
composed of the two Dirac fermions, $D_{1}$ and $D_{2}$, we resort to the
Green’s function method, and the surface LDOS is given by

$\rho(\omega)=-\frac{1}{\pi}\textrm{Im}G^{r}_{11,\sigma}(\omega)-\frac{1}{\pi}\textrm{Im}G^{r}_{22,\sigma}(\omega),$
(S5)

where the retarded Green’s function $G^{r}(\omega)$ is given by
$G^{r}(\omega)=(\omega+i\eta-H^{\prime})^{-1}$.

In Fig. S2, we present the results of surface LDOS (first column) and three
representative Fermi contours in ascending order of the three energy cuts in
the LDOS for the nonmagnetic layered topological insulators (TIs) (last row)
and three stages in the interlayer vdW gap expansion process of layered AFM
TIs, namely, without vdW expansion (first row with a representative
$\Delta_{23}/\Delta_{12}=2$), at the critical gap expansion value exhibiting
the gapless Dirac cone (second row with $\Delta_{23}/\Delta_{12}\approx 0.95$
for $h/\Delta_{12}=0.3$), and, in the surface-detached limit with
$\Delta_{23}/\Delta_{12}=0$ (third row). From the third column [Figs. S2(c, g,
k)], it can be clearly seen that, with increasing vdW gap expansion, the Fermi
contours gradually change from trigonal shape to hexagonal shape, which
results from the gradually recovered inversion symmetry of the detached
surface system. It should be emphasized that the results predicted by the
effective three-Dirac-fermion model are in qualitative consistence with the
first-principles results in Fig. 4 in the main text.

We remark that, in real materials, there also exist Rashba-like bands on the
surface, e.g., due to the broken inversion symmetry on the surface. However,
since we focus on the Dirac surface states originating from nontrivial bulk
topology, which suffice to explain the trigonal or hexagonal warping effect in
magnetic TIs, we did not include these Rashba surface bands in the above
theoretical discussion by three-Dirac-fermion model.

##  IV Results from different vdW functionals

To further validate the emergence of a gapless point in the expansion process
of the topmost interlayer vdW gap of a $(7+1)$-SL MnBi2Te4, we have chosen
different vdW functionals to plot the evolution of the surface gap, as shown
in Fig. S3. Obviously, the gapless point stably emerges in all cases.

Fig. S 3: The evolution of the surface band gap in the expansion process of
the topmost interlayer vdW gap of a (7+1)-SL MnBi2Te4 with different vdW
functionals. All curves exhibit a gapless point around $0.18\ \AA$ in the
expansion process.

##  V Results for other vdW magnetic topological insulators

In this section, to further show the functionality and universality of the
proposed three-Dirac-fermion model, we apply it to some other representative
examples of vdW magnetic TIs in the MnBi2Te4 family, namely, MnBi4Te7,
MnBi6Te10, MnSb2Te4 and MnSb4Te7. Since the basic physics is similar, we here
focus on MnBi4Te7 and present results from both first-principles calculations
and three-Dirac-fermion model analysis, which are consistent with each other.
Whereas, we only present the first-principles results for the rest three
materials.

###  V.1 MnBi4Te7

As shown in Fig. S4(a), MnBi4Te7 consists of one SL MnBi2Te4 and one
quintuple-layer (QL) Bi2Te3 as the building block. Similar to MnBi2Te4, the
magnetic ground state features out-of-plane FM couplings within each SL, and
AFM couplings between adjacent SLs. MnBi4Te7 is proposed to be a nontrivial
$Z_{2}$ AFM TI, which is expected to exhibit gapped Dirac surface states for
boundaries along the out-of-plane (z-) direction. However, experimental
observations show that Wu et al. (2020); Vidal et al. (2021); Hu et al.
(2020a, b), depending on the surface terminations, there exist two types of
topological surface states for MnBi4Te7, namely, gapless (gapped) surface
states for SL (QL) termination. The gapped surface states for the QL
termination case have been explained to result from the hybridization between
surface and bulk bands. However, the gapless surface states for the SL
termination still remain elusive.

Here, we will take advantage of the three-Dirac-fermion model to show that in
the vdW expansion process of the topmost vdW gap distance, a gapless
transition point in the surface band structure can emerge only for the SL-
termination case, whereas the surface bands always remain gapped for the QL
termination. It should be emphasized that our results are in consistence with
the experimental observations, and can provide further insight into the
properties of the surface state in MnBi4Te7.

