aainstitutetext: Department of Physics and Astronomy, University of Waterloo,
Waterloo, Ontario, N2L 3G1, Canada

# Quantum Otto engine driven by quantum fields

Kensuke Gallock-Yoshimura a Vaishant Thakur a and Robert B. Mann
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider a quantum Otto engine using an Unruh-DeWitt particle detector
model which interacts with a quantum scalar field in curved spacetime. We
express a generic condition for extracting positive work in terms of the
effective temperature of the detector. This condition reduces to the well-
known positive work condition in the literature under the circumstances where
the detector reaches thermal equilibrium with the field. We then evaluate the
amount of work extracted by the detector in two scenarios: an inertial
detector in a thermal bath and a circulating detector in the Minkowski vacuum,
which is inspired by the Unruh quantum Otto engine.

##  1 Introduction

Quantum thermodynamics is a study of thermodynamic phenomena from the
perspective of quantum mechanics. Within this field quantum heat engines,
where the working substance is quantum matter that interacts with thermal
baths, have emerged as a crucial area of research Scovil.heat.engine.maser .
Notable among quantum heat engines are the quantum Carnot engine and the
quantum Otto engine (QOE). The quantum Carnot engine operates through two
isothermal processes and two quantum adiabatic processes. The QOE, on the
other hand, is composed of two isochoric thermalization processes and two
quantum adiabatic processes Scovil.heat.engine.maser ; Feldmann.QOE.2000 ;
Kieu.secondlaw.demon.otto ; Kieu.heat.engine.2006 ; Rostovtsev.Otto.2003 ;
Quan.multilevel.engine.2005 ; Quan.multilevel.heat.engine . For a
comprehensive review, see Myers.quantumThermo.review .

Our interest lies in the QOE using a two-level quantum system in the realm of
relativistic quantum information (RQI), where relativity is taken into
account. In particular, we consider a specific model for a qubit known as the
Unruh-DeWitt (UDW) particle detector model Unruh1979evaporation ; DeWitt1979 ,
which interacts with a quantum field in (curved) spacetime. The UDW detector
model exhibits the essential features of the light-atom interaction as long as
angular momenta are not exchanged PhysRevD.87.064038 ; PhysRevA.89.033835 . A
UDW detector reveals interesting phenomena such as the Unruh effect
Unruh1979evaporation , where a uniformly accelerating UDW detector perceives a
thermal bath at the Unruh temperature $T_{\text{U}}=a/2\pi$ (where $a$ is the
magnitude of the acceleration), whilst an observer at rest sees a vacuum.

Investigations have been carried out previously on the QOE employing the UDW
detector model. In UnruhOttoEngine , it was shown that a linearly accelerating
UDW detector can be utilized to extract work in a QOE. This process is
facilitated by the Unruh effect, which generates hot and cold thermal baths by
varying the magnitude of acceleration $a$. Such an engine is commonly referred
to as the Unruh QOE. The amount of work extracted is determined by a response
function of the UDW detector, which is the probability of a transition from
the ground state to the excited state, and vice versa. Subsequent studies have
further expanded on the Unruh QOE in various scenarios, including a fermionic
field Finn.UnruhOtto , a degenerate detector Xu.UnruhOtto.degenerate ,
entangled detectors Kane.entangled.Unruh.Otto ; Barman.entangled.UnruhOtto ,
and in the presence of a reflecting boundary Mukherjee.UnruhOtto.boundary .
However, these studies focus on a specific acceleration trajectory, the state
of a field, and a switching function. A comprehensive investigation of the QOE
in a more general setting still remains largely unexplored.

In this paper, we explore the QOE within the framework of RQI in a generalized
setting. We consider a pointlike UDW detector following an arbitrary
trajectory in curved spacetime, interacting with a quantum scalar field in a
quasi-free state. Thus, the thermalization of the detector is not necessarily
required. By employing a perturbative method, we derive general expressions
(35) and (40) for the work extracted through the quantum Otto cycle, which are
written in terms of the response function of the detector. In addition, we
identify conditions for extracting positive work in a generalized setting,
given by (36) and (42). These conditions are expressed in terms of the
effective temperature, an estimator for temperature, as perceived by the
detector. Since thermality is not a prerequisite, the expressions for work
extracted and the positive work condition are applicable to any situation.
Notably, when the field is in a thermal state, known as the Kubo-Martin-
Schwinger (KMS) state, and the interaction duration of a rapidly decreasing
switching function is sufficiently long, the effective temperature becomes the
KMS temperature. In such a case, the condition for positive work (42) becomes
identical to the original condition found in Feldmann.QOE.2000 ;
Kieu.secondlaw.demon.otto .

Our derived expressions are demonstrated concretely within two scenarios: an
inertial detector immersed in a KMS state of a quantum field in Minkowski
spacetime and a detector in circular motion in the Minkowski vacuum. The first
example is the most basic instance of thermality within QFT, for which we
employ a Gaussian switching function and explore the effects of interaction
duration.

The second example is inspired by the Unruh QOE previously examined in the
literature, which has some challenges. A linearly accelerating detector
requires an immense amount of acceleration to reach a temperature of 1 Kelvin.
This can be seen from the expression of the Unruh temperature in SI units,
$T_{\text{U}}=\hbar a/2\pi k_{\text{B}}c$, where $\hbar,k_{\text{B}}$, and $c$
are respectively the reduced Planck constant, the Boltzmann constant, and the
speed of light. It is well known that an acceleration of $a\sim 10^{20}$ m/s2
is required to achieve $T_{\text{U}}\sim 1$ K. This implies that significant
work must be performed on the detector to extract a small amount of work from
the Unruh QOE. Moreover, the duration of interaction is almost instantaneous.
Typically, the Unruh QOE assumes that the detector initially travels at a
constant speed $v$ and then accelerates until it comes back to the same speed
in the opposite direction. The time interval in this process (in terms of the
detector’s proper time $\tau$) is given by
$\Delta\tau=(2c/a)\text{arctanh}(v/c)$, and inserting $a=10^{20}$ m/s2, it
requires $v\approx c$ to yield $\Delta\tau\sim 1$ s. Otherwise, if we demand a
slower speed, say $v=c/2$, then the interaction duration amounts to
$\Delta\tau\sim 10^{-12}$ seconds, which is an extremely short period of time.
The detector certainly cannot thermalize within such a time scale. Finally,
executing such a protocol demands considerable space. The procedure in the
Unruh QOE consists of linear acceleration and uniform motion at speed $v$.
This means that it requires the detector to consistently move in one
direction, which cannot be done in a confined laboratory.

Instead, one can consider a UDW detector in circular motion, a concept that is
motivated by Bell.circular.1983 ; Bell.Leinaas.1987 ; BEC_Unruh , where an
experimental setup is proposed to measure the circular Unruh effect. See also
Retzker.BEC.2008 ; Marino.zero-point.circular ; CircularTemperaturesUnruh ;
bunney2023sound . A circulating UDW detector overcomes the shortcomings
described above, though the temperature induced by the acceleration is no
longer $T_{\text{U}}$. In fact, the detector cannot be thermalized. Instead,
one should define the effective temperature $T_{\text{circ}}$ as observed by
the circulating detector CircularTemperaturesUnruh ; Good.Unruh-
like.effective.temp ; Unruh.acceleration.rad.electron.1998.arxiv . This is
where our proposed framework excels, as our expressions remain valid even in
the absence of thermality.

The outline of this paper is as follows. After we introduce the basic aspects
of the UDW detector and thermality in quantum field theory in 2, we review the
mechanism of QOE in 3.1. Our main results are shown in 3.2, followed by
examples in 4.1 and 4.2.

Throughout this paper, we use $\hbar=c=k_{\text{B}}=1$ and the mostly-positive
signature convention: $(-+++)$.

##  2 Setup

###  2.1 Unruh-DeWitt detector model

Consider a pointlike UDW detector, which is a two-level quantum system with an
energy gap $\Omega$ between ground $\ket{g}$ and excited states $\ket{e}$
locally interacting with the quantum Klein-Gordon field
$\hat{\phi}(\mathsf{x})$. Here, the background spacetime is not restricted to
a specific geometry but is assumed to be a (curved) spacetime that
accommodates a well-behaved Klein-Gordon equation.

In the Schrödinger picture, the total Hamiltonian $\hat{H}_{\text{S,tot}}$
reads

$\displaystyle\hat{H}_{\text{S,tot}}$
$\displaystyle=\hat{H}_{\text{S,D}}+\hat{H}_{\text{S,$\phi$}}+\hat{H}_{\text{S,int}}\,,$
(1)

where $\hat{H}_{\text{S,D}}$ and $\hat{H}_{\text{S,$\phi$}}$ are the free
Hamiltonians for the detector and the field, respectively, and
$\hat{H}_{\text{S,int}}$ is the interaction Hamiltonian. These are explicitly
given by

$\displaystyle\hat{H}_{\text{S,D}}$
$\displaystyle=\Omega\ket{e}\bra{e}\otimes\mathds{1}_{\phi}\,,$ (2a)
$\displaystyle\hat{H}_{\text{S,$\phi$}}$
$\displaystyle=\mathds{1}_{\text{D}}\otimes\int_{\mathbb{R}^{n}}\text{d}^{n}k\,\omega_{{\bm{k}}}\hat{a}_{{\bm{k}}}^{\dagger}\hat{a}_{{\bm{k}}}\,,$
(2b) $\displaystyle\hat{H}_{\text{S,int}}$
$\displaystyle=\lambda\chi(\tau/\sigma)(\ket{e}\bra{g}+\ket{g}\bra{e})\otimes\hat{\phi}({\bm{x}}(\tau))\,,$
(2c)

where $\lambda$ is the coupling constant, $\tau$ is the proper time of the
detector, and $\chi(\tau/\sigma)$ is the switching function (with $\sigma$
being the typical time scale of interaction), which determines the time
dependence of coupling. In this paper, we assume that the $L^{2}$ norm of
$\chi(\tau)$ is unity:

$\displaystyle||\chi||_{2}\coloneqq\sqrt{\int_{\mathbb{R}}\text{d}\tau\,|\chi(\tau)|^{2}}=1\,.$
(3)

Let $\hat{H}_{\text{I}}^{\tau}(\tau)$ be the interaction Hamiltonian in the
interaction picture. Here, the Hamiltonian is the generator of time-
translation with respect to the proper time $\tau$, which is indicated by the
superscript. The explicit form is given by

$\displaystyle\hat{H}_{\text{I}}^{\tau}(\tau)$
$\displaystyle=\lambda\chi(\tau/\sigma)\hat{\mu}(\tau)\otimes\hat{\phi}(\mathsf{x}(\tau))\,,$
(4)

where $\hat{\mu}(\tau)$ is the monopole moment describing the internal
dynamics of the detector:

