# Assisted harvesting and catalysis of coherence from scalar fields

Nikolaos K. Kollas<EMAIL_ADDRESS>Division of Theoretical and Mathematical
Physics, Astronomy and Astrophysics, Department of Physics, University of
Patras, 26504, Patras, Greece  Dimitris Moustos<EMAIL_ADDRESS>Division
of Theoretical and Mathematical Physics, Astronomy and Astrophysics,
Department of Physics, University of Patras, 26504, Patras, Greece

###### Abstract

Recently it has been demonstrated that it is possible to harvest quantum
resources other than entanglement from a coherent scalar field. Employing
time-dependent perturbation theory, we present a complete analysis of the
conditions under which a spatially extended Unruh-DeWitt detector coupled to
the proper time derivative of the field can harvest coherence for any initial
state of the field, as well as the energy cost that is required for each
harvest. By studying harvesting under repeatable extractions it is proven that
when the detector interacts with the field through a delta coupling coherence
is catalytic. For a Gaussian smeared detector it is shown that harvesting from
a coherent field depends on the phase of its amplitude distribution and its
initial energy as well as on the mean radius of the detector and the mean
interaction duration between the two. For a detector moving at a constant
velocity and with a mean radius of the same order as its transition
wavelength, we observe that, for relativistic speeds, coherence swelling
effects are present the intensity of which depends on the dimension of the
underlying Minkowski spacetime.

##  I Introduction

Superposition is one of the most striking phenomena which distinguishes
quantum from classical physics. The degree to which a system is superposed
between different orthogonal states is known as _coherence_ Streltsov  _et
al._ (2017); Aberg (2006); Theurer  _et al._ (2017). Much like entanglement
Horodecki  _et al._ (2009), coherence is considered to be a valuable resource
in quantum information processes. In Quantum computing Nielsen and Chuang
(2002); Preskill (2018), where information is encoded in the states of two-
level systems, algorithms designed to operate in superposition, are
exponentially faster than their classical counterparts Shor (1997); Grover
(1997); Arute  _et al._ (2019). Coherence is so central to the development of
a universal quantum computer that it is used as a metric for the quality of a
quantum processor. The time that it takes for a qubit to effectively decohere
due to noise is known as the _dephasing time_ with current processors
achieving times of a few hundred microseconds. Coherent phenomena are
important in other fields of research, such as quantum metrology Marvian and
Spekkens (2016) and thermodynamics Lostaglio  _et al._ (2015a); Ćwikliński
_et al._ (2015); Lostaglio  _et al._ (2015b); Narasimhachar and Gour (2015);
Korzekwa  _et al._ (2016) for example. Surprisingly it has been suggested that
these phenomena might also be present in biological processes and more
specifically in the efficiency of energy transport during photosynthesis Lloyd
(2011).

A simple method of obtaining coherence is by extracting it from another
system. When this process involves a quantum field as the source then it is
known as a _coherence harvesting protocol_. Despite an extensive amount of
research on entanglement harvesting protocols (see, e.g., Valentini (1991);
Reznik (2003); Reznik  _et al._ (2005); Salton  _et al._ (2015); Pozas-
Kerstjens and Martín-Martínez (2015); Simidzija and Martín-Martínez (2017a, b,
2018); Cong  _et al._ (2020); Tjoa and Mann (2020)) and the deep connection
that exists between entanglement and coherence Streltsov  _et al._ (2015);
Chitambar and Hsieh (2016); Zhu  _et al._ (2017), coherent harvesting has not
received any attention. By employing the Unruh-DeWitt (UDW) particle detector
model Unruh (1976); DeWitt (1979); Birrell and Davies (1982), it was shown
recently that a two-level pointlike detector, initially in its ground energy
state, interacting with a coherent massless scalar field in $1+1$ flat
spacetime, can harvest a small amount of coherence Kollas  _et al._ (2020). As
it turns out, this amount depends on the initial energy of the field, the mean
interaction duration and the detector’s state of motion. For a detector moving
at relativistic speeds and initial field energies lower than the gap between
its energy levels, it is possible to extract a larger amount of coherence than
when it is static, a phenomenon dubbed by the authors as “swelling”.

In this article, we provide a thorough study of the conditions under which
coherence harvesting is possible for any initial state of the field in $n+1$
dimensional Minkowski spacetime. In order to achieve this and to avoid the
problem of IR divergences that are present in the $1+1$ dimensional case of a
linear coupling between detector and field Juárez-Aubry and Louko (2014), we
instead consider an interaction in which the former is coupled to the proper
time derivative of the latter. Both models contain all the essential features
of matter interacting with radiation Martín-Martínez  _et al._ (2013); Pozas-
Kerstjens and Martín-Martínez (2016), so they provide a useful benchmark for
studying possible applications of relativistic effects in quantum information
processing. Acknowledging the fact that a pointlike detector is not a physical
system–an atom or an elementary particle, for example, has finite size– and to
make our results as relevant as possible we will take into consideration the
spatial extension of the detector.

We show that when the interaction is instantaneous harvesting is catalytic
Åberg (2014). At the cost of some energy, which assists in the extraction
process, it is possible to repeatedly extract the same amount of coherence
each time. For an inertial detector moving at a constant velocity and under
suitable conditions, it is proven that this is also the maximum amount that
can be obtained. As an example we consider the case of harvesting coherence
from a coherent scalar field and find that the process depends on the phase of
its coherent amplitude distribution, its initial energy, the mean radius of
the detector and the mean interaction duration between the two. For a mean
radius comparable to the inverse of its transition frequency, it is shown that
although the amount of coherence extracted is of the same order as the
coupling constant the process can be repeated to obtain a single unit of
coherence in a very short time. We conclude that even in the case of a
spatially extended detector swelling effects are still present but these are
weaker in a $3+1$ compared to a $1+1$ dimensional spacetime.

##  II Quantum coherence

From a physical point of view coherence reflects the degree of superposition
that a quantum system exhibits when it simultaneously occupies different
orthogonal eigenstates of an observable of interest Theurer  _et al._ (2017).
Coherent systems are considered to be valuable resources in quantum
information processes, because with their help it is possible, at the cost of
consuming some of the coherence that they contain, to simulate transformations
that violate conservation laws associated with the corresponding observable.

Mathematically, let $\\{\ket{i}\\}$ denote a set of basis states spanning a
finite discrete Hilbert space $\mathcal{H}$, which correspond to the
eigenstates of an observable $\hat{O}$. Any state $\rho$ which is diagonal in
this basis

$\rho=\sum_{i}p_{i}\outerproduct{i}{i}$ (1)

is called _incoherent_ and commutes with the observable. If $\rho$ contains
non-diagonal elements then it is called _coherent_ Streltsov  _et al._ (2017).
In this case $[\rho,\hat{O}]\neq 0$ 111assuming that the spectrum of $\hat{O}$
is non-degenerate, and the state changes under the action of the one parameter
group of symmetry transformations $U(s)=\text{exp}(-is\hat{O})$ generated by
the observable. This makes coherent systems useful as reference frames and
reservoirs for the implementation of non-symmetric transformations Bartlett
_et al._ (2007); Gour and Spekkens (2008); Marvian and Spekkens (2014a, b).
For example, for a fixed Hamiltonian $\hat{H}$, any system that possesses
coherence with respect to the energy basis can be used as a clock since in
this case its rate of change is non-zero, $\dot{\rho}(t)\neq 0$, so it
necessarily changes with the passage of time. The same system could
alternatively be utilised as a coherent energy reservoir with the help of
which it is possible to perform incoherent transformations on other systems
Åberg (2014).

The amount of coherence present in a system can be quantified with the help of
a _coherence measure_. This is a real valued function $C(\cdot)$ on the set of
density matrices $\mathcal{D}$ such that

$C(\rho)\geq 0,\quad\forall\rho\in\mathcal{D}$ (2)

with equality if and only if $\rho$ is incoherent. A simple example of such a
function is given by the $\ell_{1}$-norm of coherence Streltsov  _et al._
(2017), which is equal to the sum of the modulus of the system’s non-diagonal
elements

$C(\rho)=\sum_{i\neq j}\absolutevalue{\rho_{ij}}$ (3)

with values ranging between $0$ for an incoherent state and $d-1$ for the
maximally coherent $d$-dimensional pure state

$\ket{\psi}=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{i}.$ (4)

In order to extract coherence from a coherent system $\sigma$ to an incoherent
system $\rho$ it is necessary to bring the two in contact and make them
interact through a completely positive and trace preserving quantum operation.
When the latter obeys the conservation law associated with the observable and
is strictly incoherent (in the sense that it maps incoherent states to
incoherent states) the process is called _faithful_ Kollas and Blekos (2020).
When this is no longer the case the operation generates extra coherence, which
increases the amount stored in the combined system and can assist in the
extraction process Mani and Karimipour (2015); Bu  _et al._ (2017), in much
the same way that a quantum operation which is non-local can create
entanglement between two spacelike separated systems.

We shall now demonstrate how to construct such an assisted protocol for
harvesting coherence onto an UDW detector from a scalar field. In what follows
we shall assume a flat $n+1$ dimensional spacetime with metric signature
$(-+\cdots+)$. We will denote spacetime vectors by sans-serif characters, and
the scalar product of vectors $\mathsf{x}$ and $\mathsf{y}$ as
$\mathsf{x}\cdot\mathsf{y}$. Boldface letters represent spatial n-vectors.
Throughout, we make use of natural units in which $\hbar=c=1$ and employ the
interaction picture for operators and states.

##  III Unruh-DeWitt detector model

To study the amount of coherence harvested from a massless scalar field we
will employ an UDW detector coupled to the proper time derivative of the field
Hinton (1983); Takagi (1986); Moustos (2018). In the simplest case considered
here, the latter is modeled as a qubit with two energy levels, ground
$\ket{g}$ and excited $\ket{e}$ and energy gap equal to $\Omega$, with
Hamiltonian

$\hat{H}_{\text{\tiny$D$}}=\frac{\Omega}{2}(\outerproduct{e}{e}-\outerproduct{g}{g})$
(5)

which is moving along a worldline $\mathsf{x}(\tau)$ parametrized by its
proper time $\tau$. The detector is interacting with a massless scalar field
in $n+1$ dimensions

$\hat{\phi}(\mathsf{x})=\int\frac{d^{n}\mathbf{k}}{\sqrt{(2\pi)^{n}2\absolutevalue{\mathbf{k}}}}\left[\hat{a}_{\mathbf{k}}e^{i\mathsf{k}\cdot\mathsf{x}}+\text{H.c.}\right],$
(6)

with a normal-ordered Hamiltonian of the form

$\hat{H}_{\phi}=\int\absolutevalue{\mathbf{k}}\hat{a}^{\dagger}_{\mathbf{k}}\hat{a}_{\mathbf{k}}d^{n}\mathbf{k},$
(7)

where $\hat{a}_{\mathbf{k}}$, and $\hat{a}^{\dagger}_{\mathbf{k}}$ are the
creation and annihilation operators of the mode with momentum $\mathbf{k}$
that satisfy the canonical commutation relations

$[\hat{a}_{\mathbf{k}},\hat{a}_{\mathbf{k^{\prime}}}]=[\hat{a}^{\dagger}_{\mathbf{k}},\hat{a}^{\dagger}_{\mathbf{k^{\prime}}}]=0,\quad[\hat{a}_{\mathbf{k}},\hat{a}^{\dagger}_{\mathbf{k^{\prime}}}]=\delta(\mathbf{k}-\mathbf{k^{\prime}}).$
(8)

The interaction between detector and field is constructed by coupling the
former’s monopole moment operator

$\hat{\mu}(\tau)=e^{i\Omega\tau}\outerproduct{e}{g}+e^{-i\Omega\tau}\outerproduct{g}{e},$
(9)

to the momentum degrees of freedom of the latter through the following
interaction Hamiltonian

$\hat{H}_{\text{int}}(\tau)=\lambda\chi(\tau)\hat{\mu}(\tau)\otimes\partial_{\tau}\hat{\phi}_{f}(\mathsf{x}(\tau)).$
(10)

Here $\lambda$ is a coupling constant with dimensions
$(\mbox{length})^{\frac{n+1}{2}}$, $\chi(\tau)$ is a real valued _switching
function_ that describes the way the interaction is switched on and off; and
$\hat{\phi}_{f}(\mathsf{x}(\tau))$ is a smeared field on the detector’s center
of mass worldline $\mathsf{x}(\tau)=(t(\tau),\mathbf{x}(\tau))$,

$\hat{\phi}_{f}(\mathsf{x}(\tau))=\int_{\mathcal{S}(\tau)}f(\boldsymbol{\xi})\hat{\phi}(\mathsf{x}(\tau,\boldsymbol{\xi}))d^{n}\boldsymbol{\xi},$
(11)

where

$\mathsf{x}(\tau,\boldsymbol{\xi})=\mathsf{x}(\tau)+\boldsymbol{\xi}$ (12)

are the Fermi-Walker coordinates Misner  _et al._ (2017) on the simultaneity
hyperplane $S(\tau)$, which is defined by all those space-like vectors
$\boldsymbol{\xi}$ normal to the detector’s four-velocity,
$\mathcal{S}(\tau)=\left\\{\boldsymbol{\xi}|\mathsf{u}\cdot\boldsymbol{\xi}=0\right\\}$
(see Fig. 1).