####  V.1.1 SL termination case

We first consider the SL termination case, where the three-Dirac-fermion
models consists of two Dirac surface states from the topmost SL and the other
from the top surface state from the QL below, as shown in Fig. S4(a). The
model Hamiltonian is then given by

$H=\left(\begin{array}[]{cccccc}h_{\textrm{SL}}&-ivk^{-}&\Delta_{12}&0&0&0\\\
ivk^{+}&-h_{\textrm{SL}}&0&\Delta_{12}&0&0\\\
\Delta_{12}&0&h_{\textrm{SL}}&ivk^{-}&\Delta_{23}&0\\\
0&\Delta_{12}&-ivk^{+}&-h_{\textrm{SL}}&0&\Delta_{23}\\\
0&0&\Delta_{23}&0&h_{\textrm{QL}}&-ivk^{-}\\\
0&0&0&\Delta_{23}&ivk^{+}&-h_{\textrm{QL}}\end{array}\right).$ (S6)

Here the Zeeman term $h_{\textrm{SL}}$ results from the magnetic moments
within the SL, while the Zeeman term $h_{\textrm{QL}}$ comes from the FM
proximity effect on the QL from neighboring SLs. Note that $h_{\textrm{SL}}$
can be treated as a constant, whereas $h_{\textrm{QL}}$ depends on the
interlayer distance between the QL and neighboring SLs. Since a QL has two
neighboring SLs, $h_{\textrm{QL}}$ in Eq. (S6) consists of two contributions,
$h_{\textrm{QL}}^{I}$ and $h_{\textrm{QL}}^{II}$, from the upper and lower
SLs, respectively, which depend on the distance between the QL and neighboring
SLs. For simplicity and without loss of generality, in the unexpanded case, we
assume the proximity-induced Zeeman term for the top surface state of the QL
to be $h_{\textrm{QL}}^{I}=0.5h_{\textrm{SL}}$
($h_{\textrm{QL}}^{II}=-0.3h_{\textrm{SL}}$) from the upper (lower) SL. In the
expansion process of the topmost interlayer vdW gap between the QL and the SL
above, $h_{\textrm{QL}}^{II}$ obviously keeps unchanged, while
$h_{\textrm{QL}}^{I}$ gradually decreases until it vanishes in the decoupled
limit, which can be assumed to be proportional to the distance-dependent
coupling $\Delta_{23}$ between the QL and the SL. As a result,
$h_{\textrm{QL}}$ is then given by

$h_{\textrm{QL}}=h_{\textrm{QL}}^{I}+h_{\textrm{QL}}^{II}=0.5h_{\textrm{SL}}(\Delta_{23}/\Delta_{23}^{0})-0.3h_{\textrm{SL}},$
(2)

where $\Delta_{23}^{0}$ denote the coupling $\Delta_{23}$ in the unexpanded
case with $d=d_{0}$.

With the above assumptions, the evolution of the SL-terminated surface band
structure at the $\Gamma$ point in the topmost vdW gap expansion process can
be obtained, as shown in Fig. S4(b), where $\Delta_{12}$ is chosen as the
energy unit and the other parameters are representatively set to be
$h_{\textrm{SL}}/\Delta_{12}=0.5$ and $\Delta_{23}^{0}/\Delta_{12}=2$. A gap
closing-and-reopening process is discernable with a gapless point separating
two phases with $C=1/2$ and $C=-1/2$, respectively. Similar to the discussion
in the main text, the $C=1/2$ phase can be understood from the
$\Delta_{23}\gg\Delta_{12}$ limit, where the total Chern number is reduced to
that of the gapped Dirac fermion $D_{1}$ on top of the SL with $C=(1/2)\
\mathrm{sgn}(h_{\textrm{SL}})=1/2$. Whereas $C=-1/2$ comes from the other
limit of $\Delta_{23}=0$, where the total Chern number is given as the sum of
the isolated SL and the top surface state D3 of the QL. For the isolated SL
with $h_{\textrm{SL}}<\Delta_{12}$, its Chern number equals zero, so the total
Chern number is reduced to that of D3 with $C=(1/2)\
\mathrm{sgn}(h_{\textrm{QL}}^{II})=(1/2)\
\mathrm{sgn}(-h_{\textrm{SL}})=-1/2$. Consequently, such a gapless point is
topologically protected and thus inevitable for the SL termination case in the
vdW gap expansion process.