$\displaystyle\hat{\mu}(\tau)$
$\displaystyle=\ket{e}\bra{g}e^{\mathsf{i}\Omega\tau}+\ket{g}\bra{e}e^{-\mathsf{i}\Omega\tau}\,.$
(5)

Let us obtain the final state of the detector. The time-evolution operator in
the interaction picture is given by

$\displaystyle\hat{U}_{\text{I}}$
$\displaystyle=\mathcal{T}_{\tau}\exp\left(-\mathsf{i}\int_{\mathbb{R}}\text{d}\tau\,\hat{H}_{\text{I}}^{\tau}(\tau)\right)\,,$
(6)

where $\mathcal{T}_{\tau}$ is a time-ordering symbol with respect to the
proper time $\tau$. Carrying out the perturbative method, this operator can be
expanded as

$\displaystyle\hat{U}_{\text{I}}$
$\displaystyle=\mathds{1}+\hat{U}_{\text{I}}^{(1)}+\hat{U}_{\text{I}}^{(2)}+\mathcal{O}(\lambda^{3})\,,$
(7a) $\displaystyle\hat{U}_{\text{I}}^{(1)}$
$\displaystyle=-\mathsf{i}\int_{-\infty}^{\infty}\text{d}\tau\,\hat{H}_{\text{I}}^{\tau}(\tau)\,,$
(7b) $\displaystyle\hat{U}_{\text{I}}^{(2)}$
$\displaystyle=-\int_{-\infty}^{\infty}\text{d}\tau\int_{-\infty}^{\tau}\text{d}\tau^{\prime}\,\hat{H}_{\text{I}}^{\tau}(\tau)\hat{H}_{\text{I}}^{\tau}(\tau^{\prime})\,.$
(7c)

Consider the initial state of the total system,

$\displaystyle\rho_{\text{tot,0}}$
$\displaystyle=\rho_{\text{D,0}}\otimes\rho_{\phi}\,,$ (8)

where $\rho_{\phi}$ is the initial state of the field and $\rho_{\text{D,0}}$
is the detector’s initial state given by

$\displaystyle\rho_{\text{D,0}}$
$\displaystyle=p\ket{e}\bra{e}+(1-p)\ket{g}\bra{g}\,.\quad p\in[0,1]$ (9)

The final total state $\rho_{\text{tot}}$ can be obtained as

$\displaystyle\rho_{\text{tot}}$
$\displaystyle=\hat{U}_{\text{I}}\rho_{\text{tot,0}}\hat{U}_{\text{I}}^{\dagger}=\sum_{m,n=0}^{\infty}\hat{U}_{\text{I}}^{(m)}\rho_{\text{tot,0}}\hat{U}_{\text{I}}^{(n)\dagger}$
(10)

and by tracing out the field degree of freedom we obtain the final state of
the detector, $\rho_{\text{D}}$. In general, $\rho_{\text{D}}$ contains
$n$-point correlation functions
$\braket{\hat{\phi}(\mathsf{x}_{1})...\hat{\phi}(\mathsf{x}_{n})}_{\rho_{\phi}}$,
where $\braket{\cdot}_{\rho_{\phi}}$ is the expectation value with respect to
$\rho_{\phi}$. In this paper, we are particularly interested in a quantum
field in a quasi-free state,111Sometimes quasi-free states are called Gaussian
states in the literature. in which the one-point correlator
$\braket{\hat{\phi}(\mathsf{x})}_{\rho_{\phi}}$ vanishes and every correlation
function can be written in terms of two-point correlators
$W(\mathsf{x},\mathsf{x}^{\prime})\coloneqq\braket{\hat{\phi}(\mathsf{x})\hat{\phi}(\mathsf{x}^{\prime})}_{\rho_{\phi}}$,
also known as the Wightman function. Examples of quasi-free states include the
vacuum state $\ket{0}$ and the Kubo-Martin-Schwinger state, which we will
describe in the next subsection 2.2. Assuming that $\rho_{\phi}$ is a quasi-
free state, the post-interaction state of the detector to the leading order in
$\lambda$ can be calculated as follows.

$\displaystyle\rho_{\text{D}}$
$\displaystyle=\operatorname{Tr}_{\phi}[\rho_{\text{tot}}]=(p+\delta
p)\ket{e}\bra{e}+(1-p-\delta p)\ket{g}\bra{g}+\mathcal{O}(\lambda^{4})\,,$
(11)

where

$\displaystyle\delta p$
$\displaystyle=\lambda^{2}\sigma\big{[}(1-p)\mathcal{F}(\Omega)-p\mathcal{F}(-\Omega)\big{]}\,,$
(12)

with

$\displaystyle\mathcal{F}(\Omega)=\dfrac{1}{\sigma}\int_{\mathbb{R}}\text{d}\tau\int_{\mathbb{R}}\text{d}\tau^{\prime}\,\chi(\tau/\sigma)\chi(\tau^{\prime}/\sigma)e^{-\mathsf{i}\Omega(\tau-\tau^{\prime})}W(\mathsf{x}(\tau),\mathsf{x}(\tau^{\prime}))$
(13)

being the response function, i.e. the probability to excite the ground state
$\ket{g}$ to $\ket{e}$. Similarly, $\mathcal{F}(-\Omega)$ is the probability
for the transition $\ket{e}\to\ket{g}$. The quantity
$W(\mathsf{x}(\tau),\mathsf{x}(\tau^{\prime}))\coloneqq\braket{\hat{\phi}(\mathsf{x}(\tau))\hat{\phi}(\mathsf{x}(\tau^{\prime}))}_{\rho_{\phi}}$
is the pullback of the Wightman function along the detector’s trajectory
$\mathsf{x}(\tau)$. Note that the odd-powers in $\lambda$ vanish since the
one-point correlators $\braket{\hat{\phi}(\mathsf{x})}_{\rho_{\phi}}$ are
zero.

We remark that $\rho_{\text{D}}$ is the state of the detector after
interacting with the field once, while in QOE, the detector interacts with the
field twice. Fortunately, Eq. (11) tells us that the density matrix
$\rho_{\text{D}}$ stays diagonal after the interaction, meaning the final
density matrix also remains diagonal after the second interaction.
Nevertheless, careful attention must be paid to the switching timings of these
two interactions.

Figure 1: The relationship between the KMS state and the response function.

###  2.2 KMS condition

In quantum mechanics, a thermal state is given by the Gibbs state
$\rho=e^{-\beta\hat{H}}/Z$, where $\beta$ is the inverse temperature,
$\hat{H}$ is the Hamiltonian, and
$Z\coloneqq\operatorname{Tr}[e^{-\beta\hat{H}}]$ is the partition function. In
quantum field theory, the notion of thermality is rigorously defined by the
Kubo-Martin-Schwinger (KMS) condition Kubo1957thermality ; Martin-
Schwinger1959thermality ; haag1967equilibrium . The argument given in this
section is summarized in Fig. 1.

Consider a Wightman function in a field state $\rho_{\phi}$:

$\displaystyle W(\tau,\tau^{\prime})$
$\displaystyle=\braket{\hat{\phi}(\mathsf{x}(\tau))\hat{\phi}(\mathsf{x}(\tau^{\prime}))}_{\rho_{\phi}}\,.$
(14)

Note that the vacuum Wightman function
$\braket{0}{\hat{\phi}(\mathsf{x}(\tau))\hat{\phi}(\mathsf{x}(\tau^{\prime}))}{0}$
is a special case corresponding to $\rho_{\phi}=\ket{0}\bra{0}$. The Wightman
function satisfies the KMS condition with respect to time $\tau$ at the KMS
temperature $\beta_{\text{KMS}}^{-1}\equiv T_{\text{KMS}}$ if

$\displaystyle W(\tau-\mathsf{i}\beta_{\text{KMS}},\tau^{\prime})$
$\displaystyle=W(\tau^{\prime},\tau)\,.$ (15)

In addition, if the KMS condition is satisfied, then the Wightman function is
stationary, i.e., it only depends on the difference in time:
$W(\tau,\tau^{\prime})=W(\Delta\tau)$. We identify the KMS temperature
$T_{\text{KMS}}$ as the temperature of the quantum field.

The KMS condition is related to the so-called detailed balance condition. For
any function $f(\omega)$, the detailed balance condition at temperature
$\beta^{-1}$ is given by

$\displaystyle f(-\omega)=e^{\beta\omega}f(\omega)\,.$ (16)

It is known Fewster.Waiting.Unruh  that, assuming the Wightman function is
stationary, the KMS condition with respect to $\tau$ at temperature
$\beta_{\text{KMS}}^{-1}$ is satisfied if and only if the Fourier transformed
Wightman function,

$\displaystyle\tilde{W}(\omega)$
$\displaystyle\coloneqq\int_{\mathbb{R}}\text{d}\Delta\tau\,W(\Delta\tau)e^{-\mathsf{i}\omega\Delta\tau}\,,$
(17)

satisfies the detailed balance condition (16).

The detailed balance condition can be implemented into the response function
of a UDW detector. Suppose the Wightman function satisfies the KMS condition
and the switching function $\chi(\tau/\sigma)$ has a typical interaction
duration time scale $\sigma$. The response function (13) can be written as

$\displaystyle\mathcal{F}_{\sigma}(\Omega)$
$\displaystyle=\int_{\mathbb{R}}\dfrac{\text{d}\bar{\omega}}{2\pi}\,|\tilde{\chi}(\bar{\omega})|^{2}\tilde{W}(\Omega+\bar{\omega}/\sigma)\,,$
(18)

where $\bar{\omega}$ is a dimensionless variable and
$\tilde{\chi}(\bar{\omega})$ is the Fourier transform of the switching
function $\chi(\tau/\sigma)$. One can show that if
$\tilde{\chi}(\bar{\omega})$ decays sufficiently fast as $\bar{\omega}$
increases (such as a Gaussian switching), then the response function satisfies
the detailed balance condition in the long interaction limit
Fewster.Waiting.Unruh ; Garay2016anti-unruh :

$\displaystyle\lim_{\sigma\to\infty}\dfrac{\mathcal{F}_{\sigma}(-\Omega)}{\mathcal{F}_{\sigma}(\Omega)}=e^{\beta_{\text{KMS}}\Omega}\,.$
(19)

Inspired by the detailed balance condition for the response function (19), one
may define the effective temperature $T_{\text{eff}}$:

$\displaystyle T_{\text{eff}}^{-1}$
$\displaystyle\coloneqq\dfrac{1}{\Omega}\ln\dfrac{\mathcal{F}_{\sigma}(-\Omega)}{\mathcal{F}_{\sigma}(\Omega)}\,.$
(20)

The effective temperature is an estimator for the KMS temperature of the
field. In particular, if the Wightman function obeys the KMS condition and the
long interaction limit ($\sigma\to\infty$) is taken, then the effective
temperature becomes the KMS temperature due to (19).

Figure 2: The quantum Otto cycle.