Figure 1: Any point in the neighbourhood of the detector’s worldline can be
described by its Fermi-Walker coordinates $(\tau,\boldsymbol{\xi})$, where the
proper time $\tau$ indicates its position along the trajectory and
$\boldsymbol{\xi}$ is the displacement vector from this point lying on the
simultaneity hyperplane consisting of all those space-like vectors normal to
its four-velocity $\mathsf{u}$.

The real valued function $f(\boldsymbol{\xi})$ in Eq. (11) is known as the
_smearing function_ and is a physical reflection of the finite size and shape
of the detector Schlicht (2004); Louko and Satz (2006); Martín-Martínez  _et
al._ (2013); Pozas-Kerstjens and Martín-Martínez (2016).

Compared to the usual UDW interaction in which the detector is linearly
coupled to the field, the derivative coupling is free of the issue of IR
divergences in the $1+1$ dimensional case which arise due to the massless
nature of the field Juárez-Aubry and Louko (2014). The Hamiltonian in Eq. (10)
resembles closely the dipole interaction between an atom with dipole moment
$\mathbf{d}$ and an external electromagnetic field, since in this case the
electric field operator is defined, in the Coulomb gauge, by means of the
vector potential $\hat{\mathbf{A}}(t,\boldsymbol{x})$ as
$\hat{\mathbf{E}}(t,\boldsymbol{x})=-\partial_{t}\hat{\mathbf{A}}(t,\boldsymbol{x})$
Scully and Zubairy (1997).

Combining Εq. (6) with Εqs. (11)-(12) the smeared field operator reads

$\hat{\phi}_{f}(\mathsf{x}(\tau))=\int\frac{d^{n}\mathbf{k}}{\sqrt{(2\pi)^{n}2\absolutevalue{\mathbf{k}}}}\left[F(\mathsf{k},\tau)\hat{a}_{\mathbf{k}}e^{i\mathsf{k}\cdot\mathsf{x}(\tau)}+\text{H.c.}\right],$
(13)

where

$F(\mathsf{k},\tau)=\int_{\mathcal{S}(\tau)}f(\boldsymbol{\xi})e^{i\mathsf{k}\cdot\boldsymbol{\xi}}d^{n}\boldsymbol{\xi}$
(14)

is the Fourier transform of the smearing function. Now $\mathsf{k}$ can always
be decomposed as

$\mathsf{k}=(\mathsf{k}\cdot\mathsf{u})\mathsf{u}+(\mathsf{k}\cdot{\mathsf{\boldsymbol{\zeta}}}){\mathsf{\boldsymbol{\zeta}}}$
(15)

for some unit vector ${\mathsf{\boldsymbol{\zeta}}}\in\mathcal{S}(\tau)$.
Since for a massless scalar field $\mathsf{k}$ is light-like, it follows that
$(\mathsf{k}\cdot\mathsf{u})^{2}=(\mathsf{k}\cdot{\mathsf{\boldsymbol{\zeta}}})^{2}$.
This means that for a spherically symmetric smearing function the Fourier
transform in Eq. (14) is real and depends only on
$\absolutevalue{\mathsf{k}\cdot\mathsf{u}}$,

$F(\mathsf{k},\tau)=F(\absolutevalue{\mathsf{k}\cdot\mathsf{u}}).$ (16)

##  IV Assisted harvesting and catalysis of quantum coherence

Suppose now that before the interaction is switched on at a time
$\tau_{\text{on}}$, the combined system of detector and field starts out in a
separable state of the form

$\outerproduct{g}{g}\otimes\sigma_{\phi},$ (17)

where the detector occupies its lowest energy level and the field is in a
state $\sigma_{\phi}$. The final state of the system after a time
$\tau_{\text{off}}$ at which the interaction is switched off, can be obtained
by evolving Eq. (17) with the unitary operator

$\hat{U}=\mathcal{T}\text{exp}\left(-i\int\limits_{\tau_{\text{on}}}^{\tau_{\text{off}}}\hat{H}_{\text{int}}(\tau)d\tau\right),$
(18)

where $\mathcal{T}$ denotes time ordering. Assuming that the switching
function has a compact support we can extend the limits over $\pm\infty$.
Setting

$\hat{\Phi}=\int\limits_{-\infty}^{+\infty}\chi(\tau)e^{-i\Omega\tau}\partial_{\tau}\hat{\phi}_{f}(\mathsf{x}(\tau))d\tau,$
(19)

Eq. (18) can then be rewritten as

$\hat{U}=\text{exp}\left[-i\lambda(\outerproduct{e}{g}\otimes\hat{\Phi}^{\dagger}+\outerproduct{g}{e}\otimes\hat{\Phi})\right].$
(20)

Tracing out the field degrees of freedom, one can obtain the state of the
detector after the interaction which in this case is equal to

$\rho_{\text{\tiny$D$}}=\left(\begin{array}[]{cc}1-\lambda^{2}\tr(\hat{\Phi}^{\dagger}\sigma_{\phi}\hat{\Phi})&i\lambda\tr(\hat{\Phi}\sigma_{\phi})\\\
-i\lambda\tr(\hat{\Phi}^{\dagger}\sigma_{\phi})&\lambda^{2}\tr(\hat{\Phi}^{\dagger}\sigma_{\phi}\hat{\Phi})\end{array}\right)+\mathcal{O}(\lambda^{3}).$
(21)

In a similar fashion, by taking the partial trace over the detector’s Hilbert
space, we can obtain the state of the field after harvesting,

$\sigma_{\phi}^{\prime}=\sigma_{\phi}+\lambda^{2}\left(\hat{\Phi}^{\dagger}\sigma_{\phi}\hat{\Phi}-\frac{1}{2}\left\\{\hat{\Phi}\hat{\Phi}^{\dagger},\sigma_{\phi}\right\\}\right)+\mathcal{O}(\lambda^{4}).$
(22)

With the help of Eqs. (3) and (21) the amount of coherence harvested to the
detector to lowest order in the coupling constant is equal to

$C=2\lambda\absolutevalue{\tr(\hat{\Phi}\sigma_{\phi})}.$ (23)

Defining

$\mathcal{F_{\pm}}(\mathbf{k})=\int\limits_{-\infty}^{+\infty}\chi(\tau)e^{\pm
i\Omega\tau}\partial_{\tau}\left(F(\mathsf{k},\tau)e^{i\mathsf{k}\cdot\mathsf{x}(\tau)}\right)d\tau,$
(24)

Eq. (23) can be written as

$C=2\lambda\absolutevalue{\int\frac{d^{n}\mathbf{k}}{\sqrt{(2\pi)^{n}2\absolutevalue{\mathbf{k}}}}\Big{(}\mathcal{F}_{-}(\mathbf{k})a(\mathbf{k})+\mathcal{F}^{*}_{+}(\mathbf{k})a^{*}(\mathbf{k})\Big{)}},$
(25)

where

$a(\mathbf{k})=\tr(\hat{a}_{\mathbf{k}}\sigma_{\phi})$ (26)

is the _coherent amplitude distribution_ of the field.

Suppose that we wish to repeat the process and extract coherence onto a fresh
detector copy. It is straightforward to see that for the $m$-th harvest one
can extract an amount of

$C^{(m)}=2\lambda\absolutevalue{\tr(\hat{\Phi}\sigma_{\phi}^{(m)})}$ (27)

units of coherence from a perturbed field in the state

$\sigma_{\phi}^{(m)}=\sigma_{\phi}^{(m-1)}+\lambda^{2}\left(\hat{\Phi}^{\dagger}\sigma_{\phi}^{(m-1)}\hat{\Phi}-\frac{1}{2}\left\\{\hat{\Phi}\hat{\Phi}^{\dagger},\sigma_{\phi}^{(m-1)}\right\\}\right).$
(28)

By combining Eqs. (27) and (28) and exploiting the cyclic property of the
trace as well as the fact that $[\hat{\Phi},\hat{\Phi}^{\dagger}]$ is a
$c$-number (for proof see Appendix A) it follows that

$C^{(m+1)}=C^{(m)}\absolutevalue{1+\frac{\lambda^{2}}{2}\left[\hat{\Phi},\hat{\Phi}^{\dagger}\right]},$
(29)

so to lowest order in the coupling constant the amount of coherence harvested
each time remains the same.

Figure 2: Assisted harvesting of quantum coherence. A moving two-level system,
initially in its ground state at some time $t<t_{\text{on}}$, interacts with a
massless scalar field through a derivative coupling. The process requires an
external flow of energy which assists harvesting by increasing the combined
system’s coherence. After the interaction is switched off at a time
$t_{\text{off}}$ the detector will find itself in a superposition between its
energy levels.

Let’s focus our attention on normalised smearing and switching functions such
that

$\int\limits_{-\infty}^{+\infty}\chi(\tau)d\tau=\int_{\mathcal{S}(\tau)}f(\boldsymbol{\xi})d^{n}\boldsymbol{\xi}=1,$
(30)

and define

$R=\int_{\mathcal{S}(\tau)}\absolutevalue{\xi}f(\boldsymbol{\xi})d^{n}\boldsymbol{\xi}$
(31)

as the mean radius of the detector and

$T=\int\limits_{-\infty}^{+\infty}\absolutevalue{\tau}\chi(\tau)d\tau$ (32)

as the mean interaction duration respectively. This will make it easier to
compare different setups and will allow the study, in a unified way, of the
effects that different sizes and finite interaction durations have on
harvesting as well as the limiting case of an instantaneous interaction in
which $\chi(\tau)=\delta(\tau)$. In this limit,
$[\hat{\Phi},\hat{\Phi}^{\dagger}]=0$ and the amount harvested each time is
exactly the same to any order (for more details see Appendix B). It seems that
when the detector interacts with the field through a delta coupling, coherence
harvesting is catalytic Åberg (2014). Even though in principle this is allowed
for infinite dimensional systems that act as coherence reservoirs Lostaglio
and Müller (2019); Marvian and Spekkens (2019), it is not certain if this is
the case here. Since the interaction Hamiltonian does not commute with the
unperturbed part, $\hat{H}_{\text{\tiny$D$}}+\hat{H}_{\phi}$, of the total
Hamiltonian, the process requires an outside supply of positive energy $\Delta
E$ each time Hackl and Jonsson (2019); Bény  _et al._ (2018). Energy non-
conserving unitaries like the one in Eq. (18) can increase the coherence of
the combined system assisting in the extraction process Mani and Karimipour
(2015); Bu  _et al._ (2017) (see Fig. 2). Nonetheless a necessary condition
for extracting a non trivial amount of coherence is for the field to be in a
state with a non-zero coherent amplitude distribution.

##  V Inertial detectors

We will now consider an inertial detector which is moving along a worldline
with a constant velocity $\boldsymbol{\upsilon}$, and whose center of mass
coordinates is given by

$\mathsf{x}(\tau)=\mathsf{u}\tau,$ (33)

where $\mathsf{u}=\gamma(1,\boldsymbol{\upsilon})$ is its four-velocity, with
$\gamma=1/\sqrt{1-\upsilon^{2}}$ the Lorentz factor. For a spherically
symmetric smearing function with a positive Fourier transform, it can be
proven that

######  Theorem.

For a suitable choice of the coherent amplitude distribution’s phase the
maximum amount of harvested coherence to lowest order, is obtained by a
detector interacting instantaneously with the field.