As a further evidence, in Fig. S4(c), we plot the surface band gap evolution
from first-principles calculations of a AFM MnBi4Te7 with 4 (SL+QL) layers and
SL termination, where the topmost 1 SL is gradually detached from the bulk.
The emergence of the gap closing point in Fig. S4(c) is consistent with the
above three-Dirac-fermion model analysis.

Fig. S 4: Schematic of the three-Dirac model of the SL termination case (a),
and QL termination case (d) of AFM MnBi4Te7. Bands at $\Gamma$ as a function
of $\Delta_{23}/\Delta_{12}$, corresponding to increasing the interlayer vdW
gap from the three-Dirac-fermion model for the SL termination case (b) and the
QL termination case (e). First-principles results of the surface gap evolution
in the topmost vdW expansion process with increasing $\delta d$ for a
4-(SL+QL) MnBi4Te7 with the SL termination case (c) and the QL termination
case (f). Gapless point only appears for the SL termination case.

####  V.1.2 QL termination case

For the QL termination case, as shown in Fig. S4(e), the three-Dirac-fermion
model is composed of two Dirac fermions D1 and D2 from the two surface states
of the QL, and D3 from the top surface state of the SL. The Hamiltonian can
then be written as

$H=\left(\begin{array}[]{cccccc}h_{\textrm{QL}}^{t}&-ivk^{-}&\Delta_{12}&0&0&0\\\
ivk^{+}&-h_{\textrm{QL}}^{t}&0&\Delta_{12}&0&0\\\
\Delta_{12}&0&h_{\textrm{QL}}^{b}&ivk^{-}&\Delta_{23}&0\\\
0&\Delta_{12}&-ivk^{+}&-h_{\textrm{QL}}^{b}&0&\Delta_{23}\\\
0&0&\Delta_{23}&0&h_{\textrm{SL}}&-ivk^{-}\\\
0&0&0&\Delta_{23}&ivk^{+}&-h_{\textrm{SL}}\end{array}\right).$ (S7)

Here, $h_{\textrm{QL}}^{t}$ and $h_{\textrm{QL}}^{b}$ represent the proximity-
effect-induced Zeeman term of the top and bottom surface states of the topmost
QL from the SL below. Similar to the discussion above, both
$h_{\textrm{QL}}^{t}$ and $h_{\textrm{QL}}^{b}$ depend on the interlayer
distance between the topmost QL and the below SL, and they are also
proportional to the interlayer coupling $\Delta_{23}$. For simplicity,
$h_{\textrm{QL}}^{t}$ and $h_{\textrm{QL}}^{b}$ are reasonably assumed to be
$0.3h_{\textrm{SL}}(\Delta_{23}/\Delta_{23}^{0})$ and
$0.5h_{\textrm{SL}}(\Delta_{23}/\Delta_{23}^{0})$, respectively. The evolution
of the band structure at $\Gamma$ in the expansion process of the topmost
interlayer vdW gap distance can then be obtained, shown in Fig. S4(e).
Obviously, the surface band gap remains open and the Chern number keeps
unchanged as $1/2$. This result can also be understood by inspecting the two
limits of $\Delta_{23}/\Delta_{12}\rightarrow\infty$ and
$\Delta_{23}/\Delta_{12}\rightarrow 0$. In the
$\Delta_{23}/\Delta_{12}\rightarrow\infty$
($\Delta_{23}/\Delta_{12}\rightarrow 0$) limit, the total Chern number is
reduced to that of D1 (D3) on top of the QL (SL), which is simply given as
$C=(1/2)\ \mathrm{sgn}(0.3h_{\textrm{SL}})=1/2$ [$C=(1/2)\
\mathrm{sgn}(h_{\textrm{SL}})=1/2$]. It follows that the band gap should
remain open with no gapless transition point. This is also further confirmed
by the first-principles calculation of the surface gap evolution of a 4
(SL+QL) MnBi4Te7 with QL termination, as shown in Fig. S4(f).