##  3 Relativistic quantum Otto engine

###  3.1 Review: quantum Otto engine

In this subsection, we review the QOE following the paper by Kieu
Kieu.secondlaw.demon.otto . Note that the whole system is composed of a qubit
and thermal baths in the Gibbs state. In the next subsection, we will take the
following protocol and apply it to the system composed of a UDW detector
interacting with a quantum scalar field in a quasi-free state, namely, we will
omit the assumption that the baths are in a thermal state.

  1. 1.

Prepare a qubit with energy gap $\Omega_{1}$ in an initial state
$\rho_{0}=p\ket{e}\bra{e}+(1-p)\ket{g}\bra{g}$, where $p\in[0,1]$, as shown in
Fig. 2.

  2. 2.

Let the qubit undergo an adiabatic expansion (zero heat exchange:
$\braket{Q_{1}}=0$) to enlarge the energy gap from $\Omega_{1}$ to
$\Omega_{2}$, where $\Omega_{2}>\Omega_{1}$. The Hamiltonian of the qubit is
given by $\hat{H}(t)=\Omega(t)\ket{e}\bra{e}$ and so the work done on the
qubit is

$\displaystyle\braket{W_{1}}$
$\displaystyle=\int\text{d}t\,\operatorname{Tr}\left[\rho_{0}\dfrac{\text{d}\hat{H}(t)}{\text{d}t}\right]=p\Delta\Omega\,,$
(21)

where $\Delta\Omega\coloneqq\Omega_{2}-\Omega_{1}$. The qubit’s state remains
as $\rho_{0}$ due to the quantum adiabatic theorem.

  3. 3.

Let the qubit interact with a bath at temperature $T_{\text{H}}$. The qubit’s
Hamiltonian is time-independent $\hat{H}=\Omega_{2}\ket{e}\bra{e}$, and so the
work done on the qubit is zero: $\braket{W_{2}}=0$. After the interaction, the
state becomes $\rho_{1}=(p+\delta p_{\text{H}})\ket{e}\bra{e}+(1-p-\delta
p_{\text{H}})\ket{g}\bra{g}$. The first law of thermodynamics gives the
dissipated heat:

$\displaystyle\braket{Q_{2}}$
$\displaystyle=\operatorname{Tr}[\rho_{1}\hat{H}]-\operatorname{Tr}[\rho_{0}\hat{H}]=\Omega_{2}\delta
p_{\text{H}}\,.$ (22)

This is an isochoric thermalization process.

  4. 4.

After isolating the qubit from the bath, perform an adiabatic contraction of
the energy gap from $\Omega_{2}$ to $\Omega_{1}$, while the state remains to
be $\rho_{1}$. From the same calculation as in step-2, we have

$\displaystyle\braket{Q_{3}}$ $\displaystyle=0\,,$ (23)
$\displaystyle\braket{W_{3}}$ $\displaystyle=-(p+\delta
p_{\text{H}})\Delta\Omega\,.$ (24)

The work is extracted from the qubit in this process.

  5. 5.

Next another isochoric thermalization process is carried out: the qubit
interacts with a bath at a colder temperature $T_{\text{C}}(<T_{\text{H}})$.
At the end of the process, the state becomes $\rho_{2}=(p+\delta
p_{\text{H}}+\delta p_{\text{C}})\ket{e}\bra{e}+(1-p-\delta
p_{\text{H}}-\delta p_{\text{C}})\ket{g}\bra{g}$ and so

$\displaystyle\braket{Q_{4}}$ $\displaystyle=\Omega_{1}\delta p_{\text{C}}\,,$
(25) $\displaystyle\braket{W_{4}}$ $\displaystyle=0\,.$ (26)

The total extracted work and dissipated heat read

$\displaystyle\braket{W_{\text{ext}}}$
$\displaystyle=-\sum_{j=1}^{4}\braket{W_{j}}=\delta
p_{\text{H}}\Delta\Omega\,,$ (27) $\displaystyle\braket{Q}$
$\displaystyle=\sum_{j=1}^{4}\braket{Q_{j}}=\Omega_{2}\delta
p_{\text{H}}+\Omega_{1}\delta p_{\text{C}}\,.$ (28)

Assuming $\Omega_{2}>\Omega_{1}$, the extracted work is positive if $\delta
p_{\text{H}}>0$.

  6. 6.

Impose $\delta p_{\text{H}}+\delta p_{\text{C}}=0$ to complete a thermodynamic
cycle. From this condition, Eqs. (27) and (28) yield
$\braket{Q}=\braket{W_{\text{ext}}}$, which obeys the first law of
thermodynamics.

It is known that the positive work condition reads Feldmann.QOE.2000 ;
Kieu.secondlaw.demon.otto  $T_{\text{C}}/\Omega_{1}<T_{\text{H}}/\Omega_{2}$
and the efficiency of the QOE is
$\eta_{\text{O}}\coloneqq\braket{W_{\text{ext}}}/Q_{2}=1-\Omega_{1}/\Omega_{2}$.

Figure 3: Each step in a relativistic QOE. The red and blue stripes indicate
the isochoric process, i.e., the interaction between the detector and the
field.

###  3.2 Work extracted by a UDW detector

Now let us employ a UDW detector as the qubit in this process. As mentioned
before, we will not assume that the baths are thermal. Instead, we only assume
that the quantum field is in a quasi-free state, i.e., the one-point
correlators $\braket{\hat{\phi}(\mathsf{x})}_{\rho_{\phi}}$ vanish.
Nevertheless, we will keep using subscripts such as $\delta p_{\text{H}}$ and
$\delta p_{\text{C}}$ for convenience.

In this subsection, we consider a general scenario in which a UDW detector
following an arbitrary path in spacetime interacts with a quantum field in a
quasi-free state. Working perturbatively, we derive the extracted work and the
condition for it to be positive.

####  3.2.1 Relativistic quantum Otto engine

We first describe our QOE in the relativistic setting as depicted in Fig. 3.

  * •

(step-1) Before the interaction, the detector with the energy gap $\Omega_{1}$
is prepared in $\rho_{\text{D,0}}$ in (9).

  * •

(Step-2) One performs a quantum adiabatic expansion $\Omega_{1}\to\Omega_{2}$
with $\Omega_{2}>\Omega_{1}$. During this process, the detector does not
interact with the field.

  * •

(Step-3) Isochoric process. The detector interacts with the field with a time-
dependence specified by a compactly supported switching function
$\chi_{\text{H}}[(\tau-\tau_{\text{H}})/\sigma_{\text{H}}]$, where
$\tau_{\text{H}}$ is the center of interaction and $\sigma_{\text{H}}(>0)$ is
the interaction duration; its support is
$\text{supp}(\chi_{\text{H}})=[\tau_{\text{H}}-\sigma_{\text{H}}/2,\tau_{\text{H}}+\sigma_{\text{H}}/2]$.
The state changes from $\rho_{\text{D,0}}\to\rho_{\text{D,1}}$, where
$\rho_{\text{D,1}}$ is given by Eqs. (11)-(13), that is,

$\displaystyle\rho_{\text{D,1}}=(p+\delta
p_{\text{H}})\ket{e}\bra{e}+(1-p-\delta
p_{\text{H}})\ket{g}\bra{g}+\mathcal{O}(\lambda^{4})\,,$ (29a)
$\displaystyle\delta
p_{\text{H}}=\lambda^{2}\sigma_{\text{H}}\big{[}(1-p)\mathcal{F}_{\text{H}}(\Omega_{2})-p\mathcal{F}_{\text{H}}(-\Omega_{2})\big{]}\,,$
(29b)
$\displaystyle\mathcal{F}_{\text{H}}(\Omega_{2})=\dfrac{1}{\sigma_{\text{H}}}\int_{\mathbb{R}}\text{d}\tau\int_{\mathbb{R}}\text{d}\tau^{\prime}\,\chi_{\text{H}}\left(\frac{\tau-\tau_{\text{H}}}{\sigma_{\text{H}}}\right)\chi_{\text{H}}\left(\frac{\tau^{\prime}-\tau_{\text{H}}}{\sigma_{\text{H}}}\right)e^{-\mathsf{i}\Omega_{2}(\tau-\tau^{\prime})}W(\mathsf{x}(\tau),\mathsf{x}(\tau^{\prime}))\,.$
(29c)

While we identified the switching function as a compactly supported function,
it is also possible to employ rapidly decreasing functions such as Gaussian or
Lorentzian functions. These are not compactly supported, but the results
remain unchanged when the exponentially decreasing tails of these functions
are disregarded. We will make use of these functions later in this subsection.

  * •

(Step-4) Adiabatic compression $\Omega_{2}\to\Omega_{1}$. Again, the detector
does not interact during this process and so the state remains as
$\rho_{\text{D,1}}$.

  * •

(Step-5) Implement another isochoric process using
$\chi_{\text{C}}[(\tau-\tau_{\text{C}})/\sigma_{\text{C}}]$ for the switching
function. To ensure that the interaction does not overlap with the previous
isochoric process, we make sure that the supports of the two switching
functions are disjoint by implementing
$\tau_{\text{H}}+\sigma_{\text{H}}/2<\tau_{\text{C}}-\sigma_{\text{C}}/2$. The
state of the detector changes as $\rho_{\text{D,1}}\to\rho_{\text{D,2}}$ where
$\rho_{\text{D,2}}=(p+\delta p_{\text{H}}+\delta
p_{\text{C}})\ket{e}\bra{e}+(1-p-\delta p_{\text{H}}-\delta
p_{\text{C}})\ket{g}\bra{g}$. The quantity $\delta p_{\text{C}}$ is

$\displaystyle\delta
p_{\text{C}}=\lambda^{2}\sigma_{\text{C}}\big{[}(1-p-\delta
p_{\text{H}})\mathcal{F}_{\text{C}}(\Omega_{1})-(p+\delta
p_{\text{H}})\mathcal{F}_{\text{C}}(-\Omega_{1})\big{]}+\mathcal{O}(\lambda^{6})\,,$
(30a)
$\displaystyle\mathcal{F}_{\text{C}}(\Omega_{1})=\dfrac{1}{\sigma_{\text{C}}}\int_{\mathbb{R}}\text{d}\tau\int_{\mathbb{R}}\text{d}\tau^{\prime}\,\chi_{\text{C}}\left(\frac{\tau-\tau_{\text{C}}}{\sigma_{\text{C}}}\right)\chi_{\text{C}}\left(\frac{\tau^{\prime}-\tau_{\text{C}}}{\sigma_{\text{C}}}\right)e^{-\mathsf{i}\Omega_{1}(\tau-\tau^{\prime})}W(\mathsf{x}(\tau),\mathsf{x}(\tau^{\prime}))\,.$
(30b)

Notice that $\delta p_{\text{C}}$ contains $\lambda^{4}$ terms since $\delta
p_{\text{H}}$ is of order $\lambda^{2}$. As in the traditional QOE, the work
extracted reads $\braket{W_{\text{ext}}}=\delta p_{\text{H}}\Delta\Omega$.