###### Proof.

Taking the absolute value inside the integral in Eq. (25) we find that

$C\leq
2\lambda{\int\frac{d^{n}\mathbf{k}}{\sqrt{(2\pi)^{n}2\absolutevalue{\mathbf{k}}}}\absolutevalue{a(\mathbf{k})}\left(\absolutevalue{\mathcal{F}_{-}(\mathbf{k})}+\absolutevalue{\mathcal{F}_{+}(\mathbf{k})}\right)}.$
(34)

For a detector moving with a constant velocity the Fourier transform of the
smearing function no longer depends on its proper time, in this case

$\mathcal{F}_{-}(\mathbf{k})=i(\mathsf{k}\cdot\mathsf{u})F(\absolutevalue{\mathsf{k}\cdot\mathsf{u}})X^{*}(\Omega-\mathsf{k}\cdot\mathsf{u})$
(35)

and

$\mathcal{F}_{+}(\mathbf{k})=i(\mathsf{k}\cdot\mathsf{u})F(\absolutevalue{\mathsf{k}\cdot\mathsf{u}})X(\Omega+\mathsf{k}\cdot\mathsf{u})$
(36)

where

$X(\Omega\pm\mathsf{k}\cdot\mathsf{u})=\int\limits_{-\infty}^{+\infty}\chi(\tau)e^{i(\Omega\pm\mathsf{k}\cdot\mathsf{u})\tau}d\tau.$
(37)

Because of the normalization property in Eq. (30),
$\absolutevalue{X(\Omega\pm\mathsf{k}\cdot\mathsf{u})}\leq 1$ so finally

$C\leq
4\lambda\int\frac{(-\mathsf{k}\cdot\mathsf{u})}{\sqrt{(2\pi)^{n}2\absolutevalue{\mathbf{k}}}}F(\absolutevalue{\mathsf{k}\cdot\mathsf{u}})\absolutevalue{a(\mathbf{k})}d^{n}\mathbf{k},$
(38)

where equality holds for $\chi(\tau)=\delta(\tau)$ and a coherent amplitude
distribution with phase, $\text{arg}(a(\mathbf{k}))=\frac{\pi}{2}$ 222In the
Unruh-DeWitt interaction where the factor $(-\mathsf{k}\cdot\mathsf{u})$ in
the numerator is absent, the Theorem holds for an arbitrary motion of the
detector as long as $\mathsf{x}(0)=0$..

Note that if the Fourier transform of the smearing function is not positive
then Eq. (38) is only an upper bound on the amount of harvested coherence. ∎

If the amplitude distribution is also spherically symmetric then

$C=2\lambda\left|\int\frac{(-\mathsf{k}\cdot\mathsf{u})F(\absolutevalue{\mathsf{k}\cdot\mathsf{u}})}{\sqrt{(2\pi)^{n}2\absolutevalue{\mathbf{k}}}}\left[a(\absolutevalue{\mathbf{k}})X^{*}(\Omega-\mathsf{k}\cdot\mathsf{u})\right.\right.\\\
\left.-{a}^{*}(\absolutevalue{\mathbf{k}})X(\Omega+\mathsf{k}\cdot\mathsf{u})\right]d^{n}\mathbf{k}\Bigg{|},$
(39)

which for a static detector reduces to

$C=\frac{2\lambda
s_{n}}{\sqrt{2(2\pi)^{n}}}\left|\int_{0}^{\infty}k^{n-\frac{1}{2}}F(k)\left[a(k)X^{*}(\Omega+k)\right.\right.\\\
\left.-a^{*}(k)X(\Omega-k)\right]dk\Bigg{|},$ (40)

where $s_{n}=\frac{2\pi^{n/2}}{\Gamma(n/2)}$ is the surface area of the unit
$n$-sphere. By boosting the four-momentum $\mathsf{k}$ to the detector’s frame
of reference it can be shown that Eq. (39) is equivalent to Eq. (40) with a
symmetric coherent amplitude distribution of the form

$a_{\upsilon}(k)=\frac{1}{s_{n}}\int
a\left(\frac{k}{\gamma(1-\mathbf{\boldsymbol{\upsilon}}\cdot\hat{\mathbf{k}})}\right)\frac{d\hat{\mathbf{k}}}{{[\gamma(1-\mathbf{\boldsymbol{\upsilon}}\cdot\hat{\mathbf{k}})]^{n-\frac{1}{2}}}}.$
(41)

From the detector’s point of view, the field’s coherent amplitude is
equivalent to a mixture of Doppler shifted distributions with weight equal to
$[s_{n}\gamma(1-\boldsymbol{\upsilon}\cdot\hat{\mathbf{k}})^{n-\frac{1}{2}}]^{-1}$.
For a similar result regarding the interaction of an inertial detector with a
heat bath see Papadatos and Anastopoulos (2020).

##  VI Assisted harvesting and catalysis from a coherent field

For a coherent state $\ket{a}$ of the field, the coherent amplitude
distribution in Eq. (26) is equal to the eigenvalue of the annihilation
operator with mode $\mathbf{k}$

$\hat{a}_{\mathbf{k}}\ket{a}=a(\mathbf{k})\ket{a},$ (42)

in this case the amount of harvested coherence to lowest order is given by the
expectation value of the field operator $\hat{\Phi}$

$C=2\lambda|{\bra{a}\hat{\Phi}\ket{a}}|.$ (43)

The energetic cost associated with harvesting is equal to the energy
difference between the final and initial states of the combined system of
detector and field

$\Delta
E=\tr(\hat{H}_{\text{\tiny$D$}}(\rho_{\text{\tiny$D$}}-\outerproduct{g}{g}))+\tr(\hat{H}_{\phi}(\sigma_{\phi}^{\prime}-\outerproduct{a}{a})).$
(44)

To lowest order this splits into two contributions

$\Delta E=\Delta E_{\text{coh}}+\Delta E_{\text{vac}},$ (45)

where

$\Delta
E_{\text{coh}}=\frac{C^{2}}{4}\left(\Omega+4\real\left[\frac{\bra{a}[\hat{\Phi},\hat{H}_{\phi}]\ket{a}}{\bra{a}\hat{\Phi}\ket{a}}\right]\right)$
(46)

is the cost associated with harvesting and

$\Delta
E_{\text{vac}}=\frac{\lambda^{2}}{2(2\pi)^{n}}\int\left(1+\frac{\Omega}{\absolutevalue{\mathbf{k}}}\right)\absolutevalue{\mathcal{F}_{-}(\mathbf{k})}^{2}d^{n}\mathbf{k}.$
(47)

is the cost of interacting with the vacuum Simidzija and Martín-Martínez
(2017a).

Let us consider an inertial detector and a harvesting process in which the
switching and smearing functions are respectively given by the following
Gaussians

$\chi(\tau)=\frac{\text{exp}\left(-\frac{\tau^{2}}{\pi T^{2}}\right)}{\pi T}$
(48)
$f(\boldsymbol{\xi})=\frac{\text{exp}\left(-\frac{\boldsymbol{\xi}^{2}}{\pi
R_{n}^{2}}\right)}{(\pi R_{n})^{n}},$ (49)

while the state of the field is described by a coherent amplitude distribution
with a unit average number of excited quanta of the form

$a(\mathbf{k})=\frac{\text{exp}(-\frac{k^{2}}{2\pi{E}_{n}^{2}}+i\frac{\pi
r}{2})}{(\pi{E}_{n})^{n/2}},\quad r=0,1$ (50)

where

$E_{n}=\frac{s_{n+1}}{\pi s_{n}}E\quad\text{and}\quad R_{n}=\frac{s_{n+1}}{\pi
s_{n}}R,$ (51)

with $E=\bra{a}\hat{H}_{\phi}\ket{a}$ the mean initial energy of the field.
Note that even though the support of Eq. (48) is no longer compact, as was
originally required, the analysis is expected to present a good approximation
to a compact switching function of the form

$\chi(\tau)=\begin{cases}{\text{exp}(-\frac{\tau^{2}}{\pi T^{2}})}/(\pi
T),&|\tau|\leq\mathcal{T}\\\ 0,&\text{otherwise}\end{cases}$ (52)

provided that $\mathcal{T}\geq 4\sqrt{\pi}T$. We will now treat the static and
moving cases separately.

###  VI.1 Static detector

For $\upsilon=0$ the Fourier transforms of the switching and smearing
functions are equal to

$X(\Omega\pm k)=\text{exp}\left[{-\frac{\pi(\Omega\pm k)^{2}T^{2}}{4}}\right]$
(53)

and

$F(\mathsf{k})=\text{exp}\left[-\frac{\pi k^{2}R_{n}^{2}}{4}\right]$ (54)

respectively. Inserting these into Eq. (40) we obtain that the amount of
harvested coherence, which now depends on the initial energy of the field, the
mean interaction duration and the mean radius of the detector is

$C(E,T,R)=\frac{4\lambda
s_{n}}{\sqrt{2(2\pi^{2}{E}_{n})^{n}}}e^{-\frac{\pi\Omega^{2}T^{2}}{4}}\\\
\times\int_{0}^{\infty}k^{n-\frac{1}{2}}e^{-\mathrm{a}{k^{2}}}\sinh^{1-r}(bk)\cosh^{r}(bk)dk,$
(55)

with

$\mathrm{a}=\frac{1}{2\pi{E}_{n}^{2}}\left[1+\frac{\pi^{2}E_{n}^{2}(R^{2}_{n}+T^{2})}{2}\right],\quad
b=\frac{\pi\Omega T^{2}}{2}.$ (56)

The integral on the right hand side is equal to

(a) $r=1$

(b) $r=0$

Figure 3: Amount of harvested coherence $C/\bar{\lambda}$ from a coherent
scalar field in $1+1$ dimensions and a Gaussian amplitude distribution with
phase a) $\phi=\frac{\pi}{2}$ and b) $\phi=0$, as a function of the mean
initial energy of the field (in units $\Omega$) and the mean interaction
duration (in units $1/\Omega$), for a detector with mean radius $R=1/\Omega$.

(a) $r=1$

(b) $r=0$

Figure 4: Amount of harvested coherence $C/\bar{\lambda}$ from a coherent
scalar field in $3+1$ dimensions and a Gaussian amplitude distribution with
phase a) $\phi=\frac{\pi}{2}$ and b) $\phi=0$, as a function of the mean
initial energy of the field (in units $\Omega$) and the mean interaction
duration (in units $1/\Omega$), for a detector with mean radius $R=1/\Omega$

$\int\limits_{0}^{\infty}k^{n-\frac{1}{2}}e^{-\mathrm{a}k^{2}}\sinh^{1-r}(bk)\cosh^{r}(bk)dk=\frac{\Gamma(n+\frac{1}{2})}{2(2\mathrm{a})^{\frac{n}{2}+\frac{1}{4}}}e^{\frac{b^{2}}{8\mathrm{a}}}\left[D_{-n-\frac{1}{2}}\left(-\frac{b}{\sqrt{2\mathrm{a}}}\right)-(-1)^{r}D_{-n-\frac{1}{2}}\left(\frac{b}{\sqrt{2\mathrm{a}}}\right)\right],\quad
b>0$ (57)

where $D_{p}(z)$ denotes the _parabolic cylinder function_ Gradshteyn and
Ryzhik (2014). In a similar way it can be shown that

$\Delta
E_{\text{coh}}=\frac{C^{2}}{4}\left[\Omega-\frac{4(n+\frac{1}{2})}{\sqrt{2\mathrm{a}}}\frac{D_{-n-\frac{3}{2}}\left(-\frac{b}{\sqrt{2\mathrm{a}}}\right)+(-1)^{r}D_{-n-\frac{3}{2}}\left(\frac{b}{\sqrt{2\mathrm{a}}}\right)}{D_{-n-\frac{1}{2}}\left(-\frac{b}{\sqrt{2\mathrm{a}}}\right)-(-1)^{r}D_{-n-\frac{1}{2}}\left(\frac{b}{\sqrt{2\mathrm{a}}}\right)}\right]$
(58)

and

$\Delta E_{\text{vacuum}}=\frac{\lambda^{2}\pi
s_{n}\Gamma(n+1)}{(8\pi^{2}\mathrm{a}^{\prime})^{\frac{n+1}{2}}}e^{-\frac{\pi\Omega^{2}T^{2}}{2}+\frac{b^{2}}{8\mathrm{a}^{\prime}}}\left[\frac{n+1}{\sqrt{2\mathrm{a}^{\prime}}}D_{-n-2}\left(\frac{2b}{\sqrt{2\mathrm{a}^{\prime}}}\right)+\Omega
D_{-n-1}\left(\frac{2b}{\sqrt{2\mathrm{a}^{\prime}}}\right)\right],$ (59)

where

$\mathrm{a}^{\prime}=\frac{\pi(R_{n}^{2}+T^{2})}{2}.$ (60)

In Figs. 4 and 4 we present the amount of coherence harvested, scaled by the
dimensionless coupling constant $\bar{\lambda}=\lambda\Omega^{\frac{n+1}{2}}$,
as a function of the initial mean energy $E$ of the field (in units $\Omega$)
and the interaction duration $T$ (in units $1/\Omega$) for a $1+1$ and a $3+1$
dimensional Mikowski spacetime respectively. In order to simplify the
situation we will tacitly assume from now on that the mean radius of the qubit
is equal to its transition wavelength $R=1/\Omega$. It is clear from both
figures that the harvesting profile depends strongly on the phase of the
coherent amplitude distribution. For $r=1$ and for a fixed initial field
energy, the maximum amount that can be harvested is obtained through the use
of an instantaneous interaction ($T=0$), in agreement with the Theorem of Sec.
V. When $r=0$ it is impossible to harvest coherence to a qubit interacting
instantaneously with the field, in this case the maximum is obtained for
interaction durations comparable to the mean radius. In both settings, if the
initial energy of the field is zero the amount of coherence harvested
vanishes. This is also true for very large energy values. Qualitatively,
harvesting is more efficient for field energies comparable to the energy gap.
For a resonant energy of the field, $E=\Omega$, it is possible to extend the
process to greater interaction times compared to other energies and still
extract a small amount of coherence.