Fig. S 5: (a) Schematic of MnBi6Te10 with three different kinds of surface
terminations. Surface band gap evolutions in the expansion process of the
topmost interlayer vdW gap from first-principles calculations for 3-(SL+QL+QL)
AFM MnBi6Te10 with the three surface terminations (b), (5+1)-SL MnSb2Te4 with
both FM and AFM configurations (c), and 4-(SL+QL) AFM MnSb4Te7 with SL and QL
terminations, respectively (d).

###  V.2 MnBi6Te10, MnSb2Te4 and MnSb4Te7

Considering the similar underlying physics concerning the three-Dirac-fermion
model, here, we only present the results of the surface band gap evolutions in
the topmost interlayer vdW gap expansion process from first-principles
calculations for MnBi6Te10, MnSb2Te4 and MnSb4Te7.

As schematically shown Fig. S5(a), MnBi6Te10 has three types of surface
terminations, namely, SL-, QL1-, QL2- terminations, respectively. In Fig.
S5(b), we choose 3-(SL+QL+QL) AFM MnBi6Te10 with different surface
terminations to plot the surface band gap as a function of the expansion
$\delta d$ of the topmost interlayer vdW gap. It can be seen that similar to
MnBi4Te7, only the SL termination of MnBi6Te10 exhibit a gapless point, while
the gap remains for both QL1 and QL2 terminations.

Figure S5(c) presents the results of the band gap evolution for both the AFM
and FM cases of a (5+1)-SL MnSb2Te4. Since MnSb2Te4 is topologically trivial
without any Dirac surface states, the band gap persists in the topmost vdW gap
expansion process for both cases.

In fig. S5(d), we plot the evolutions of the surface band gap in the topmost
vdW gap expansion process of a 4-(QL+SL) AFM MnSb4Te7 films with SL and QL
terminations, respectively. Because MnSb2Te4 is topologically nontrivial,
analogous to MnBi4Te7, the gapless point appears only in the SL termination
case.

##  VI Wavefunction profile of topological surface states

To further support the validity of the three-Dirac-fermion model in the
description of layered TIs with vdW gap modulation, we plot the wavefunction
profile of the topological surface state in the vdW gap expansion process. The
wavefunction of the Dirac state at the $\Gamma$ point is simply assumed as
$\psi_{i}\left(z\right)\propto e^{-\lambda_{i}\left|z-z_{i}\right|}$
($1/\lambda_{i}$ characterizes the material-dependent decay length and
$\lambda_{i}=1$ is chosen for simplicity in numerical calculations), with
$i=1,2$, and $3$ corresponding to $D_{1}$, $D_{2}$, and $D_{3}$ Dirac states,
respectively, and $z_{i}$ denotes the position of the peak of each Dirac
state. When there exist finite overlaps between wavefunctions of these Dirac
states, e.g., when the surface layer is sufficiently thin (e.g., a few SLs for
MnBi2Te4), finite couplings will be induced between them. This underlies the
coupling terms of $\Delta_{12}$ between $D_{1}$ and $D_{2}$, and $\Delta_{23}$
between $D_{2}$ and $D_{3}$ in the construction of the coupled three-Dirac-
fermion model, shown in Fig. S6(a). From this coupling picture, it is obvious
that the final topological surface state should be formed by the hybridization
of all the three Dirac states.

Here, we take the case of $\Delta_{12}>h$ as an example, corresponding to Fig.
2(e) in the main text, where the coupling between $D_{1}$ and $D_{2}$ is
significantly larger than the magnetic Zeeman coupling strength. We plot the
wavefunction profile of the final topological surface state (red lines) for
three representative cases, namely, $\Delta_{23}/\Delta_{12}=2$ [Fig. S6(b)],
$\Delta_{23}/\Delta_{12}=0.92$ [Fig. S6(c)], and $\Delta_{23}/\Delta_{12}=0.2$
[Fig. S6(d)], at different expansion distances of the interlayer vdW gap
between the surface layer (shaded in pink) and blow bulk (shaded in blue),
corresponding to the pristine case with strongly-coupled surface and bulk
systems, the critical case near the gapless point with moderate expansion, and
the significantly expanded case with weakly-coupled surface and bulk systems.
As expected, in the strongly-coupled (weakly-coupled) case, the wavefunction
distribution of the topological surface state locates mainly within the
surface layer (around the top of the below bulk). Whereas for the moderate
expansion case, the topological surface state exhibit similar weights on the
surface layer and the top of the below bulk. The overall feature of the above
topological surface state is consistent with previous works on MnBi2Te4.