  * •

(Step-6) Finally, we impose the condition $\delta p_{\text{H}}+\delta
p_{\text{C}}=0$ to complete a thermodynamic cycle.

Since $\braket{W_{\text{ext}}}$ only depends on $\delta p_{\text{H}}$, it may
appear that the cyclicity condition does not affect this value. However there
is a subtlety. If we assume that an experimenter can freely choose a value of
$p\in[0,1]$ at the beginning of the cycle, this means that the experimenter
has to adjust the response functions $\mathcal{F}_{\text{H}}(\pm\Omega_{2})$
and $\mathcal{F}_{\text{C}}(\pm\Omega_{1})$ in such a way that the cyclicity
condition is satisfied. On the other hand, if we allow the response functions
to take any value, then the population $p$ has to be adjusted accordingly.

####  3.2.2 When $p$ is free to choose

We first require the cyclicity condition, $\delta p_{\text{H}}+\delta
p_{\text{C}}=0$, to close the thermodynamic cycle. Since $\delta p_{\text{C}}$
contains terms of order $\mathcal{O}(\lambda^{4})$, we omit these by imposing
$1\gg\delta p_{\text{H}}/p$ for a given $p(\neq 0)$. This reduces to

$\displaystyle
p\gg\dfrac{\lambda^{2}\sigma_{\text{H}}\mathcal{F}_{\text{H}}(\Omega_{2})}{1+\lambda^{2}\sigma_{\text{H}}[\mathcal{F}_{\text{H}}(\Omega_{2})+\mathcal{F}_{\text{H}}(-\Omega_{2})]}\,.$
(31)

In particular, if the long-interaction limit $\sigma_{\text{H}}\to\infty$ is
taken, we obtain

$\displaystyle
p\gg\dfrac{1}{1+\displaystyle\lim_{\sigma_{\text{H}}\to\infty}\mathcal{F}_{\text{H}}(-\Omega_{2})/\mathcal{F}_{\text{H}}(\Omega_{2})}=\dfrac{1}{1+\displaystyle\lim_{\sigma_{\text{H}}\to\infty}e^{\Omega_{2}/T_{\text{H}}^{\text{eff}}}}\,.$
(32)

Thus, in the long interaction duration,
$T_{\text{H}}^{\text{eff}}/\Omega_{2}\ll 1$ is required. Moreover, as a
special case of a detector with a rapidly decreasing switching function
interacting with the quantum field in the KMS state, this reads

$\displaystyle p\gg\dfrac{1}{1+e^{\Omega_{2}/T_{\text{H}}^{\text{KMS}}}}\,.$
(33)

Nevertheless, if the interaction duration is finite, then $\lambda\ll 1$ is
sufficient for our perturbative analysis.222To be precise, one has to make
$\lambda$ a dimensionless quantity since it has units of
$[\text{Length}]^{(n-3)/2}$ in $(n+1)$ dimensions.

Under this restriction, the cyclicity condition for any $p\in(0,1]$ leads to

$\displaystyle\sigma_{\text{C}}\mathcal{F}_{\text{C}}(\Omega_{1})\left(\dfrac{1-p}{p}-\dfrac{\mathcal{F}_{\text{C}}(-\Omega_{1})}{\mathcal{F}_{\text{C}}(\Omega_{1})}\right)=-\sigma_{\text{H}}\mathcal{F}_{\text{H}}(\Omega_{2})\left(\dfrac{1-p}{p}-\dfrac{\mathcal{F}_{\text{H}}(-\Omega_{2})}{\mathcal{F}_{\text{H}}(\Omega_{2})}\right)\,.$
(34)

That is, one has to tune the response functions so that this equality is
satisfied.

We then consider the positive work condition in the case where the value of
$p$ is chosen freely by an experimenter. The work extracted (27) is

$\displaystyle\braket{W_{\text{ext}}}$
$\displaystyle=\lambda^{2}\sigma_{\text{H}}\big{[}(1-p)\mathcal{F}_{\text{H}}(\Omega_{2})-p\mathcal{F}_{\text{H}}(-\Omega_{2})\big{]}\Delta\Omega\,,$
(35)

and the positive work condition is
$(1-p)\mathcal{F}_{\text{H}}(\Omega_{2})-p\mathcal{F}_{\text{H}}(-\Omega_{2})>0$,
i.e.,
$(1-p)/p>\mathcal{F}_{\text{H}}(-\Omega_{2})/\mathcal{F}_{\text{H}}(\Omega_{2})$.
We note that
$\mathcal{F}_{\text{H}}(-\Omega_{2})/\mathcal{F}_{\text{H}}(\Omega_{2})>1$
since the de-excitation probability is higher than the excitation probability,
thereby implying $p\in(0,1/2)$. This is consistent with previous work
UnruhOttoEngine , where the Unruh QOE with a free $p$ is able to extract work
only when $0<p<1/2$. The condition can be written in terms of the effective
temperature (20):

$\displaystyle\left(\ln\dfrac{1-p}{p}\right)^{-1}<\dfrac{T_{\text{H}}^{\text{eff}}}{\Omega_{2}}\,,\quad(0<p<1/2)\,.$
(36)

Recall that in the long interaction duration,
$T_{\text{H}}^{\text{eff}}/\Omega_{2}\ll 1$ is required, which means that (36)
tells us that $p\ll 1$ needs to be chosen if we wish to extract positive work
in the long-interaction limit within the perturbation theory.

Notice that if positive work is extracted while the cyclicity condition is
satisfied, then $\mathcal{F}_{\text{C}}(\pm\Omega_{1})$ has to obey
$(1-p)/p<\mathcal{F}_{\text{C}}(-\Omega_{1})/\mathcal{F}_{\text{C}}(\Omega_{1})$
since the response functions are non-negative and the positive work condition
requires
$(1-p)/p>\mathcal{F}_{\text{H}}(-\Omega_{2})/\mathcal{F}_{\text{H}}(\Omega_{2})$.
Therefore, within the perturbation theory,

$\displaystyle\braket{W_{\text{ext}}}>0~{}\text{from
QOE}\Rightarrow\dfrac{T_{\text{C}}^{\text{eff}}}{\Omega_{1}}<\left(\ln\dfrac{1-p}{p}\right)^{-1}<\dfrac{T_{\text{H}}^{\text{eff}}}{\Omega_{2}}\,.$
(37)

The opposite is not always true since the inequality does not necessarily
imply the cyclicity condition (34). In addition, the statement (37) does not
hold without the cyclicity condition. Nevertheless, by contraposition, if
$T_{\text{H}}^{\text{eff}}/\Omega_{2}>T_{\text{C}}^{\text{eff}}/\Omega_{1}$ is
violated then positive work cannot be extracted or the cycle is not closed.

The inequality
$T_{\text{H}}^{\text{eff}}/\Omega_{2}>T_{\text{C}}^{\text{eff}}/\Omega_{1}$
can be thought of as the generalization of the positive work condition found
in Feldmann.QOE.2000 ; Kieu.secondlaw.demon.otto  to the effective
temperature. In fact, in the long-interaction limit, the effective
temperatures become the KMS temperatures. In this particular scenario, the
work extracted becomes

$\displaystyle\braket{W_{\text{ext}}}$
$\displaystyle\approx\lambda^{2}\sigma_{\text{H}}\mathcal{F}_{\text{H}}(\Omega_{2})\left[1-p\left(1+e^{\Omega_{2}/T_{\text{H}}^{\text{KMS}}}\right)\right]\Delta\Omega\,.$
(38)

####  3.2.3 When $p$ is adjusted

Let us focus on the case where $p$ is initially adjusted and derive the work
extracted. We again assume that $p\neq 0$ and $p\gg\delta p_{\text{H}}$ so
that the terms higher than $\lambda^{4}$ can be omitted in $\delta
p_{\text{C}}$. From the cyclicity condition, we can solve for $p$, which reads

$\displaystyle p$
$\displaystyle=\dfrac{\sigma_{\text{H}}\mathcal{F}_{\text{H}}(\Omega_{2})+\sigma_{\text{C}}\mathcal{F}_{\text{C}}(\Omega_{1})}{\sigma_{\text{H}}[\mathcal{F}_{\text{H}}(\Omega_{2})+\mathcal{F}_{\text{H}}(-\Omega_{2})]+\sigma_{\text{C}}[\mathcal{F}_{\text{C}}(\Omega_{1})+\mathcal{F}_{\text{C}}(-\Omega_{1})]}\,.$
(39)

By inserting $p$ into (27) we obtain

$\displaystyle\braket{W_{\text{ext}}}=\lambda^{2}\sigma_{\text{H}}\sigma_{\text{C}}\dfrac{\mathcal{F}_{\text{H}}(\Omega_{2})\mathcal{F}_{\text{C}}(-\Omega_{1})-\mathcal{F}_{\text{H}}(-\Omega_{2})\mathcal{F}_{\text{C}}(\Omega_{1})}{\sigma_{\text{H}}[\mathcal{F}_{\text{H}}(\Omega_{2})+\mathcal{F}_{\text{H}}(-\Omega_{2})]+\sigma_{\text{C}}[\mathcal{F}_{\text{C}}(\Omega_{1})+\mathcal{F}_{\text{C}}(-\Omega_{1})]}\Delta\Omega\,.$
(40)

The extract work is positive if and only if

$\displaystyle\mathcal{F}_{\text{H}}(\Omega_{2})\mathcal{F}_{\text{C}}(-\Omega_{1})>\mathcal{F}_{\text{H}}(-\Omega_{2})\mathcal{F}_{\text{C}}(\Omega_{1})\,.$
(41)

We find that the positive work condition (41) can be written in terms of the
effective temperature defined by (20) as

$\displaystyle\dfrac{T_{\text{C}}^{\text{eff}}}{\Omega_{1}}<\dfrac{T_{\text{H}}^{\text{eff}}}{\Omega_{2}}\,.$
(42)

Again, we are not necessarily assuming that the field is in the KMS state, and
so the effective temperature is not necessarily the KMS temperature. If the
field is in the KMS state and the detector with a rapidly decreasing switching
function interacts with the field for a long time
($\sigma_{\text{H}},\sigma_{\text{C}}\to\infty$), the effective temperature
becomes the KMS temperature, leading to

$\displaystyle\dfrac{T_{\text{C}}^{\text{KMS}}}{\Omega_{1}}<\dfrac{T_{\text{H}}^{\text{KMS}}}{\Omega_{2}}\,,$
(43)

which is the condition found in Feldmann.QOE.2000 ; Kieu.secondlaw.demon.otto
. The extracted work in this case is

$\displaystyle\braket{W_{\text{ext}}}=\lambda^{2}\sigma_{\text{H}}\sigma_{\text{C}}\dfrac{(e^{\Omega_{1}/T_{\text{C}}}-e^{\Omega_{2}/T_{\text{H}}})\mathcal{F}_{\text{H}}(\Omega_{2})\mathcal{F}_{\text{C}}(\Omega_{1})}{\sigma_{\text{H}}(1+e^{\Omega_{2}/T_{\text{H}}})\mathcal{F}_{\text{H}}(\Omega_{2})+\sigma_{\text{C}}(1+e^{\Omega_{1}/T_{\text{C}}})\mathcal{F}_{\text{C}}(\Omega_{1})}\Delta\Omega\,.$
(44)

Therefore, (43) is a special case of (41) [and equivalently (42)] when the
response function obeys the detailed balance condition (19). In other words,
the condition (41) (or alternatively (42)) is applicable to any scenario, even
when the detector is not thermalized.