Now with the help of Eq. (69) of Appendix A, Eqs. (53)-(54) and Eq. (57) it
can be shown that

$\lambda^{2}[\hat{\Phi},\hat{\Phi}^{\dagger}]=-\frac{2ns_{n}\bar{\lambda}^{2}}{s_{2n}[4\pi\Omega^{2}(R_{n}^{2}+T^{2})]^{\frac{n+1}{2}}}e^{-\frac{\pi\Omega^{2}T^{2}(2R^{2}_{n}+T^{2})}{4(R^{2}_{n}+T^{2})}}\\\
\times\left[D_{-n-1}\left(\\!-\sqrt{\frac{\pi\Omega^{2}T^{4}}{R^{2}_{n}+T^{2}}}\right)\\!-D_{-n-1}\left(\sqrt{\frac{\pi\Omega^{2}T^{4}}{R^{2}_{n}+T^{2}}}\right)\right].$
(61)

From Fig. 5 it can be seen that for $\bar{\lambda}<<1$ and $R=1/\Omega$ this
term is negligible. Since the maximum amount of harvested coherence is of the
same order as $\bar{\lambda}$ then, according to Eq. (29), we can repeat the
process $m$ times for a total of $C_{\text{tot}}=\mathcal{O}(m\bar{\lambda})$
units of coherence. Assuming that for a phase-less coherent amplitude
distribution obtaining the maximum in each harvest requires a time of
approximately $T=1/\Omega$ it follows the total duration is of the order
$\mathcal{O}(m/\Omega)$. To extract a single unit of coherence requires
therefore approximately $\mathcal{O}(1/\bar{\lambda}\Omega)$ seconds. For a
transition frequency in the optical spectrum and $\bar{\lambda}=10^{-3}$ this
time is of the order of $10^{-12}$ seconds.

Figure 5: $\lambda^{2}[\hat{\Phi},\hat{\Phi}^{\dagger}]/\bar{\lambda}^{2}$ as
a function of the mean interaction duration (in units $1/\Omega$), for a
detector with mean radius $R=1/\Omega$.

###  VI.2 Detector moving at a constant velocity

(a) $C_{0.8}/\bar{\lambda}$ ($r=1$)

(b) $C_{0}/C_{0.8}$ ($r=1$)

(c) $E=0.1\Omega$ ($r=1$)

(d) $C_{0.8}/\bar{\lambda}$ ($r=0$)

(e) $C_{0}/C_{0.8}$ ($r=0$)

(f) $E=0.1\Omega$ ($r=0$)

Figure 6: Left: Amount of harvested coherence, $C_{0.8}/\bar{\lambda}$, in
$1+1$ dimensions. Center: Amount of swelling $C_{0}/C_{0.8}$. Right:
Comparison between a static and a moving detector for an initial energy of the
field $E=0.1\Omega$.

(a) $C_{0.8}/\bar{\lambda}$ ($r=1$)

(b) $C_{0}/C_{0.8}$ ($r=1$)

(c) $E=0.2\Omega$ ($r=1$)

(d) $C_{0.8}/\bar{\lambda}$ ($r=0$)

(e) $C_{0}/C_{0.8}$ ($r=0$)

(f) $E=0.2\Omega$ ($r=0$)

Figure 7: Left: Amount of harvested coherence, $C_{0.8}/\bar{\lambda}$, in
$3+1$ dimensions. Center: Amount of swelling $C_{0}/C_{0.8}$. Right:
Comparison between a static and a moving detector for an initial energy of the
field $E=0.2\Omega$.

According to Eq. (41), a detector moving at a constant velocity still
perceives the field as a coherent state but in a mixture of static coherent
amplitude distributions of the form (50) with Doppler shifted energies equal
to

$E(\boldsymbol{\upsilon})=E\gamma(1-\boldsymbol{\upsilon}\cdot\hat{\mathbf{k}}).$
(62)

The amount of harvested coherence in this case is given by

$C_{\upsilon}(E,T,R)=\frac{1}{s_{n}}\int\frac{C(E(\boldsymbol{\upsilon}),T,R)}{\gamma(1-\boldsymbol{\upsilon}\cdot\hat{\mathbf{k}})^{\frac{n-1}{2}}}d\hat{\mathbf{k}}.$
(63)

In Figs. 7 and 7 we numerically evaluate this amount for a detector moving at
a constant relativistic speed of $\upsilon=0.8$, in $1+1$ and $3+1$ dimensions
respectively. We observe that close to resonance the amount of coherence
harvested decreases with an increasing value of the detector’s speed. As in
Kollas  _et al._ (2020), for lower and higher initial energies of the field
there exist “swelling” regions, where it is possible to extract more coherence
to a moving than to a static detector. However, this effect becomes less
intense for a higher spacetime dimension.

###  VI.3 Assisted catalysis

For an instantaneous interaction coherence harvesting is catalytic. Despite
the fact that after each harvest the state of the field has changed, it is
possible to extract the same amount of coherence to a sequence of detectors.
Ignoring the trivial case of $r=0$, for a coherent amplitude distribution with
phase $\phi=\frac{\pi}{2}$ each detector will harvest

$C_{\upsilon}(E)=\frac{2\lambda\Gamma(3/4)}{(2\pi)^{\frac{1}{4}}}\left[\frac{E_{+}}{\left(1+\frac{\pi^{2}E_{+}^{2}}{\Omega^{2}}\right)^{\frac{3}{4}}}+\frac{E_{-}}{\left(1+\frac{\pi^{2}E_{-}^{2}}{\Omega^{2}}\right)^{\frac{3}{4}}}\right]$
(64)

units of coherence in $1+1$ and

$C_{\upsilon}(E)=\\\
\frac{16\bar{\lambda}\Gamma(3/4)}{(2\pi^{9})^{\frac{1}{4}}\gamma\upsilon}\left[\left(1+\frac{\pi^{2}E_{-}^{2}}{32\Omega^{2}}\right)^{-\frac{3}{4}}-\left(1+\frac{\pi^{2}E_{+}^{2}}{32\Omega^{2}}\right)^{-\frac{3}{4}}\right]$
(65)

in $3+1$ dimensions, where $E_{\pm}=E\gamma(1\pm\upsilon)$ denote the field’s
relativistic Doppler shifted energies. As has already been mentioned in Sec.
IV, catalysis is an energy consuming process. The cost of each extraction to
lowest order in this case is equal to

$\Delta
E=\begin{cases}\frac{C_{\upsilon}^{2}(E)\Omega}{4}+\frac{\bar{\lambda}^{2}\Omega}{\pi^{2}}(1+\frac{\gamma}{\sqrt{2}}),&n=1\\\
&\\\
\frac{C_{\upsilon}^{2}(E)\Omega}{4}+\frac{8\bar{\lambda}^{2}\Omega}{\pi^{4}}\left(1+\frac{3\gamma}{\sqrt{2}}\right),&n=3.\end{cases}$
(66)

In Fig. 8 we plot the amount of coherence harvested through catalysis along
with its energy cost (in units $\Omega$) as a function of the initial energy
of the field. For field energies close to resonance the amount obtained is
maximized. Once again it can be seen that this amount decreases for an
increasing value of the detector’s speed. This is also true for the energy
cost associated with harvesting. On the other hand, the cost associated with
the vacuum remains relatively constant.

##  VII Conclusions

(a) $\upsilon=0$

(b) $\upsilon=0.6$

(c) $\upsilon=0.8$

(d) $\upsilon=0$

(e) $\upsilon=0.6$

(f) $\upsilon=0.8$

Figure 8: Amount of harvested coherence $C_{\upsilon}/\bar{\lambda}$ and cost
in energy $\Delta E/\Omega\bar{\lambda}^{2}$ as a function of the initial
energy of the field (in units $\Omega$) for various detector speeds. Upper:
1+1 dimensions. Lower: 3+1 dimensions.

We have thoroughly investigated the conditions under which an UDW detector,
coupled to a massless scalar field through a derivative coupling, succeeds in
harvesting quantum coherence. It was proven that for an instantaneous
interaction between detector and field, harvesting is catalytic, i.e., the
same amount can be repeatedly extracted. For a suitable choice of the field’s
coherent amplitude distribution and an inertial detector, when the Fourier
transform of the smearing function is positive this is also the maximum amount
that can be obtained. By considering as an example a harvesting protocol in
which the switching, smearing and coherent amplitude functions are Gaussian,
it was demonstrated that for a coherent state of the field the process depends
on the phase of the amplitude, the mean initial field energy, the mean
interaction duration and the mean radius of the detector. We observed that,
for a resonant energy of the field, it is possible to extend the process to
longer interaction durations. It was also shown that the total time required
to harvest, through repeated applications of the protocol, a single unit of
coherence to a sequence of detectors is very short.

For a detector moving at a constant velocity and for a mean radius equal to
the inverse of its transition frequency we verify the presence of swelling
affects as was reported in Kollas  _et al._ (2020). Nonetheless, since energy
non-conserving interactions such as the one considered here are coherence
generating Mani and Karimipour (2015); Bu  _et al._ (2017), it is possible
that this increase is due to the interaction. To avoid this possibility and in
order to be able to determine how different parameters which are intrinsic to
the combined system of qubit and field affect harvesting, we will study, in
future work , protocols under energy conserving interactions such as the one
given by the Glauber photodetection model Glauber (1963); Svaiter and Svaiter
(1992) for example.

###### Acknowledgements.

The authors wish to thank Lena S. Peurin for fruitful discussions during
preparation of this manuscript. D. M.’s research is co-financed by Greece and
the European Union (European Social Fund-ESF) through the Operational
Programme “Human Resources Development, Education and Lifelong Learning” in
the context of the project “Reinforcement of Postdoctoral Researchers - 2nd
Cycle” (MIS-5033021), implemented by the State Scholarships Foundation (IKY).