Fig. S 6: For the case of $\Delta_{12}>h$ in the three-Dirac fermion model
illustrated in (a), wavefunction profile at $\Gamma$ point of the surface
states for three representative cases, $\Delta_{23}/\Delta_{12}=2$ (b),
$\Delta_{23}/\Delta_{12}=0.92$ (c), and $\Delta_{23}/\Delta_{12}=0.2$ (d), in
the vdW gap expansion process with gradually increased $d$ and decreased
$\Delta_{23}/\Delta_{12}$, corresponding to the the pristine case with
strongly coupled surface and bulk systems, the critical case near the gapless
point with moderate expansion, and the significantly expanded case with weakly
coupled surface and bulk systems. $h=0.4$ is chosen in the numerical
calculations.

##  VII First-principles calculation details

First-principles calculations were performed using the Vienna ab initio
simulation package (VASP) Kresse and Furthmüller (1996). The Perdew-Burke-
Ernzerhof (PBE) functional Perdew et al. (1996), within the projector
augmented wave (PAW) Blöchl (1994) are used to describe the exchange-
correlation potential and energy. Lattice constants are adopted from
experimental data, $a=4.334\text{ \AA}$ and $c=13.637\text{ \AA}$ for each
septuple layer. Then we relax atoms positions. A Hubbard-like $U=5$ eV is used
to account for strongly localized Mn 3d orbitals. A kinetic cutoff energy of
410 eV and a 10$\times$10$\times$1 $\Gamma$-centered k points mesh is used in
a self-consistent field. The unit cells are stacked along $c$ direction to
generate multi-septuple layers films with a $20\text{ \AA}$ vacuum space for
the thick slab system. In all calculations, we keep G-type AFM with a magnetic
axis along the $c$ direction. By replacing Mn in MnBi2Te4 with Ca, and then
relaxing the atoms positions, we simulate the non-magnetic case in CaBi2Te4.
To account for the surface layer vdW expansion effect, we construct the
maximally localized Wannier function (MLWF) Marzari and Vanderbilt (1997) from
first-principles calculations with Mn $d$, Bi $p$, and Te $p$ orbitals. The
tight-binding Hamiltonian is divided into surface and bulk parts. The spectral
functions and Fermi surface are calculated with the surface Green’s functions
of the semi-infinite system. Both spectral functions and the Fermi surface are
projected to the upmost septuple layer.

## References

  * Naselli et al. (2022) G. Naselli, A. G. Moghaddam, S. Di Napoli, V. Vildosola, I. C. Fulga, J. van den Brink, and J. I. Facio, Phys. Rev. Res. 4, 033198 (2022).
  * Liu et al. (2010) C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C. Zhang, Phys. Rev. B 82, 045122 (2010).
  * Wu et al. (2020) X. Wu, J. Li, X.-M. Ma, Y. Zhang, Y. Liu, C.-S. Zhou, J. Shao, Q. Wang, Y.-J. Hao, Y. Feng, et al., Phys. Rev. X 10, 031013 (2020).
  * Vidal et al. (2021) R. C. Vidal, H. Bentmann, J. I. Facio, T. Heider, P. Kagerer, C. I. Fornari, T. R. F. Peixoto, T. Figgemeier, S. Jung, C. Cacho, et al., Phys. Rev. Lett. 126, 176403 (2021).
  * Hu et al. (2020a) Y. Hu, L. Xu, M. Shi, A. Luo, S. Peng, Z. Y. Wang, J. J. Ying, T. Wu, Z. K. Liu, C. F. Zhang, et al., Phys. Rev. B 101, 161113 (2020a).
  * Hu et al. (2020b) C. Hu, K. N. Gordon, P. Liu, J. Liu, X. Zhou, P. Hao, D. Narayan, E. Emmanouilidou, H. Sun, Y. Liu, et al., Nat. Commun. 11, 97 (2020b).
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
  * Perdew et al. (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * Blöchl (1994) P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
  * Marzari and Vanderbilt (1997) N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997).