In summary, we showed that within the perturbation theory, the positive work
condition is associated with
$T_{\text{C}}^{\text{eff}}/\Omega_{1}<T_{\text{H}}^{\text{eff}}/\Omega_{2}$.
If $p$ is not fixed and the cycle is closed by adjusting the response
functions,
$T_{\text{C}}^{\text{eff}}/\Omega_{1}<T_{\text{H}}^{\text{eff}}/\Omega_{2}$ is
a necessary condition to extract positive work. On the other hand, if we
instead adjust $p$ to close the cycle while the response functions are not
fixed, then this condition is necessary and sufficient. This recovers the
traditional QOE positive work condition Feldmann.QOE.2000 ;
Kieu.secondlaw.demon.otto  as a special case. Note that our results hold not
only for a quantum field in the KMS state but also for any quasi-free state in
curved spacetime. The use of the effective temperature is similar to other
papers such as Huang.squeezing.work .

##  4 Examples

We now demonstrate our results in $(3+1)$-dimensional Minkowski spacetime.
Throughout this section, we use a Gaussian switching function

$\displaystyle\chi[(\tau-\tau_{j})/\sigma)]$
$\displaystyle=e^{-(\tau-\tau_{j})^{2}/2\sigma^{2}}\,,\quad
j\in\\{\text{H},\text{C}\\}$ (45)

where $\tau_{j}$ is the center of the Gaussian and $\sigma>0$ is the
characteristic Gaussian width, which has the units of time.

###  4.1 Inertial detector in thermal quantum field

Consider $(3+1)$-dimensional Minkowski spacetime. We wish to examine the
extracted work (40) by coupling an inertial UDW detector to a KMS state of the
field to explore the basic properties of the work extraction.

The Wightman function in a thermal state of a massless field can be written as
simidzija2018harvesting

$\displaystyle W_{\beta}(\mathsf{x},\mathsf{x}^{\prime})$
$\displaystyle=W_{\text{M}}(\mathsf{x},\mathsf{x}^{\prime})+W_{\text{th}}(\mathsf{x},\mathsf{x}^{\prime})\,,$
(46)

where $W_{\text{M}}$ is the Wightman function in the Minkowski vacuum and
$W_{\text{th}}$ is the contribution from the thermal state:

$\displaystyle W_{\text{M}}(\mathsf{x},\mathsf{x}^{\prime})$
$\displaystyle=-\dfrac{1}{4\pi^{2}}\dfrac{1}{(t-t^{\prime}-\mathsf{i}\epsilon)^{2}-|{\bm{x}}-{\bm{x}}^{\prime}|^{2}}\,,$
(47a) $\displaystyle W_{\text{th}}(\mathsf{x},\mathsf{x}^{\prime})$
$\displaystyle=\int_{\mathbb{R}^{3}}\dfrac{\text{d}^{3}k}{(2\pi)^{3}2|\bm{k}|}\dfrac{e^{\mathsf{i}|\bm{k}|(t-t^{\prime})-\mathsf{i}{\bm{k}}\cdot({\bm{x}}-{\bm{x}}^{\prime})}+\text{c.c.}}{e^{\beta|\bm{k}|}-1}\,,$
(47b)

Here, $\epsilon$ is the UV cutoff and $\beta=T^{-1}$ is the inverse KMS
temperature of the field.

Assuming the Gaussian switching (45), the response function
$\mathcal{F}_{\beta}(\Omega)$ of an inertial detector in the thermal bath
reads simidzija2018harvesting

$\displaystyle\mathcal{F}_{\beta}(\Omega)$
$\displaystyle=\mathcal{F}_{\text{M}}(\Omega)+\mathcal{F}_{\text{th}}(\Omega)\,,$
(48)

where

$\displaystyle\mathcal{F}_{\text{M}}(\Omega)$
$\displaystyle=\dfrac{1}{4\pi\sigma}\left[e^{-\Omega^{2}\sigma^{2}}-\sqrt{\pi}\Omega\sigma\text{erfc}(\Omega\sigma)\right]\,,$
(49a) $\displaystyle\mathcal{F}_{\text{th}}(\Omega)$
$\displaystyle=\dfrac{e^{-\Omega^{2}\sigma^{2}}\sigma}{\pi}\int_{0}^{\infty}\text{d}k\,\dfrac{ke^{-k^{2}\sigma^{2}}\cosh(2\Omega\sigma^{2}k)}{e^{\beta
k}-1}$ (49b)

are respectively the response functions for an inertial detector in the
Minkowski vacuum and the thermal contribution. Note that it does not depend on
the center of the Gaussian switching since the Wightman function is time-
translation invariant. Below, we assume that the interaction durations are the
same: $\sigma_{\text{H}}=\sigma_{\text{C}}\equiv\sigma$.

Figure 4: The extracted work and conditions for a UDW detector in a KMS state
of a thermal quantum field for (a) $T_{\text{H}}=2$ and (b) $T_{\text{H}}=4$.
Here, $\Omega_{2}=3,T_{\text{C}}=1$, and $p=0.315$ in units of $\Omega_{1}$.
(a-i) and (b-i) depict $\braket{W_{\text{ext}}}/\lambda^{2}\sigma$ as a
function of the interaction duration $\sigma$ in units of $\Omega_{1}$. The
dashed line indicates the asymptotic value. The positive work condition is
examined in (a-ii) and (b-ii), and the cyclicity condition is investigated in
(a-iii) and (b-iii).

####  4.1.1 Case: free choice of $p$

We first consider the case where $p$ is freely chosen as discussed in Sec.
3.2.2. We plot $\braket{W_{\text{ext}}}/\lambda^{2}\sigma$ defined in (35) as
a function of interaction duration $\sigma$ when
$T_{\text{H}}=2,\Omega_{2}=3,p=0.315$, and $\Omega_{1}=1$ in Fig. 4(a-i). The
dashed line represents the asymptotic value,
$\lim_{\sigma\to\infty}\braket{W_{\text{ext}}}/\lambda^{2}\sigma$, given by
(38). We see that the work can be extracted when the interaction duration is
short, but as $\sigma$ increases extraction becomes exponentially difficult.
This behavior can be explained by examining the positive work condition (36)
in Fig. 4(a-ii). The figure shows that the positive work condition is
satisfied when $\sigma$ is small enough. When $\sigma\to\infty$ is taken, the
effective temperature becomes the KMS temperature:
$T_{\text{H}}^{\text{eff}}/\Omega_{2}\to T_{\text{H}}/\Omega_{2}=2/3$ in
4(a-ii). Moreover, based on the discussion in Sec. 3.2.2, the perturbative
analysis is valid in this limit when $T_{\text{H}}/\Omega_{2}\ll 1$, which is
not the case in our scenario. Nevertheless, the perturbative analysis is valid
in the figure as long as $\lambda\ll 1$.

Although $\braket{W_{\text{ext}}}/\lambda^{2}\sigma>0$ for some $\sigma$, the
thermodynamic cycle is not closed. The cyclicity condition (34) when
$\sigma_{\text{H}}=\sigma_{\text{C}}$ reads

$\displaystyle\mathcal{F}_{\text{C}}(\Omega_{1})\left(\dfrac{1-p}{p}-\dfrac{\mathcal{F}_{\text{C}}(-\Omega_{1})}{\mathcal{F}_{\text{C}}(\Omega_{1})}\right)=-\mathcal{F}_{\text{H}}(\Omega_{2})\left(\dfrac{1-p}{p}-\dfrac{\mathcal{F}_{\text{H}}(-\Omega_{2})}{\mathcal{F}_{\text{H}}(\Omega_{2})}\right)\,.$
(50)

We plot each side of this equation in Fig. 4(a-iii), fixing $T_{\text{C}}=1$
and other parameters (except $\sigma$), so as to find an optimal interaction
duration $\sigma$ such that the two curves cross. This point in the figure
represents the scenario when the cyclicity condition is satisfied. As one can
see, the positive work condition in 4(a-ii) and the cyclicity condition in
4(a-iii) are not satisfied at the same time; hence the work extracted is not
coming from a genuine QOE. In fact, Fig. 4(a-ii) tells us that
$T_{\text{C}}^{\text{eff}}/\Omega_{1}>T_{\text{H}}^{\text{eff}}/\Omega_{2}$,
and the statement (37) implies that either work cannot be extracted or that
the thermodynamic cycle is not closed.

Meanwhile, we can find a set of parameters that enables us to extract positive
work from a closed QOE. This scenario is demonstrated in Fig. 4(b), where
$T_{\text{H}}=4$ is chosen. Figure 4(b-i) shows that the extracted work is
positive for all $\sigma$, which is consistent with the positive work
condition examined in 4(b-ii). This time, from Fig. 4(b-iii), there exists a
value of $\sigma$ that satisfies the cyclicity condition while
$\braket{W_{\text{ext}}}/\lambda^{2}\sigma>0$. Moreover, Figs. 4(b-ii) and
(b-iii) exemplify the statement (37): if positive work is extracted from a
genuine QOE, then the effective temperatures satisfy
$T_{\text{C}}^{\text{eff}}/\Omega_{1}<1/\ln[(1-p)/p]<T_{\text{H}}^{\text{eff}}/\Omega_{2}$,
but the opposite is not always true.

####  4.1.2 Case: adjusted $p$

We now consider the QOE described in Sec. 3.2.3. We first note that an
inertial detector in the Minkowski vacuum (i.e., zero temperature:
$T_{\text{H}}=T_{\text{C}}=0$) cannot extract work from vacuum fluctuations
since the condition (41) cannot be satisfied no matter what $\Omega$ and
$\sigma$ are. We then consider the KMS temperatures
$T_{\text{H}},T_{\text{C}}\neq 0$ and compute the extracted work
$\braket{W_{\text{ext}}}/\lambda^{2}\sigma$. The interaction duration $\sigma$
is divided so that the perturbative analysis is valid—otherwise, the extracted
work $\braket{W_{\text{ext}}}/\lambda^{2}$ monotonically increases with
$\sigma$ without an upper bound. In Fig. 5(a), we plot the extracted work
$\braket{W_{\text{ext}}}/\lambda^{2}\sigma$ as a function of $\sigma$ with
various $T_{\text{H}}$. Here, quantities are in units of the energy gap
$\Omega_{1}$ at the beginning of the cycle, and we fix
$\Omega_{2}/\Omega_{1}=3$ and $T_{\text{C}}/\Omega_{1}=1$. We observe that the
detector cannot extract positive work when the interaction duration is too
short (i.e., $\sigma\Omega_{1}\ll 1$), while
$\braket{W_{\text{ext}}}/\lambda^{2}\sigma$ asymptotes to a number at
$\sigma\Omega_{1}\gg 1$. This is the opposite behavior of the result shown in
Fig. 4, and can be attributed to the difference between the positive work
conditions given by (36) and (42). In the case of free $p$, as illustrated in
Fig. 4(b), $T_{\text{H}}^{\text{eff}}/\Omega_{2}$ is compared to a constant
value. In contrast, for Fig. 5, $T_{\text{H}}^{\text{eff}}/\Omega_{2}$ is
compared to another effective temperature
$T_{\text{C}}^{\text{eff}}/\Omega_{1}$ as depicted in 5(b).