##  Appendix A Useful relations

Let

$\mathcal{F_{\pm}}(\mathbf{k})=\int\limits_{-\infty}^{+\infty}\chi(\tau)e^{\pm
i\Omega\tau}\partial_{\tau}\left(F(\mathsf{k},\tau)e^{i\mathsf{k}\cdot\mathsf{x}(\tau)}\right)d\tau.$
(67)

Taking advantage of the commutation relations between the creation and
annihilation operators in Eq. (8) and rewriting $\hat{\Phi}$ as

$\hat{\Phi}=\int\frac{d^{n}\mathbf{k}}{\sqrt{(2\pi)^{n}2\absolutevalue{\mathbf{k}}}}\left(\mathcal{F}_{-}(\mathbf{k})\hat{a}_{\mathbf{k}}+\mathcal{F}_{+}^{*}(\mathbf{k})\hat{a}_{\mathbf{k}}^{\dagger}\right)$
(68)

we can easily compute the following commutators

$[\hat{\Phi},\hat{\Phi}^{\dagger}]=\int\frac{d^{n}\mathbf{k}}{{(2\pi)^{n}2\absolutevalue{\mathbf{k}}}}\left(\absolutevalue{\mathcal{F}_{-}(\mathbf{k})}^{2}-\absolutevalue{\mathcal{F}_{+}(\mathbf{k})}^{2}\right)$
(69)
$[\hat{\Phi},\hat{H}_{\phi}]=\int\frac{\absolutevalue{\mathbf{k}}d^{n}\mathbf{k}}{\sqrt{(2\pi)^{n}2\absolutevalue{\mathbf{k}}}}\left(\mathcal{F}_{-}(\mathbf{k})\hat{a}_{\mathbf{k}}-\mathcal{F}_{+}^{*}(\mathbf{k})\hat{a}_{\mathbf{k}}^{\dagger}\right)$
(70)
$\left[[\hat{\Phi},\hat{H}_{\phi}],(\hat{\Phi}^{\dagger})^{m}\right]=mc^{2}(\hat{\Phi}^{\dagger})^{m-1}$
(71)

where

$c^{2}=\frac{1}{{2(2\pi)^{n}}}\int\left(\absolutevalue{\mathcal{F}_{-}(\mathbf{k})}^{2}+\absolutevalue{\mathcal{F}_{+}(\mathbf{k})}^{2}\right)d^{n}\mathbf{k}.$
(72)

##  Appendix B Assisted catalysis for instantaneous interactions

For $\chi(\tau)=\delta(\tau)$ it is easy to see from Eq. (19) that
$\hat{\Phi}=\hat{\Phi}^{\dagger}$. The unitary evolution operator in Eq. (18)
can then be written as Simidzija and Martín-Martínez (2017b)

$\hat{U}=I\otimes\cos(\lambda\hat{\Phi})-i\sigma_{x}\otimes\sin(\lambda\hat{\Phi})$
(73)

where $\sigma_{x}=\outerproduct{e}{g}+\outerproduct{g}{e}$. Evolving the
separable state of the combined system of detector and field in Eq. (17) and
tracing out the field degrees of freedom we find that the state of the
detector after the interaction is equal to

$\rho_{\text{\tiny$D$}}=\left(\begin{array}[]{cc}\tr(\cos^{2}(\lambda\hat{\Phi})\sigma_{\phi})&\frac{i}{2}\tr(\sin(2\lambda\hat{\Phi})\sigma_{\phi})\\\
-\frac{i}{2}\tr(\sin(2\lambda\hat{\Phi})\sigma_{\phi})&\tr(\sin^{2}(\lambda\hat{\Phi})\sigma_{\phi})\end{array}\right).$
(74)

Similarly the state of the field after harvest is given by

$\sigma^{\prime}_{\phi}=\cos(\lambda\hat{\Phi})\sigma_{\phi}\cos(\lambda\hat{\Phi})+\sin(\lambda\hat{\Phi})\sigma_{\phi}\sin(\lambda\hat{\Phi}).$
(75)

From Eqs. (74) and (75) and the definition of the $\ell_{1}$-norm of coherence
it can be seen that the amount of harvested coherence extracted the second
time is equal to

$\displaystyle C^{\prime}$
$\displaystyle=\absolutevalue{\tr(\sin(2\lambda\hat{\Phi})\sigma_{\phi}^{\prime})}$
$\displaystyle=\absolutevalue{\tr(\sin(2\lambda\hat{\Phi})\sigma_{\phi})}$
(76)

where in the last equality we have taken advantage of the cyclic property of
the trace and the fact that
$\cos^{2}(\lambda\hat{\Phi})+\sin^{2}(\lambda\hat{\Phi})=I_{\phi}$.

We will now compute the energy difference $\Delta E$ between the initial and
final states of the combined system of field plus detector and show that it is
always positive. This means that catalysis is an energy consuming process so
it cannot be repeated indefinitely.

From Eqs. (74) and (75) it is easy to see that the difference in energy before
and after extraction is

$\displaystyle\Delta E$
$\displaystyle=\tr(\hat{H}_{\text{\tiny$D$}}(\rho_{\text{\tiny$D$}}-\outerproduct{g}{g}))+\tr(\hat{H}_{\phi}(\sigma_{\phi}^{\prime}-\sigma_{\phi}))$
$\displaystyle=\Omega\tr(\sin^{2}(\lambda\hat{\Phi})\sigma_{\phi})$
$\displaystyle+\frac{1}{2}\tr(\left[[\cos(\lambda\hat{\Phi}),\hat{H}_{\phi}],\cos(\lambda\hat{\Phi})\right]\sigma_{\phi})$
$\displaystyle+\frac{1}{2}\tr(\left[[\sin(\lambda\hat{\Phi}),\hat{H}_{\phi}],\sin(\lambda\hat{\Phi})\right]\sigma_{\phi}).$
(77)

The first term on the right hand side as a product of two positive matrices is
evidently positive, indeed this must be the case since the qubit started out
in its ground state and can only gain energy. On the other hand from Eq. (71)
it can be shown by iteration that

$\left[[\hat{\Phi}^{\ell},\hat{H}_{\phi}],\hat{\Phi}^{m}\right]=\ell
mc^{2}\hat{\Phi}^{\ell+m-2}.$ (78)

This means that

$\left[[\cos(\lambda\hat{\Phi}),\hat{H}_{\phi}],\cos(\lambda\hat{\Phi})\right]=c^{2}\lambda^{2}\sin^{2}(\lambda\hat{\Phi})$
(79)

and

$\left[[\sin(\lambda\hat{\Phi}),\hat{H}_{\phi}],\sin(\lambda\hat{\Phi})\right]=c^{2}\lambda^{2}\cos^{2}(\lambda\hat{\Phi})$
(80)

so finally

$\Delta
E=\Omega\tr(\sin^{2}(\lambda\hat{\Phi})\sigma_{\phi})+\frac{c^{2}\lambda^{2}}{2}$
(81)

which is always positive.

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Gauge-invariant coefficients in perturbative quantum gravity

Fiorenzo Bastianelli${}^{\,a,b}$, Roberto Bonezzi${}^{\,c}$, Marco
Melis${}^{\,a}$

aDipartimento di Fisica e Astronomia “Augusto Righi”, Università di Bologna,
via Irnerio 46, I-40126 Bologna, Italy

bINFN, Sezione di Bologna, via Irnerio 46, I-40126 Bologna, Italy

cInstitute for Physics, Humboldt University Berlin,
Zum Großen Windkanal 6, D-12489 Berlin, Germany

###### Abstract

Heat kernel methods are useful for studying properties of quantum gravity. We
recompute here the first three heat kernel coefficients in perturbative
quantum gravity with cosmological constant to ascertain which ones are
correctly reported in the literature. They correspond to the counterterms
needed to renormalize the one-loop effective action in four dimensions. They
may be evaluated at arbitrary dimensions $D$, in which case they identify only
a subset of the divergences appearing in the effective action for $D\geq 6$.
Generically, these coefficients depend on the gauge-fixing choice adopted in
quantizing the Einstein-Hilbert action. However, they become gauge-invariant
once evaluated on-shell, i.e. using Einstein’s field equations with
cosmological constant. We identify them and use them as a benchmark for
checking alternative approaches to perturbative quantum gravity. One such
approach describes the graviton in first-quantization through the use of the
action of the ${\cal N}=4$ spinning particle, characterized by four
supersymmetries on the worldline and a set of worldline gauge invariances.
This description has been used for computing the gauge-invariant coefficients
as well. We verify their correctness at $D=4$, but find a mismatch at
arbitrary $D$ when comparing with the benchmark fixed earlier. We interpret
this result as signaling that the path integral quantization of the ${\cal
N}=4$ spinning particle should be amended. We perform this task by fixing the
correct counterterm that must be used in the worldline path integral
quantization of the ${\cal N}=4$ spinning particle to make it consistent in
arbitrary dimensions.

##  1 Introduction

A well-known approach to studying perturbative quantum gravity is to apply
covariant quantization schemes and the background field method to the
Einstein-Hilbert action, and then use heat kernel techniques for calculations,
as pioneered by DeWitt [1, 2]. An alternative approach is to treat the
graviton in first quantization, as in string theory. A model recently proposed
for the graviton makes use of the ${\cal N}=4$ spinning particle and related
BRST structure [3]. It has been used in a path integral approach in [4] to
study certain gauge-invariant coefficients corresponding to the on-shell, one-
loop divergences of the effective action of quantum gravity with cosmological
constant. These coefficients have been checked to be correct in $D=4$
dimensions. Here, we wish to study the general case of arbitrary $D$.

The most straightforward way to obtain these coefficients is to use the heat
kernels of the differential operators appearing in the gauge-fixed Einstein-
Hilbert action, and compute the total heat kernel coefficients of quantum
gravity out of them.

We are interested in the first three coefficients, which we denote by $a_{0}$,
$a_{1}$, $a_{2}$. They are associated with the divergences of the one-loop
effective action and should be renormalized away (additional divergences are
present for $D\geq 6$). The first one is universal and measures the number of
physical degrees of freedom of the graviton. As for the remaining ones, we
have found different, inequivalent expressions in the literature. Most likely
the different versions signal misprints. In any case, to ascertain which ones
are correctly reported, we have decided to recompute them, verifying that the
ones given in [1] and [2] which also differ between themselves are incorrect,
while finding agreement with the ones computed more recently in [5].

These coefficients computed for a generic background metric depend on the
gauge-fixing procedure adopted. They are expected to become gauge-invariant
once restricted to Einstein spaces. This is particularly evident for $a_{0}$,
which is independent of the background metric and measures directly the number
of physical polarizations of the graviton. We calculate these gauge-invariant
coefficients explicitly, to have at hand a benchmark that any alternative
attempt to perturbative quantum gravity should be able to reproduce.

With these coefficients at hand, we compare them with the ones obtained in [4]
through the ${\cal N}=4$ spinning particle and discover that there is
agreement at $D=4$, while a mismatch appears for $D\neq 4$. We take this
result as suggesting that the path integral quantization of the ${\cal N}=4$
particle should be reconsidered and improved. We do this by modifying the
counterterm used in the path integral quantization. The counterterm used in
[4] was selected by taking into account the algebra of the four
supersymmetries on the worldline that seemed to indicate a specific form of
the hamiltonian constraint, and it was found that the chosen counterterm could
reproduce correctly the results at $D=4$. However, the founding principle of
the ${\cal N}=4$ spinning particle description of the graviton is the BRST
symmetry, which requires a specific coupling to the curvature to achieve
nilpotency and differs from the one used in [4]. Modifying appropriately the
counterterm, and precisely in the way dictated by the BRST symmetry, we find
that the gauge-invariant coefficients are correctly reproduced by the path
integral of the ${\cal N}=4$ spinning particle at arbitrary $D$.

In this paper, we proceed as follows. In section 2, we describe the
calculation of the heat kernel coefficients in perturbative quantum gravity at
one loop, keeping the background metric arbitrary. In section 3, we identify
the corresponding gauge-invariant coefficients by using Einstein’s equations
with cosmological constant. In section 4, we reconsider the path integral for
the ${\cal N}=4$ spinning particle and fix the correct counterterm that must
go along with the path integral in worldline dimensional regularization. This
way the correct gauge-invariant coefficients are reproduced at arbitrary $D$
by the path integral on the circle of the ${\cal N}=4$ spinning particle. We
present our conclusions and outlook in section 5, leaving appendix A for
collecting useful formulae on one-loop effective actions and heat kernels.

##  2 Heat kernel coefficients in perturbative quantum gravity

Let us consider the Einstein-Hilbert action with cosmological constant
$\Lambda$ for the metric $G_{\mu\nu}$ in $D$ euclidean dimensions

$S[G_{\mu\nu}]=-\frac{1}{\kappa^{2}}\int
d^{D}x\sqrt{G}\,\Big{[}R(G)-2\Lambda\Big{]}$ (1)

where $\kappa^{2}=16\pi G_{N}$ is the gravitational coupling constant. The
effective action can be studied using the background field method. One splits
the metric as

$G_{\mu\nu}(x)=g_{\mu\nu}(x)+h_{\mu\nu}(x)$ (2)

where $g_{\mu\nu}$ is an arbitrary background metric and $h_{\mu\nu}$ the
quantum fluctuations. To obtain the one-loop effective action, it is enough to
expand the action at quadratic order in $h_{\mu\nu}$

$S[g+h]=\frac{1}{\kappa^{2}}\Big{[}S_{0}+S_{1}+S_{2}+\sum_{n=3}^{\infty}S_{n}\Big{]}$
(3)

where one finds

$\displaystyle\begin{split}S_{0}&=-\int
d^{D}x\sqrt{g}\big{[}R-2\Lambda\big{]}\;,\\\ S_{1}&=\int
d^{D}x\sqrt{g}\bigg{[}h^{\mu\nu}\biggl{(}R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+g_{\mu\nu}\Lambda\biggr{)}\bigg{]}\;,\\\
S_{2}&=\int
d^{D}x\sqrt{g}\bigg{[}-\frac{1}{4}h^{\mu\nu}(\nabla^{2}+2\Lambda-R)h_{\mu\nu}+\frac{1}{8}h(\nabla^{2}+2\Lambda-R)h-\frac{1}{2}\biggl{(}\nabla^{\nu}h_{\nu\mu}-\frac{1}{2}\partial_{\mu}h\biggr{)}^{2}\\\
&\hskip
66.86414pt-\frac{1}{2}\big{(}h^{\mu\lambda}h^{\nu}_{\lambda}-hh^{\mu\nu}\big{)}R_{\mu\nu}-\frac{1}{2}h^{\mu\lambda}h^{\nu\rho}R_{\mu\nu\lambda\rho}\bigg{]}\;,\end{split}$
(4)

with $h\equiv g^{\mu\nu}h_{\mu\nu}$. Indices are raised and lowered with the
background metric $g_{\mu\nu}$ and all curvature tensors and covariant
derivatives are constructed using $g_{\mu\nu}$.

The gauge symmetries acting on $h_{\mu\nu}$ leave the background metric
$g_{\mu\nu}$ invariant and must be gauge-fixed. It is useful to employ BRST
methods and maintain the background gauge symmetry in the gauge-fixing terms
as well. This can be done by choosing a weighted gauge based on the de Donder
gauge-fixing function
$f_{\mu}=\nabla^{\nu}h_{\nu\mu}-\frac{1}{2}\partial_{\mu}h$, which is a tensor
under the background gauge symmetry. It gives rise to a total gauge-fixed
action $S_{tot}$ for the graviton $h_{\mu\nu}$ and ghost fields
$b_{\mu},c_{\mu}$ that at quadratic order reads

$S_{2,tot}[h,c,b]=S_{h}[h]+S_{gh}[b,c]$ (5)

where

$\displaystyle S_{h}$ $\displaystyle=\int
d^{D}x\sqrt{g}\biggl{[}-\frac{1}{4}h^{\mu\nu}(\nabla^{2}+2\Lambda-R)h_{\mu\nu}+\frac{1}{8}h(\nabla^{2}+2\Lambda-R)h$
$\displaystyle\hskip
65.44142pt-\frac{1}{2}\big{(}h^{\mu\lambda}h^{\nu}_{\lambda}-hh^{\mu\nu}\big{)}R_{\mu\nu}-\frac{1}{2}h^{\mu\lambda}h^{\nu\rho}R_{\mu\nu\lambda\rho}\biggr{]}\;,$
(6) $\displaystyle S_{gh}$ $\displaystyle=\int
d^{D}x\sqrt{g}\,b_{\mu}(-\delta^{\mu}_{\nu}\nabla^{2}-R^{\mu}_{\nu})c^{\nu}\;.$
(7)

The gauge-fixing procedure just described is standard, see for example [1, 2].
Here we have followed [6], where additional details on the BRST procedure may
be found.

From these quadratic actions one can extract the invertible kinetic operators
that lead to the one-loop effective action through the heat kernel. To this
end, one can introduce a one-parameter family of metrics on the space of
symmetric tensors $h_{\mu\nu}$:

$\gamma_{(k)}^{\lambda\rho\mu\nu}=\frac{1}{2}\Big{(}g^{\lambda\mu}g^{\rho\nu}+g^{\lambda\nu}g^{\rho\mu}-k\,g^{\lambda\rho}g^{\mu\nu}\Big{)}\;,$
(8)

which allow to write a norm for the graviton fluctuations as

$||h||_{(k)}^{2}=\frac{1}{4}\int
d^{D}x\sqrt{g}\,h_{\lambda\rho}\gamma_{(k)}^{\lambda\rho\mu\nu}h_{\mu\nu}\;.$
(9)

The metric (8) has inverse

$\gamma_{(k)\mu\nu\sigma\tau}=\frac{1}{2}\Big{(}g_{\mu\sigma}g_{\nu\tau}+g_{\nu\sigma}g_{\mu\tau}-\frac{2k}{kD-2}g_{\mu\nu}g_{\sigma\tau}\Big{)}\;,$
(10)

which shows that $k=\frac{2}{D}$ is not allowed, as expected, since
$\gamma_{\scaleobj{0.7}{\frac{2}{D}}}^{\lambda\rho\mu\nu}$ is the projector on
the traceless subspace. From now on we will choose $k=1$ following [1] and
work with

$\gamma^{\lambda\rho\mu\nu}=\frac{1}{2}\Big{(}g^{\lambda\mu}g^{\rho\nu}+g^{\lambda\nu}g^{\rho\mu}-g^{\lambda\rho}g^{\mu\nu}\Big{)}\;,\quad\gamma_{\mu\nu\sigma\tau}=\frac{1}{2}\Big{(}g_{\mu\sigma}g_{\nu\tau}+g_{\nu\sigma}g_{\mu\tau}-\frac{2}{D-2}g_{\mu\nu}g_{\sigma\tau}\Big{)}\;.$
(11)

This choice has the advantage of making the derivative part of the kinetic
operator proportional to the identity in the tensor indices. This is also
reflected in the fact that in flat space
$\frac{\gamma_{\mu\nu\sigma\tau}}{p^{2}}$ is the graviton propagator in the
Feynman-de Donder gauge.

Thus, we rewrite the quadratic actions (6) and (7) in the form

$\displaystyle S_{h}$ $\displaystyle=\int
d^{D}x\sqrt{g}\,\frac{1}{4}\,h_{\lambda\rho}\,\gamma^{\lambda\rho\mu\nu}\,F_{\mu\nu}{}^{\sigma\tau}\,h_{\sigma\tau}$
(12) $\displaystyle S_{gh}$ $\displaystyle=\int
d^{D}x\sqrt{g}\,b_{\mu}\,\mathfrak{F}^{\mu}{}_{\nu}\,c^{\nu}$ (13)

to identify the differential operators

$\displaystyle F_{\mu\nu}{}^{\sigma\tau}$
$\displaystyle=-\frac{1}{2}(\delta_{\mu}{}^{\sigma}\delta_{\nu}{}^{\tau}+\delta_{\mu}{}^{\tau}\delta_{\nu}{}^{\sigma})(\nabla^{2}+2\Lambda-R)$
$\displaystyle\hskip
14.22636pt-\frac{1}{D-2}g_{\mu\nu}g^{\sigma\tau}R+\frac{2}{D-2}g_{\mu\nu}R^{\sigma\tau}+g^{\sigma\tau}R_{\mu\nu}$
$\displaystyle\hskip
14.22636pt-\frac{1}{2}(\delta_{\mu}{}^{\sigma}R_{\nu}{}^{\tau}+\delta_{\mu}{}^{\tau}R_{\nu}{}^{\sigma}+\delta_{\nu}{}^{\sigma}R_{\mu}{}^{\tau}+\delta_{\nu}{}^{\tau}R_{\mu}{}^{\sigma})-R_{\mu}{}^{\sigma}{}_{\nu}{}^{\tau}-R_{\mu}{}^{\tau}{}_{\nu}{}^{\sigma}\;,$
(14) $\displaystyle\mathfrak{F}^{\mu}{}_{\nu}$
$\displaystyle=-(\delta^{\mu}{}_{\nu}\nabla^{2}+R^{\mu}{}_{\nu})$ (15)

whose determinants appear in the effective action for quantum gravity.

Thus, let us consider the one-loop effective action. It is obtained by
computing the functional determinants of the above operators, which we denote
simply by $F$ and $\mathfrak{F}$. Using standard formulae, reviewed in
appendix A, we find for the one-loop effective action $\Gamma$

$\displaystyle e^{-\Gamma}={\rm Det}^{-\frac{1}{2}}F\ {\rm Det}\,\mathfrak{F}$
(16)

which in a proper-time representation leads to

$\Gamma=-\frac{1}{2}\int_{0}^{\infty}\frac{dT}{T}\left(\text{Tr}\left[\text{e}^{-FT}\right]-2\text{Tr}\big{[}\text{e}^{-{\mathfrak{F}}T}\big{]}\right)$
(17)

containing the heat kernels of the above operators. Now, we insert the heat
kernel expansions for small $T$, see eq. (63) of appendix A, and identify the
total heat kernel coefficients for quantum gravity

$\text{tr}[a_{n,tot}]=\text{tr}[a_{n}]-2\text{tr}[a_{n,gh}]$ (18)

where the first one is due to the graviton fluctuations and the second one to
the ghosts. This expansion is useful to identify the diverging terms of the
effective action, but cannot be employed to obtain the finite terms because of
infrared divergences that appear from the upper limit of the proper time
integration.

To evaluate the heat kernel coefficients, we specialize the general formulae
of appendix A to the operators $F$ and $\mathfrak{F}$, and find the following
values. The graviton heat kernel coefficients from $F$ at arbitrary dimension
$D$ are given by

$\displaystyle\begin{split}\text{tr}[a_{0}]&=\frac{1}{2}D(D+1)\end{split}$
(19)
$\displaystyle\begin{split}\text{tr}[a_{1}]&=-\frac{D(5D-7)}{12}R+D(D+1)\Lambda\end{split}$
(20)
$\displaystyle\begin{split}\text{tr}[a_{2}]&=-\frac{D(2D-3)}{30}\nabla^{2}R+\frac{25D^{3}-145D^{2}+262D+144}{144(D-2)}R^{2}\\\
&-\frac{D^{3}-181D^{2}+1438D-720}{360(D-2)}R_{\mu\nu}R^{\mu\nu}+\frac{D^{2}-29D+480}{360}R_{\mu\nu\sigma\tau}R^{\mu\nu\sigma\tau}\\\
&+D(D+1)\Lambda^{2}-\frac{D(5D^{2}-17D+14)}{6(D-2)}R\Lambda\end{split}$ (21)

and at $D=4$ reduce to

$\displaystyle\begin{split}\text{tr}[a_{0}]\ &\xrightarrow{D=4}\
10\end{split}$ (22) $\displaystyle\begin{split}\text{tr}[a_{1}]\
&\xrightarrow{D=4}\ -\frac{13}{3}R+20\Lambda\end{split}$ (23)
$\displaystyle\begin{split}\text{tr}[a_{2}]\ &\xrightarrow{D=4}\
-\frac{2}{3}\nabla^{2}R+\frac{59}{36}R^{2}-\frac{55}{18}R_{\mu\nu}R^{\mu\nu}+\frac{19}{18}R_{\mu\nu\sigma\tau}R^{\mu\nu\sigma\tau}+20\Lambda^{2}-\frac{26}{3}R\Lambda\;.\end{split}$
(24)

One can recognize that eq. (19) counts the number of degrees of freedom of a
symmetric rank-2 tensor with a non-vanishing trace. Now, one should add the
contribution of the ghosts. However, as an aside, we notice that these
coefficients are also useful as they stand. They describe the total
coefficients of the system composed of gravity coupled to a complex spin 1
field and 4 real scalars: the determinant of the complex spin 1 field in the
Feynman gauge compensates that of the gravity ghosts, while the 4 scalars
compensate the ghosts system for the complex spin 1 field. As a simple check,
one verifies that the total number of the physical degrees of freedom is
correctly reproduced by $\text{tr}[a_{0}]$.