The asymptotic value for each curve is depicted by the dashed line in Fig.
5(a), which is evaluated from (44). This confirms that, at the long
interaction duration, the detailed balance condition for the response function
(19) is satisfied. In addition, we observe that the work extracted when
$(T_{\text{H}}/\Omega_{1},\Omega_{2}/\Omega_{1})=(2,3)$ and $(3,3)$ is always
non-positive, whereas it is possible to extract positive work from the system
with $(T_{\text{H}}/\Omega_{1},\Omega_{2}/\Omega_{1})=(4,3)$. This can be
understood from the condition (43); when the detailed balance condition (19)
is satisfied ($\sigma\Omega_{1}\gg 1$ in this case), the detector can extract
positive work when the temperature of the baths and the energy gap obey (43).

Figure 5: (a) The extracted work $\braket{W_{\text{ext}}}/\lambda^{2}\sigma$
as a function of the interaction duration $\sigma$ in units of $\Omega_{1}$.
Here, $T_{\text{C}}=1$. The dashed lines are the work extracted (44) after the
detailed balance condition is satisfied. (b) The effective temperature divided
by the energy gap in the case of $T_{\text{H}}=4$ and $\Omega_{2}=3$ in (a).
The positive work condition is satisfied when
$T_{\text{C}}^{\text{eff}}/\Omega_{1}<T_{\text{H}}^{\text{eff}}/\Omega_{2}$.

###  4.2 UDW detector in circular motion

The Unruh QOE utilizes the thermality caused by the Unruh effect at the
temperatures $T_{\text{H}}=a_{\text{H}}/2\pi$ and
$T_{\text{C}}=a_{\text{C}}/2\pi$ with $a_{\text{H}}>a_{\text{C}}$ in the
protocol described in Sec. 3 UnruhOttoEngine . However, as we pointed out in
the Introduction, a linearly accelerating UDW detector is not ideal for
extracting thermodynamic work since it requires a tremendous amount of
acceleration $a$ as well as a huge space to let the detector travel long
distances. Instead, we wish to employ a UDW detector in circular motion, which
not only allows us to confine the detector in a compact space, but also less
work is required to acquire some temperatures compared to the linear
acceleration case. Moreover, the Wightman function along a circulating
detector’s trajectory does not satisfy the KMS condition. Our general
expression for the extracted work (35) and (40) as well as the positive work
condition (36) and (42) can still be applied to such a scenario.

Consider the circular trajectory of the UDW detector, which is given by

$\displaystyle\mathsf{x}(\tau)$
$\displaystyle=\\{t=\gamma\tau,x=R\cos(\omega\gamma\tau)\,,y=R\sin(\omega\gamma\tau),z=0\\}\,.$
(51)

Here, $R(>0)$ is the radius of the circle, $\omega$ is the angular velocity of
the detector, and $\gamma=1/\sqrt{1-R^{2}\omega^{2}}$. One can introduce the
proper acceleration of the detector, whose magnitude $a$ is given by
$a=R\omega^{2}\gamma^{2}$, as well as the speed of the detector
$v=R\omega(\leq 1)$. Using these relations, one can write $\gamma,\omega,v$ in
terms of $a$ and $R$ as follows.

$\displaystyle\omega$ $\displaystyle=\sqrt{\dfrac{a}{(1+aR)R}}\,,$ (52a)
$\displaystyle\gamma$ $\displaystyle=\sqrt{1+aR}\,,$ (52b) $\displaystyle v$
$\displaystyle=\sqrt{\dfrac{aR}{1+aR}}\,.$ (52c)

In order to calculate the response function (13), one needs to have an
expression of the pullback of the Wightman function along the trajectory (51).
Consider a minimally coupled, massless quantum scalar field in
$(3+1)$-dimensional Minkowski spacetime. The Wightman function in the
Minkowski vacuum is given by (47a). Inserting (51), we obtain the pullback of
the Wightman function:

$\displaystyle
W(\mathsf{x}(\tau),\mathsf{x}^{\prime}(\tau^{\prime}))=-\dfrac{1}{4\pi^{2}}\dfrac{1}{\gamma^{2}(\Delta\tau-\mathsf{i}\epsilon)^{2}-4R^{2}\sin^{2}(\omega\gamma\Delta\tau/2)}\,,$
(53)

where $\Delta\tau\coloneqq\tau-\tau^{\prime}$. Since the Wightman function is
time-translation invariant, the response functions do not depend on the
switching time.

Figure 6: The extracted work $\braket{W_{\text{ext}}}/\lambda^{2}\sigma$ by a
detector in circular motion as a function of the interaction duration $\sigma$
in units of $\Omega_{1}$. Here, $\Omega_{2}=1.01$ and $a_{\text{C}}=1$. (a)
The case where $p$ is adjustable and $p=1/20$, and (b) the case where $p$ is
tuned to complete the cycle. Each figure contains $a_{\text{H}}=2$ and
$a_{\text{H}}=3$.

####  4.2.1 Extracted work

From the Wightman function above, we can numerically calculate the response
functions and the work extracted for the detector in circular motion. In Fig.
6, we plot $\braket{W_{\text{ext}}}/\lambda^{2}\sigma$ as a function of
$\sigma$ in units of the energy gap $\Omega_{1}$ in step-1. Figures 6(a) and
6(b) are respectively the cases where $p$ is freely chosen and tuned. In both
figures, we set $\Omega_{2}/\Omega_{1}=1.01,R\Omega_{1}=1$, and
$a_{\text{C}}/\Omega_{1}=1$, and each figure includes the curves for
$a_{\text{H}}/\Omega_{1}=2$ and 3.

As in the case of the inertial detector in a thermal bath, QOE with a free $p$
(here $p=1/20$) extracts more work when the interaction duration is short
[Fig. 6(a)], whereas if $p$ is tuned to close the cycle, the opposite behavior
holds [Fig. 6(b)]. However, we find in the latter case that, unlike in Sec.
2.2, the extracted work does not reach a maximum at large $\sigma$. Instead,
there exists a maximum at $\sigma\Omega_{1}\approx 1$. Nevertheless, we
confirm that this behavior is contingent on the choice of parameters.

Figure 7: Work extracted by a circulating UDW detector as a function of the
radius $\log_{10}R$ when (a) $p$ is adjustable (chosen to be $p=1/20$), and
(b) $p$ is tuned to complete the cycle. Here, $\Omega_{2}=2,\sigma=1$, and
$a_{\text{C}}=1$ in units of $\Omega_{1}$. Each figure contains three cases of
$a_{\text{H}}$: $a_{\text{H}}=3,4$, and 5.

We then plot $\braket{W_{\text{ext}}}/\lambda^{2}\sigma$ as a function of the
radius of the circle, $\log_{10}R$, in units of $\Omega_{1}$ in Fig. 7. Here,
we set $\Omega_{2}=2,\sigma=1,a_{\text{C}}=1$, and each figure includes curves
for $a_{\text{H}}=3,4$, and 5. As before, Fig. 7(a) corresponds to QOE with
$p=1/20$, while 7(b) depicts the result with a tuned $p$. Both figures exhibit
a similar pattern; a detector orbiting with a small radius $R$ has difficulty
extracting work, and $\braket{W_{\text{ext}}}/\lambda^{2}\sigma$ is maximized
as $R$ increases, eventually asymptoting to some value at $R\to\infty$. Thus,
the radius $R$ of the orbit must be optimized to extract the maximum amount of
work.

####  4.2.2 Fixed acceleration

Although the analytic expression for the effective temperature for a
circulating detector, $T_{\text{circ}}$, cannot be obtained in general, one
can impose an additional assumption to obtain the exact form of
$T_{\text{circ}}$. Suppose that the detector is circulating at high speed
$v\approx 1$ (i.e., ultrarelativistic regime), which can be realized by taking
a large radius $R$. The effective temperature in this case is known to be
CircularTemperaturesUnruh ; Unruh.acceleration.rad.electron.1998.arxiv

$\displaystyle T_{\text{circ}}$
$\displaystyle=\dfrac{|\Omega|}{\ln\left[1+\dfrac{4\sqrt{3}|\Omega|}{a}\exp\left(\dfrac{2\sqrt{3}|\Omega|}{a}\right)\right]}\in(a/4\sqrt{3},a/2\sqrt{3})\,.$
(54)

While the effective temperature $T_{\text{circ}}$ increases with $a$, it is
also possible to enhance the temperature by enlarging the energy gap $\Omega$
with fixed acceleration. That is, for a given $a$,
$T_{\text{circ}}(\Omega_{2})>T_{\text{circ}}(\Omega_{1})$ if
$\Omega_{2}>\Omega_{1}$. We then ask if the circulating detector at high speed
with a fixed acceleration $a$ can extract positive work by manipulating the
effective temperature by adjusting the energy gap.

The answer to this inquiry is negative. To see this, it is easy to check that,
for a given $a>0$, $\Omega_{2}>\Omega_{1}\Rightarrow
T_{\text{circ}}(\Omega_{2})/\Omega_{2}<T_{\text{circ}}(\Omega_{1})/\Omega_{1}$,
which violates the necessary condition for positive work, (36) and (42). Thus,
we find that, for any $p\in(0,1)$, it is impossible to extract positive work
from a UDW detector in a circular motion at $v\approx 1$ by fixing $a$. This
is one example of the feature of quantum heat engines:
$T_{\text{C}}<T_{\text{H}}$ does not necessarily guarantee the extraction of
positive work in quantum heat engines, but rather, one should consider
$T_{\text{C}}/\Omega_{1}<T_{\text{H}}/\Omega_{2}$, where
$\Omega_{1}<\Omega_{2}$.

##  5 Conclusion

Inspired by the Unruh quantum Otto engines, we considered the quantum Otto
engine (QOE) in a general setting utilizing an Unruh-DeWitt (UDW) detector
interacting with a quantum scalar field. While the original QOE considered a
qubit thermalized in thermal baths, we assumed (i) the field is in a quasi-
free state, not necessarily in a Kubo-Martin-Schwinger (KMS) thermal state,
and (ii) the detector follows an arbitrary trajectory in curved spacetime. By
carrying out a perturbative method, we managed to derive the extracted work
(35) and (40) in terms of the response function of the UDW detector. In
addition, we found that the condition for extracting positive work can be
expressed in terms of the effective temperature perceived by the detector,
(36) and (42). These formulae are applicable to various scenarios outside
(thermal) equilibrium. As a particular case, if the field is in the KMS state
and the detector with a rapidly decreasing switching function interacts with
it for a long time, the response function is known to obey the detailed
balance condition, which makes the effective temperature identical to the KMS
temperature. The condition (42) then becomes the positive work condition (43)
originally found by Feldmann.QOE.2000 ; Kieu.secondlaw.demon.otto .

Using the formula for the extracted work, we explored two concrete examples:
an inertial detector in the KMS state and a circulating detector in the
Minkowski vacuum.

In the scenario of an inertial detector in thermal baths, we numerically
examined the effect of the interaction duration. The relationship between the
extracted work and the interaction duration is drastically changed by the
choice of the detector’s initial state. Nevertheless, the extracted work
asymptotes to the value that corresponds to thermal equilibrium as the
interaction duration increases.

Another example involves a circulating detector in the Minkowski vacuum. The
motivation for this scenario comes from the previously proposed Unruh QOE
UnruhOttoEngine , in which the thermal baths are generated by the Unruh
effect. However, utilizing a linearly accelerating UDW detector in the Unruh
QOE is not practical for several reasons such as the work necessary for
acceleration and the requirement for a large space. The circulating detector,
on the other hand, circumvents these issues. Moreover, the QOE with a
circulating detector can be adequately examined only when our general
framework is employed; The Wightman function pulled back along the circular
trajectory does not satisfy the KMS condition, which suggests that the use of
the effective temperature is necessary.

The work extracted by the circulating detector showed behavior similar to that
of the inertial detector in a thermal bath. However, we found that there exist
optimal values for the interaction duration and radius of the orbit to extract
maximum work. Furthermore, we showed that the detector is unable to extract
work with a fixed acceleration due to the violation of the positive work
condition.

An interesting extension to our work would involve exploring quantum heat
engines using non quasi-free states. In particular, coherent and squeezed
states—which are not quasi-free—are known to be particular interest in a
variety of settings. It would be interesting to see how a non-vanishing one-
point correlator contributes to the cycle.

## Acknowledgments

This work was supported in part by the Natural Sciences and Engineering
Research Council of Canada. KGY is thankful to Dr. Jorma Louko for the
insightful discussion.

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# Optical pilot-tone correction of phase errors in photo-detection chains

Alexander Schultze  Airbus DS GmbH, 88090 Immenstaad, Germany
<EMAIL_ADDRESS>0000-0002-5417-6373 Dennis Weise  Airbus DS
GmbH, 88090 Immenstaad, Germany  Claus Braxmaier  DLR, Institute of Quantum
Technologies, 89077 Ulm, Germany  Universität Ulm, Institut für
Mikroelektronik, 89069 Ulm, Germany  0000-0002-5640-1634 Alexander Sell
Airbus DS GmbH, 88090 Immenstaad, Germany

###### Abstract

The delay of photodetectors can be affected by intensity, reverse bias, and
temperature through different effects. An optical pilot tone superimposed on
the detectors allows the independent measurement of such phase errors in the
complete photodetection chain and provides an opportunity to correct them.
This allows further separate readout noise from the measurement, providing a
more performant and intensity-invariant phase readout. We test the functional
principle on a setup demonstrating an improved phase noise performance and a
reduced phase walk below 10 mHz in particular. This benefits applications
which require accurate timing or signal phase determination with photodiodes.

††journal: arxiv

##  1 Introduction

Figure 1: Setup with optical readout: A light-emitting diode (LED) is driven
by two signal generators whose signals $f_{\textrm{a}}$, $f_{\textrm{b}}$ are
combined electrically. The modulated light is equally distributed on two
photodiodes (D1/2). After conditioning with a transimpedance amplifier (TIA)
and analog-to-digital conversion (ADC), the phase of the AC signal is
recovered by IQ demodulation. A second LED is optionally used to vary the mean
intensity ($\nu$).

Picometer metrology requires a phase determination error of just ppm of a
period for light in the NIR spectrum. Typical phase noise curves for length
metrology expose a 1/f behavior for frequencies below 10 mHz, impeding
accurate long-term measurements. Consequently, setups for picometer metrology
often include elaborate stabilization of light intensities, temperature, and
phase to reduce the noise floor. However, not all measurement setups can
provide stable intensities during the measurement period, for instance, due to
changes in reflectivity, polarization losses in the measurement path, or
parasitic interferometers. Intensity-to-phase conversion in the detector
itself can thus become a limiting effect [1]. State-of-the-art phasemeters
like the prototype for the LISA mission [2] in addition employ electrical
pilot tones superimposed to the measurement in order to remove noise from
filters, amplifiers, and ADC-stages, correcting analog-to-digital converter
(ADC) timing jitter in particular. State-of-the-art stability of phasemeters
was recently investigated by [3].
Here we propose a method that allows characterization and subtraction of
errors in the photodetection by adding a pilot tone already in the optical
domain. A pilot tone is a signal of known shape and phase within the bandwidth
of the photodetector, with its frequency typically closely related to the
detected measurement signal (e.g., beat). The optical pilot tone correction
method can improve measurement performance in applications requiring a precise
phase or delay readout. These include telemeters, which measure distances by
evaluating the phase shift accumulated by an intensity-modulated light beam
during its propagation in air [4]. This principle is widely implemented, for
instance, in distance meters for geodetic applications. Other use cases
include optical time-of-flight sensors (LiDAR [5], laser ranging [6]), optical
clocks, optical frequency distribution, or phase delay characterisation of
optical coatings and fibers. We demonstrate that in a typical heterodyne
setup, using our pilot tone correction approach, optical phase detection noise
can effectively be suppressed by a factor of 100.

###  1.1 Phase noise contributions in optical signal detection

In this work, several contributors originating in the read-out chain that
couple into phase through intensity and temperature have been identified,
including:

  * •

Photodiode induced phase delays by pn junction carrier drift velocity

  * •

Electrical phase shift caused by low pass filters

  * •

Analog-to-digital-converter (ADC) phase shifts

  * •

Slew rate limitations of operational amplifiers

The photodiode transfer function is influenced by the carrier propagation
velocity in the diode substrate, depending on the internal electrical field.
This field can be affected directly by changes in the PN junction bias
voltage, or the photocurrent, as shown in [1, 7]. The relation is non-linear
and can expose working points for bias and photocurrent insensitive to minor
variations, effectively reducing phase shift susceptibility. Additionally, the
equilibrium mentioned above can be affected by temperature through changes in
the photosensor’s quantum efficiency.
A second type of coupling from temperature into phase may occur through
variations in the dark current, which typically increases logarithmically over
temperature. These two latter effects also effectively limit the performance a
photodiode-based intensity stabilization can reach, causing out-of-loop
intensity noise.
Electrical phase shifts can further be introduced by the low-pass filter that
is typically implemented before ADC conversion to sample only the first
Nyquist zone, e.g. by means of resistor-capacitor filter circuits. The phase
shift that a frequency $f$ is subjected to in a first order filter is given
as:

$\varphi=-{\rm tan}^{-1}(2\pi\cdot R\cdot C\cdot f)$ (1)

Both components will expose a thermal coefficient of several ppm/K for real
parts, e.g., temperature compensated ceramic capacitors (NP0/C0G) still
exhibit a typical tolerance of 0$\pm$30 ppm/K. The operational amplifiers used
to condition the photodiode’s electrical signal and drive the analog-to-
digital converters may also introduce phase shifts when their slew rate limit
$\hat{V}_{\rm lim}$ is exceeded by the amplitude of the electrical output
signal $\tilde{V}_{\rm o}$, i.e.

$\hat{V}_{\rm lim}\leq 2\pi\cdot f\cdot\tilde{V}_{\rm o}$ (2)

ADC phase shifts have been observed, caused by changes in the electrical DC-
offset of its input. These are described and characterized in detail in
section 5.

##  2 Characterization of phase shifts in photodiodes

Figure 2: Signal phase as function of photocurrent (top) and reverse bias
voltage (bottom) for different photodiodes.

The experimental setup sketched in Fig. (1) for characterization comprises two
LEDs, one of which is driven directly by AC signal generators that provide
sinusoidal signals at frequencies $f_{a}$ and $f_{b}$. A second LED is used to
vary the mean intensity. The modulated LED light is equally split onto two
photodiodes. The photocurrent of each photodiode is amplified by
transimpedance amplifiers, converted by analog-to-digital converters
(ADC161S626 SAR, 16-bit, 200 ksps) in differential mode, and then read into an
FPGA to perform IQ demodulation at both frequencies by multiplication with a
digital reference sine and cosine followed by a digital 4.th order low pass
filter. The setup demonstrated successfully that the phase of both reference
oscillators could be recovered without adding additional noise due to the
narrow bandwidth of the IQ demodulation. Characterization of effects as
mentioned above for an exemplary set of photodiodes was performed with the
setup; the results are presented in Fig. (2). Variation of the mean
photocurrent uses the 2nd LED to rule out cross-coupling in the signal
generator. Extensive characterizations were also performed by [7].

Diode | Type | Intensity | Bias | Temp.
---|---|---|---|---
|  | $\rm(rad/A)$ | $\rm(rad/V)$ | $\rm(rad/K)$
Hamamatsu G12180 | InGaAs PIN | -80 | 20 | -
Hamamatsu S3071 | Si PIN | -60 | 36 | 0.1
Centronic OSD15-5T | Si | 40 | 0.1 | -
Table 1: Phase coupling factors (peak) for tested photodiodes. Intensity
coupling factor measured @ 2.5V bias.

The experiments confirm phase shift contributions through bias and intensity
variation. For the low incident light levels used, all our detectors show
significant phase shifts. For all tested photodiodes, the phase contributions
are more severe for lower incident light levels. Only one type of photodiode
offered a working point insensitive to bias and intensity with neglectable
phase gradient for our given frequencies and light intensities.

##  3 Optical pilot-tone aided phase shift correction

A method for characterizing and eliminating phase shifts introduced by the
optical readout chain is proposed, where an optical pilot tone of known shape
and phase is superimposed onto the detectors to correct phase delays. It is
based on the premise that two closely related detected signals of adjacent
frequencies, the measurement and the pilot, will experience similar phase
delays in photodiode and readout. In the following, we demonstrate that two
optical tones undergo a similar phase shift for different kinds of excitation.
The setup was exposed to variations in intensity, or temperature of one of the
photodiodes using a heater. In Fig.(3a) the measured intensity-to-phase
conversion of slowly modulated mean intensity, imprinted employing an
independent 2nd LED, is shown: Differential measurements 111A high intensity
gradient change (edges of sawtooth function) would translate to a phase. Data
near turning points of intensity function is therefore omitted. (Diode 1,
Pilot A-B) of both signal phases suppress the conversion by a factor of
$\approx$ 30\. The remaining phase difference, likely caused by differing mean
intensities, corresponds to single picometer levels. Fig. (3b) shows the
temperature dependency (120 µrad/K) of a photodiode and the excellent
suppression using the pilot tone scheme while excitation with a $\approx$5 K
temperature step.