Let us now turn to the coefficients due to the ghosts operator $\mathfrak{F}$.
They are given by

$\displaystyle\begin{split}\text{tr}[a_{0,gh}]&=D\end{split}$ (25)
$\displaystyle\begin{split}\text{tr}[a_{1,gh}]&=\frac{D+6}{6}R\end{split}$
(26)
$\displaystyle\begin{split}\text{tr}[a_{2,gh}]&=\frac{D+5}{30}\nabla^{2}R+\frac{D+12}{72}R^{2}-\frac{D-90}{180}R_{\mu\nu}R^{\mu\nu}+\frac{D-15}{180}R_{\mu\nu\sigma\tau}R^{\mu\nu\sigma\tau}\end{split}$
(27)

that in $D=4$ dimensions reduce to

$\displaystyle\begin{split}\text{tr}[a_{0,gh}]\ &\xrightarrow{D=4}\
4\end{split}$ (28) $\displaystyle\begin{split}\text{tr}[a_{1,gh}]\
&\xrightarrow{D=4}\ \frac{5}{3}R\end{split}$ (29)
$\displaystyle\begin{split}\text{tr}[a_{2,gh}]\ &\xrightarrow{D=4}\
\frac{3}{10}\nabla^{2}R+\frac{2}{9}R^{2}+\frac{43}{90}R_{\mu\nu}R^{\mu\nu}-\frac{11}{180}R_{\mu\nu\sigma\tau}R^{\mu\nu\sigma\tau}\;.\end{split}$
(30)

Finally, the total coefficients for quantum gravity are obtained by evaluating
(18) and read

$\displaystyle\begin{split}\text{tr}[a_{0,tot}]&=\frac{D(D-3)}{2}\end{split}$
(31)
$\displaystyle\begin{split}\text{tr}[a_{1,tot}]&=-\frac{5D^{2}-3D+24}{12}R+D(D+1)\Lambda\end{split}$
(32)
$\displaystyle\begin{split}\text{tr}[a_{2,tot}]&=-\frac{2D^{2}-D+10}{30}\nabla^{2}R+\frac{25D^{3}-149D^{2}+222D+240}{144(D-2)}R^{2}\\\
&-\frac{D^{3}-185D^{2}+1806D-1440}{360(D-2)}R_{\mu\nu}R^{\mu\nu}+\frac{D^{2}-33D+540}{360}R_{\mu\nu\sigma\tau}R^{\mu\nu\sigma\tau}\\\
&+D(D+1)\Lambda^{2}-\frac{5D^{3}-17D^{2}+14D}{6(D-2)}R\Lambda\end{split}$ (33)

reducing at $D=4$ to

$\displaystyle\begin{split}\text{tr}[a_{0,tot}]\ &\xrightarrow{D=4}\
2\end{split}$ (34) $\displaystyle\begin{split}\text{tr}[a_{1,tot}]\
&\xrightarrow{D=4}\ -\frac{23}{3}R+20\Lambda\end{split}$ (35)
$\displaystyle\begin{split}\text{tr}[a_{2,tot}]\ &\xrightarrow{D=4}\
-\frac{19}{15}\nabla^{2}R+\frac{43}{36}R^{2}-\frac{361}{90}R_{\mu\nu}R^{\mu\nu}+\frac{53}{45}R_{\mu\nu\sigma\tau}R^{\mu\nu\sigma\tau}+20\Lambda^{2}-\frac{26}{3}R\Lambda\;.\end{split}$
(36)

Einstein’s field equations have not been used at any step in our calculations
and the results are valid for any background. However, they depend on the
choice of the gauge-fixing terms, i.e. they are not BRST invariant.

Let us compare them with the ones that are found in the literature. One may
check that, at arbitrary dimension $D$ and vanishing cosmological constant,
some of them differ from the ones reported in eqs. (16.80)-(16.82) of [1] and
in eq. (35.168) of the second volume of [2], which also differ between
themselves. Nevertheless, they are identical to the ones recently computed in
[5], which we thus consider as the correct ones.

We should stress that these terms depend in general on the gauge-fixing
procedure adopted, which however is the same in the references cited above. To
obtain truly gauge-independent coefficients one should evaluate them on-shell,
i.e. using a background that satisfies Einstein’s equations. We do this in the
next section.

##  3 Gauge-invariant coefficients in perturbative quantum gravity

To derive gauge independent coefficients, we use background metrics that
correspond to Einstein spaces, that by definition satisfy Einstein’s equations
with cosmological constant $\Lambda$

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+g_{\mu\nu}\Lambda=0$ (37)

allowing to relate the cosmological constant and the Ricci tensor to the Ricci
scalar

$\Lambda=\frac{D-2}{2D}R\;,\ \ \ \ R_{\mu\nu}=\frac{1}{D}\,g_{\mu\nu}R\;.$
(38)

Using these relations, we find that the coefficients $a_{n,tot}$ in eqs.
(31)–(33) reduce to

$\displaystyle\begin{split}\text{tr}[a_{0,tot}]&=\frac{D(D-3)}{2}\end{split}$
(39)
$\displaystyle\begin{split}\text{tr}[a_{1,tot}]&=\frac{1}{12}(D^{2}-3D-36)R\end{split}$
(40)
$\displaystyle\begin{split}\text{tr}[a_{2,tot}]&=\frac{(D+5)(5D^{2}-42D-144)}{720D}R^{2}+\frac{D^{2}-33D+540}{360}R_{\mu\nu\sigma\tau}R^{\mu\nu\sigma\tau}\end{split}$
(41)

with values at $D=4$ given by

$\displaystyle\begin{split}\text{tr}[a_{0,tot}]&\ \xrightarrow{D=4}\
2\end{split}$ (42) $\displaystyle\begin{split}\text{tr}[a_{1,tot}]&\
\xrightarrow{D=4}\ -\frac{8}{3}R\end{split}$ (43)
$\displaystyle\begin{split}\text{tr}[a_{2,tot}]&\ \xrightarrow{D=4}\
-\frac{29}{40}R^{2}+\frac{53}{45}R_{\mu\nu\sigma\tau}R^{\mu\nu\sigma\tau}\;.\end{split}$
(44)

As these terms are evaluated on-shell, they should not depend on the gauge
chosen. They identify gauge-invariant coefficients. They sit on the
divergences of the effective action and must be renormalized away. Let us
briefly comment on this point, reviewing some old literature. In $D=4$ and
setting the cosmological constant to zero, also the Ricci scalar vanishes, and
from (43) one finds that $\text{tr}[a_{2,tot}]\sim R_{\mu\nu\sigma\tau}^{2}$,
which becomes a total derivative proportional to the Euler density on Einstein
spaces. Being a total derivative, it can be dropped from the effective action.
This makes the logarithmic divergences due to (44) absent from the one-loop
effective action, thus reproducing the famous result of t’ Hooft and Veltman
[7], according to which quantum gravity is finite at one-loop (more precisely,
it is free of logarithmic divergences, as the quartic divergence from
$a_{0,tot}$ contributes to the cosmological constant; in any case, quantum
gravity remains renormalizable at one loop). This result does not hold anymore
at two loops and pure quantum gravity becomes non-renormalizable, as shown by
Goroff and Sagnotti [8] and checked by van de Ven [9]. Adding a cosmological
constant, the logarithmic divergence at $D=4$ does not vanish on-shell
anymore, even dropping the total derivative corresponding to the Euler
density. The precise coefficient from (44) coincides with the one obtained
long ago by Christensen and Duff [10]. Similarly, the term
$\text{tr}[a_{1,tot}]$ in $D=4$ coincides with the one calculated in [6].

At arbitrary $D$, and with $D\geq 6$, the above gauge-invariant coefficients
form only a subset of the possible divergences of perturbative quantum
gravity. As they are gauge-invariant, any formulation of quantum gravity
should be able to reproduce them, independently of the scheme chosen in the
calculation.

An alternative formulation treats the graviton in first quantization. A
mechanical action useful for describing the graviton is the $\mathcal{N}=4$
spinning particle. When supplemented with a set of gauge symmetries, the
$\mathcal{N}=4$ spinning particle has only the graviton in the physical
spectrum at arbitrary $D$. It has been used in [4] to compute the gauge-
invariant coefficients discussed above. A direct comparison shows that the
results are correctly reproduced in 4 dimensions, but differ at arbitrary $D$
in the terms proportional to $R$ and $R^{2}$. We take this fact as suggesting
that the path integral quantization of the $\mathcal{N}=4$ spinning particle
developed in [4] is correct only at $D=4$, but needs an improvement for
arbitrary $D$. We discuss this issue next.

##  4 Worldline path integral for the graviton

In flat space, a relativistic particle with $\cal N$-extended local
supersymmetry on the worldline describes a particle of spin $s=\frac{\cal
N}{2}$ in four dimensions, as suggested in [11] and demonstrated explicitely
in [12, 13]. One obtains spin 2, the graviton, by setting ${\cal N}=4$, but
couplings to nontrivial backgrounds have been proven difficult to achieve for
sufficiently large $\cal N$, including the case of the graviton, see refs.
[14, 15] for some attempts. The worldline path integral performed on the
circle in [16] could just reproduce the physical degrees of freedom of the
graviton. At that time it was also realized that the gauging of the full
$R$-symmetry group of the $\cal N$-extended supersymmetry, the group $SO({\cal
N})$ that rotates the ${\cal N}$ real supercharges, could have been relaxed to
a subgroup without destroying the unitarity of the model, a fact used in [17]
and [18] to describe multiplets of particles of different spins.

The understanding of allowed couplings to background fields was improved by
using BRST methods. As discussed in [19], BRST techniques give a way of
introducing the Yang-Mills couplings for the ${\cal N}=2$ spinning particle,
which describes the propagation of a particle of spin 1. Following similar
strategies, it was found in [3] how the graviton can be coupled to a curved
background that satisfies Einstein’s field equations with or without
cosmological constant, see also [20] for further extensions.

The same BRST construction was used in [4] to define a worldline path integral
for the graviton by using the ${\cal N}=4$ spinning particle. The path
integral takes into account the gauging of worldline translations,
supersymmetries, and a parabolic subgroup of the $SO({\cal N})$ $R$-symmetry
group. Gauging of translations and supersymmetries guarantees the unitarity of
the model, while the gauging of a maximal parabolic subgroup of $SO({\cal N})$
leaves only irreducible spin 2 states in the physical Hilbert space. The model
includes two additional Chern-Simons couplings fixed in such a way to describe
the graviton in arbitrary dimensions $D$. Performing the path integral on a
circle, one finds a worldline representation of the one-loop effective action
of the graviton $\Gamma[g_{\mu\nu}]$ with the schematic form

$\Gamma[g_{\mu\nu}]=\int_{S^{1}}\frac{\mathcal{D}G\,\mathcal{D}X^{\mu}}{\mathrm{Vol(Gauge)}}\,{\rm
e}^{-S[X^{\mu},G;g_{\mu\nu}]}\;.$ (45)

where the particle action $S[X^{\mu},G;g_{\mu\nu}]$ depends on the worldline
gauge fields $G=(e,\chi_{i},a_{ij})$, and on the coordinates
$X^{\mu}=(x^{\mu},\psi^{\mu}_{i})$ that contain the true worldline coordinates
$x^{\mu}$ plus their supersymmetric partners $\psi^{\mu}_{i}$ with $i=1,..,4$,
while $g_{\mu\nu}$ is the background metric. The BRST symmetry underlying the
model makes sure that the path integral is correct when the background metric
$g_{\mu\nu}$ is on-shell, meaning that it must correspond to a metric of an
Einstein space. Upon gauge fixing, and using complex combinations of the real
fermions, that we still denote by $\psi^{\mu}_{i}$ but with a redefined
internal index $i$ taking only two values ($i=1,2$), the path integral takes
the following concrete form

$\Gamma[g_{\mu\nu}]=-\frac{1}{2}\int_{0}^{\infty}\frac{dT}{T}\int_{0}^{2\pi}\frac{d\theta}{2\pi}\int_{0}^{2\pi}\frac{d\phi}{2\pi}\;\mu(\theta,\phi)\int_{P}{\cal
D}x\int_{A}D\bar{\psi}D\psi\ {\rm e}^{-S_{g}}$ (46)

where the gauge-fixed, nonlinear sigma model action for the graviton is

$S_{g}=\int_{0}^{1}d\tau\Big{[}\,\frac{1}{4T}\,g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}+\bar{\psi}^{ia}(\delta_{i}^{j}D_{\tau}-\hat{a}_{i}^{j})\psi_{ja}-TR_{abcd}\,\bar{\psi}^{a}\cdot\psi^{b}\,\bar{\psi}^{c}\cdot\psi^{d}+2T\,V_{0}\Big{]}$
(47)

where

$\hat{a}_{i}^{j}={\left(\begin{array}[]{cc}\theta&0\\\
0&\phi\end{array}\right)}$ (48)

contains the two moduli $\theta$ and $\phi$. We have used flat indices on the
worldline complex fermions $\psi^{a}_{i}$, so that $D_{\tau}$ is the covariant
derivative with the spin connection, while a dot denotes contraction on the
internal indices. The scalar potential $V_{0}\sim R$ is an order $\hbar^{2}$
improvement term to be discussed shortly. The measure $\mu(\theta,\phi)$ on
the moduli space $(\theta,\phi)$ coming from the Faddeev–Popov determinants is
given by

$\mu(\theta,\phi)=\frac{1}{2}\left(2\cos\frac{\theta}{2}\right)^{-2}\left(2\cos\frac{\phi}{2}\right)^{-2}\,2i\,\sin\tfrac{\theta+\phi}{2}\left(2i\,\sin\tfrac{\theta-\phi}{2}\right)^{2}\,{\rm
e}^{-iq(\theta+\phi)}$ (49)

and includes the Chern-Simons coupling $q=\frac{3-D}{2}$, needed to select the
graviton for arbitrary spacetime dimensions $D$. We refer to [4] for further
details on the particle action and its gauge fixing on the circle.