Figure 3: Phase noise due to conversion effects in a photodiode (S3071),
caused by intensity variation (top) and temperature variation (bottom). The
signal A of one detector ($\times$ blue) responds strongly to intensity and
temperature noise (orange). Referencing to a second pilot signal B effectively
suppresses the noise ($\ast$ green). Figure 4: Heterodyne interferometer setup
with optical frequencies $f_{1}$ and $f_{2}$, which are superimposed with a
first pilot tone $f_{a}$ on the measurement path (MEAS), and an optional
second pilot tone $f_{b}$ on the local oscillator path (LO). A polarizing
beam-splitter (PBS) and quarter-wave plate ($\lambda/4$) polarization-route
the measurement beam towards two photodetectors (D1/2) sensing the signals.

As the amplitude of each beat note can only use half of the dynamic range, the
quantization noise floor is slightly increased with this method. This can be
compensated by increasing the total dynamic range of the measurement, e.g.
with higher ADC bit-rate, oversampling, and driving the ADC input signal to
full scale.
For this characterisation, amplitude modulations at frequencies
$f_{a}=12.5\leavevmode\nobreak\ \mathrm{kHz}$ (divider 800) and $f_{b}\approx
12.690\leavevmode\nobreak\ \mathrm{kHz}$ (divider 788) are chosen as derivates
from a 10 MHz reference clock that is used to synchronize signal generators
and FPGA, such that no phase cross talk is observed between both phase
signals. As signal generator precision is limited to 8 / 11 digits (based on
operation) a post-correction of linear trend to compensate for the remainder
of the rational fraction of $f_{b}$ is applied.

Figure 5: Left: Phase of pilot tones and optical intensity vs. time on
detector D1 of a heterodyne interferometer. A single pilot tone phase on D1
$\varphi_{a,D1}$ (blue) apparently correlates with total intensity noise
(red). Subtraction of a second pilot tone phase, i.e.
$\varphi_{a,D1}-\varphi_{b,D1}$ (black), suppresses detector contributed phase
noise. Right: Linear spectral densities of $\varphi_{a,D1}$ (blue), the
corrected signal $\varphi_{a,D1}-\varphi_{b,D1}$ (black), and after correction
of signal generator noise subtracting a second detection of both pilot tones
on an additional detector D2, i.e.
$\left(\varphi_{a,D1}-\varphi_{b,D1}\right)-\left(\varphi_{a,D2}-\varphi_{b,D2}\right)$
(yellow). In a further experiment, phase noise of a pilot tone
$\varphi_{b,D1}$ (green) is recorded in presence of a free-running heterodyne
beat.

##  4 Heterodyne interferometer with pilot tone correction

The correction based on a locally generated pilot tone is here demonstrated in
a configuration representative for a heterodyne interferometer, where two
laser beams with frequencies $f_{1}$ and $f_{2}$ interfere to generate a beat
signal at frequency $f_{1}-f_{2}$. Figure (4) shows a typical Mach-Zehnder
interferometer, where additional amplitude modulated signals of frequencies
$f_{a}$ and $f_{b}$ are superimposed on the laser inputs. Either, the beams
themselves are modulated (e.g. using acousto-optical modulators (AOM), which
might already be present for generating the heterodyne optical frequencies),
or alternatively those signals are introduced by superposing additional
(modulated) light sources. In a first demonstration, the heterodyne signal is
simulated by the signal at frequency $f_{a}$, featuring a known phase
$\varphi_{a}$. Figure (5) shows the measured phase $\varphi_{a,D1}$ (blue
line), recorded via a single detector D1, which apparently is strongly
correlated to the intensity level incident on the photodiodes (red line). Now,
a constant-phase signal at frequency $f_{b}$ is used as a pilot tone,
exploiting that the excess phase noise
$\varphi_{D1}=\varphi_{a,D1}-\varphi_{a}$ affects any measurement on this
detector, i.e. $\varphi_{b,D1}\approx\varphi_{b}+\varphi_{D1}$. As shown by
the black curve, the difference $\varphi_{a,D1}-\varphi_{b,D1}$ of both
measurements indeed reduces the detection noise significantly, i.e.

$\displaystyle\varphi_{a,D1}-\varphi_{b,D1}$
$\displaystyle\approx\left(\varphi_{a}+\varphi_{D1}\right)-\left(\varphi_{b}+\varphi_{D1}\right)$
$\displaystyle=\varphi_{a}-\varphi_{b}=\varphi_{a}-{\rm const.}$ (3)

However, the phase of the reference pilot tone may be subject to phase noise
as well, which can be improved by adding more detections. The residual phase
noise in $\varphi_{a,D1}-\varphi_{b,D1}$ (black curve) might originate from
time base difference in the used (externally synchronized) signal generators,
and possibly also other delays involved in the generation of the optical
signals. We demonstrate this is by adding a detection on a second photodiode
D2, and removing this contribution by forming the difference
$\left(\varphi_{a,D1}-\varphi_{b,D1}\right)-\left(\varphi_{a,D2}-\varphi_{b,D2}\right)$
(see yellow line in Fig. 5), showing that the actual residual error
corresponds to phase stabilities of $\approx$ 120 µrad$/\sqrt{\textrm{Hz}}$ at
low frequencies, resulting in a 100-fold improvement over an uncorrected
measurement. A similar level of noise is present in actual heterodyne
detection and thus can be removed by our technique. The green curve shows the
phase noise of a pilot tone (at a frequency $f_{b}$) measured in parallel to a
heterodyne beat note (at frequency $f_{h}=f_{1}-f_{2}$). To demonstrate that
the concept does not rely on the stability of the heterodyne beat, here it was
left free-running with large phase-noise. 222Our experiments using signal
generators with synchronized time bases have shown that a 2-channel signal
generator device does not necessarily share the same time base between both
channels. Hence, using two separate signal generators linked to a master clock
yielded better performance.. During all measurements, the setup was placed in
a Styrofoam box at ambient conditions, intentionally no phase-locking,
intensity-stabilization or active thermal control was involved. The laser
sources used are Nd:YAG operating at a wavelength $\lambda=1064$ nm.
One particular use case of heterodyne interferometry is the very precise
measurements of angles with quadrant photodiodes (QPDs) using differential
wavefront sensor (DWS) evaluation, based on the phase difference between
adjacent segments. Assuming a worst-case quadrant-to-quadrant phase difference
of $\approx$240 µrad (i.e. twice the value demonstrated above for a single
channel of a Hamamatsu G6849 diode), would imply a possible angular error of
less than 200 nrad.

#### Correction of signal generation phase noise using multiple pilot tones.

In heterodyne detection as described above, each frequency $f_{1}$ and $f_{2}$
is generated by a digital-signal synthesizer, hence the measured beat note is
subject to the phase noise difference of the signal generators
$\varphi_{1}-\varphi_{2}$. This is typically dealt with by an additional local
detection of the beat phase, to which all other phase detections are
referenced.
If now each signal generator imprints its individual pilot tone, additional
information of the time bases is available on the detector (exploiting
$\varphi_{a,b}=\left(f_{a,b}/f_{1,2}\right)\cdot\varphi_{1,2}$). With this
independent measurements, the signal generator phase difference contribution
is extracted via:

$\varphi_{1}-\varphi_{2}=\left(f_{1}/f_{a}\right)\varphi_{a}-\left(f_{2}/f_{b}\right)\varphi_{b}$
(4)

##  5 Characterization of analog-to-digital phase deviation caused by DC
signal component

Phase variations, based on the dc component of the input of an analog-to-
digital converter (ADC), have been observed and are described in the
following. The experimental setup comprises of a signal generator directly fed
into a differential analog input, with a sine signal (2 Vpp) combined with a
slowly modulated mean offset ($\pm$8 V, 0.1 Hz), by direct digital synthesis
in the function generator. IQ demodulation can be considered robust against
amplitude and different working points, however an intensity variation still
couples into phase. The mean signal component is therefore varied with a
constant slope of a triangle wave. The measurements in Fig. (6a) indicate a
worst-case mean-signal-to-phase correlation of $1.9$ mrad/FS over an intensity
variation of 80% full-scale (FS), with significantly lower contribution when
the mean voltage approaches zero. The measurement is performed with a
commercial ADC333The ADC input characterized here is a AD7980 SAR 16-bit / NI
PXI-7854.. As the phasemeter is sensitive to a gradient of the mean signal, a
minor hysteresis can be observed in the measurement when both the positive and
negative slope are evaluated.

Figure 6: Phase response due to offset of differential analog-to-digital input
over 80% full-scale (FS) (top) and corresponding pilot tone phases (bottom).

One assumption is, that this is caused by the kickback charge from the ADC
input. The analog input reservoir capacities, which are sampled into sample-
and-hold circuits, are discharged more for non-zero dc offsets, and recharge
of these reservoir capacities is limited by their low-pass behavior or
bandwidth of the driving amplifiers. This theory is supported by measurements
that show an increased dc-to-phase coupling of 34% when a capacitor (66 nF) is
added between the differential input signal lines, and can possibly indicate
how capacitive changes in the electronics and transmitting line can couple
into phase. Figure (6b) shows the performance of pilot tone correction (pilot
tone amplitudes $A_{\textrm{a}}=2.0\leavevmode\nobreak\ $Vpp,
$A_{\textrm{b}}=0.6\leavevmode\nobreak\ $Vpp) for this effect. The pilot tone
suppresses the error by about 50%. As recommendation for analog input designs,
it may be therefore beneficial to separate ac and dc components of the signal
before converting to digital, or improve the analog input stage design with
high bandwidth operational amplifiers and by avoiding propagation of kickback
effects by use of voltage followers. The dc-to-phase coupling factor measured
here is likely subordinate to the analog output and input stage design used.

##  6 Conclusion

Several sources for phase delays in the optical detection via photodiodes have
been identified, and a method for characterization and correction based on a
reference pilot tone superimposed on the original signal has been proposed.
Experimental validation has shown that these effects can be mitigated, and
performance at lower frequencies can be improved with the proposed correction
scheme, enabling picometer interferometry also in the presence of intensity
noise. This in particular allows dynamic length measurements with varying
intensity to achieve increased performance, or to reduce the effort necessary
in terms of intensity, temperature and bias-voltage stabilization of photo
sensors and electronics.
Disclosures The authors declare no conflicts of interest. Data Availability
Statement Data underlying the results presented in this paper are available in
Dataset 1, Ref. [8].

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