In [4] the scalar potential $V_{0}$ in eq. (47) was taken to be given by
$V_{0}=-\tfrac{D+2}{8(D-1)}R$ upon consideration of the ${\cal N}=4$
supersymmetry algebra. In addition, the path integral for the nonlinear sigma
model must be regularized, see [21] for a review on this issue. The
counterterms needed in models with extended supersymmetries can be found in
[22]. From the latter reference, one extracts the counterterm $V_{\rm
CT}=\frac{1}{8}R$ needed in the worldline dimensional regularization for the
${\cal N}=4$ model. Once added to $V_{0}$, it gives a total potential

$\displaystyle V=V_{\rm
CT}+V_{0}=\left(\frac{1}{8}-\frac{D+2}{8(D-1)}\right)R=-\frac{3}{8(D-1)}R=:\omega
R$ (50)

that must be used in (47) instead of $V_{0}$ when using worldline dimensional
regularization. This was done in [4], where a perturbative calculation for
small $T$ delivered the divergences of the effective action in the form

$\displaystyle\Gamma[g_{\mu\nu}]=-\frac{1}{2}\int_{0}^{\infty}\frac{dT}{T}\int
d^{D}x\,\frac{\sqrt{g}}{(4\pi
T)^{\frac{D}{2}}}\,\Big{\langle}\\!\\!\Big{\langle}{\rm
e}^{-S_{int}}\Big{\rangle}\\!\\!\Big{\rangle}$ (51)

where $\langle\langle\dots\rangle\rangle$ denotes the perturbative corrections
of the path integral and subsequent modular integration, giving for Einstein
spaces the answer

$\Big{\langle}\\!\\!\Big{\langle}e^{-S_{int}}\Big{\rangle}\\!\\!\Big{\rangle}=a_{0}+a_{1}T+a_{2}T^{2}+O(T^{3})$
(52)

with

$\displaystyle\begin{split}a_{0}&=\frac{D(D-3)}{2}\end{split}$ (53)
$\displaystyle\begin{split}a_{1}&=\left[\frac{D^{2}}{24}(5-24\omega)+D\Big{(}3\omega-\frac{39}{24}\Big{)}\right]R\end{split}$
(54)
$\displaystyle\begin{split}a_{2}&=\Bigg{[}\frac{D^{2}}{576}(5-24\omega)^{2}-D\Big{(}3\omega^{2}-\frac{13}{4}\omega+\frac{1583}{2880}\Big{)}+\frac{121}{120}+\frac{2}{D}\Bigg{]}R^{2}+\frac{D^{2}-33D+540}{360}R_{\mu\nu\sigma\tau}R^{\mu\nu\sigma\tau}\;.\end{split}$
(55)

From these results, one verifies that the value of $\omega$ defined by eq.
(50) reproduces only the coefficients at $D=4$ but not those at arbitrary $D$,
just compare with eqs. (39)–(41).

One may guess that the mismatch is due to an incorrect identification of the
potential $V$ in eq. (50) used in [4] for getting the above result, as for the
rest the construction of the path integral stands on solid principles. This
conjecture is correct. One may fix $\omega$ by requiring that it reproduces
the expected coefficient $a_{1}$ in eq. (40), finding

$\omega=\frac{1}{8}-\frac{1}{D}\;.$ (56)

Then, one verifies that with this value of $\omega$ also $a_{2}$ comes out
correctly. This is a nontrivial check, as there is no left-over freedom in
defining the path integral.

The correct value of $\omega$ could have been deduced also from first
principles. The BRST analysis of ref. [3] requires a value of $V_{0}$ in the
hamiltonian constraint to be given by $V_{0}=-\frac{1}{D}R$ to achieve
nilpotency of the BRST charge on the relevant physical subspace of the full
BRST Hilbert space, see eq. (5.12) of [3]. Adding the counterterm $V_{\rm
CT}=\frac{1}{8}R$ required by the worldline dimensional regularization of the
${\cal N}=4$ model gives a total potential

$V=\left(\frac{1}{8}-\frac{1}{D}\right)R$ (57)

with the correct value of $\omega$ used above. This modification makes the
worldline path integral for the graviton correct in any spacetime dimension.

It is perhaps surprising that both values of the potential $V$ coincide at
$D=4$. This is probably due to the fact that the potential
$V_{0}=-\frac{D+2}{8(D-1)}\,R$ was found in [15] upon demanding closure of the
supersymmetry algebra on maximally symmetric spaces. Since the model analyzed
in [15] does propagate a graviton in $D=4$ (but not in arbitrary dimensions),
one may understand why the above value for $V_{0}$ coincides with the correct
one $-\frac{1}{D}\,R$ in this case.

##  5 Conclusions

Heat kernel methods have proved to be useful to study properties of QFT, like
one-loop effective actions, anomalies, dressed propagators, etc., see for
example the recent applications aimed at finding the correct trace anomalies
of chiral fermions [23, 24]. We have used them again here to recompute the
full set of divergences of the one-loop effective action of quantum gravity in
$D=4$, but keeping the spacetime dimension arbitrary (for $D\geq 6$ there are
additional divergences). Evaluating them on-shell allowed us to identify
gauge-invariant coefficients, that may be used as a benchmark for testing
alternative formulations of quantum gravity. We have used them straight away
to check the path integral construction of the graviton in first-quantization,
which employs the ${\cal N}=4$ spinning particle. A mismatch of the
coefficients at $D\neq 4$ has prompted us to improve on the path integral
construction to make it consistent in arbitrary dimensions.

This first-quantized approach to gravity extends the scope of worldline
methods [25] to include gravity as a quantum theory. Recent developments on
worldline approaches to QFT have addressed the study of dressed propagators
and more general properties in QED [26, 27, 28, 29]. We hope that additional
gravitational applications may extend the usefulness of the method further. In
that regard, it might be useful to consider the extension of quantum gravity
to complex spaces, which may be studied by using the simpler $U({\cal N})$
particles considered in [30, 31, 32, 33]. Finally, it might be useful to
extend the calculation of the gauge-invariant coefficients to include $a_{3}$,
generalizing the results of ref. [34] to arbitrary dimensions and with a
nonvanishing cosmological constant.

## Acknowledgments

R.B. would like to thank the Department of Physics and Astronomy of the
University of Bologna for kind hospitality during the initial stage of this
work. The work of R.B. is funded by the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement No 771862).

##  Appendix A Effective action and heat kernel

We recapitulate here standard formulae for the one-loop effective action and
heat kernel.

The one-loop effective action is linked to the determinant of a differential
operator $H$ that may depend on various background fields. For our purposes,
it is enough to consider the sufficiently general case of a real scalar field
$\phi$ of mass $m$ in curved space, coupled to a non-abelian gauge field and a
Lie algebra valued scalar potential. The euclidean action is taken to be
quadratic in the scalar field and of the general form

$S[\phi]=\int d^{D}x\sqrt{g}\ \frac{1}{2}\phi^{T}(H+m^{2})\phi$ (58)

with $H$ the second order differential operator mentioned above. The
corresponding one-loop effective action $\Gamma$ depends on the background
fields contained in $H$. It is obtained by a Gaussian path integral

$\displaystyle e^{-\Gamma}=\int D\phi\,e^{-S[\phi]}={\rm
Det}^{-\frac{1}{2}}(H+m^{2})=e^{-\frac{1}{2}{\rm Tr}\ln(H+m^{2})}$ (59)

which leads, using a proper-time representation and dropping a constant term,
to

$\Gamma=\frac{1}{2}{\rm
Tr}\ln(H+m^{2})=-\frac{1}{2}\int_{0}^{\infty}\frac{dT}{T}\,e^{-m^{2}T}{\rm
Tr}\,e^{-HT}$ (60)

where ${\rm Tr}\,e^{-HT}$ is the functional trace of the heat kernel of $H$.

For $H$ we consider an operator of the form

$H=-\nabla^{2}+V$ (61)

with $V$ a Lie algebra valued potential. The laplacian
$\nabla^{2}=\nabla^{\mu}\nabla_{\mu}$ is constructed with the gravitational
and gauge covariant derivative $\nabla_{\mu}=D_{\mu}+W_{\mu}$, where $D_{\mu}$
is the covariant derivative containing the usual metric connection while
$W_{\mu}$ is the Lie algebra valued gauge field. The covariant derivative
$\nabla_{\mu}$ satisfies

$[\nabla_{\mu},\nabla_{\nu}]\phi={\cal
F}_{\mu\nu}\phi\;,\qquad[\nabla_{\mu},\nabla_{\nu}]A^{\lambda}=R_{\mu\nu}{}^{\lambda}{}_{\rho}A^{\rho}$
(62)

where $\phi$ is the scalar field in a real representation of the gauge group
and $A^{\mu}$ a vector field invariant under the gauge group. From these
commutators one reads off the definition of the various curvatures. In our
conventions, $R_{\mu\nu}=R_{\lambda\mu}{}^{\lambda}{}_{\nu}$ and
$R=R^{\mu}_{\mu}>0$ on a sphere.

The trace of the heat kernel corresponding to $H$ is given in perturbation
theory for small $T$

${\rm Tr}\left[e^{-HT}\right]=\int\frac{d^{D}x\sqrt{g(x)}}{(4\pi
T)^{\frac{D}{2}}}\,{\rm tr}\,(a_{0}(x)+a_{1}(x)T+a_{2}(x)T^{2}+\cdots)$ (63)

where the symbol “tr” denotes the trace on the remaining discrete matrix
indices corresponding to the chosen representation of the gauge group and

$\displaystyle a_{0}(x)$ $\displaystyle={\mathbb{1}}$ (64) $\displaystyle
a_{1}(x)$ $\displaystyle=\frac{1}{6}R\hskip 1.13809pt{\mathbb{1}}-V$
$\displaystyle a_{2}(x)$
$\displaystyle=\frac{1}{180}(R^{2}_{\mu\nu\lambda\rho}-R_{\mu\nu}^{2}){\mathbb{1}}+\frac{1}{72}\left(R\hskip
1.13809pt{\mathbb{1}}-6V\right)^{2}+\frac{1}{30}\nabla^{2}\left(R\hskip
1.13809pt{\mathbb{1}}-5V\right)+\frac{1}{12}{\cal F}_{\mu\nu}^{2}$

are the heat kernel coefficients (also known as Seeley-DeWitt coefficients).
As $V$ is matrix valued, $\nabla_{\mu}V=\partial_{\mu}V+[W_{\mu},V]$.

After inserting (63) into (60), one finds the following derivative expansion
of the effective action

$\Gamma=-\frac{1}{2}\int
d^{D}x\sqrt{g(x)}\int_{0}^{\infty}\frac{dT}{T}\,\frac{e^{-m^{2}T}}{(4\pi
T)^{\frac{D}{2}}}\,{\rm tr}\,(a_{0}(x)+a_{1}(x)T+a_{2}(x)T^{2}+\cdots)\;.$
(65)

The mass term guarantees convergence at the upper limit of the proper-time
integral. Infrared divergences arise in the massless limit and invalidate in
that case this derivative expansion. Ultraviolet divergences arise instead
from the lower limit of the proper-time integration, i.e. at $T\to 0$. For
example, in $D=4$ one may verify that $a_{0}$, $a_{1}$, and $a_{2}$ all lead
to UV divergences, namely quartic, quadratic, and logarithmic divergences,
respectively. These UV divergences must be renormalized away. As already
mentioned, in the massless limit the above expansion is not valid for getting
finite terms of the effective action, but it is still useful to recognize the
explicit form of the UV divergences, as in the application to quantum gravity
described in the main text. In particular, ${\rm tr}\,a_{0}$ counts the number
of degrees of freedom, is normalized to 1 for a single real scalar field, and
contributes to the divergence of the cosmological constant.

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