# Decoherence of spin superposition state caused by a quantum electromagnetic
field

Kensuke Gallock-Yoshimura<EMAIL_ADDRESS>Department of Physics, Kyushu University, 744 Motooka, Nishi-Ku, Fukuoka
819-0395, Japan  Yuuki Sugiyama<EMAIL_ADDRESS>Institute
for Solid State Physics, the University of Tokyo, Kashiwa, Chiba 277-8581,
Japan  Akira Matsumura<EMAIL_ADDRESS>Department of
Physics, Kyushu University, 744 Motooka, Nishi-Ku, Fukuoka 819-0395, Japan
Kazuhiro Yamamoto<EMAIL_ADDRESS>Department of Physics, Kyushu
University, 744 Motooka, Nishi-Ku, Fukuoka 819-0395, Japan  Research Center
for Advanced Particle Physics, Kyushu University, 744 Motooka, Nishi-ku,
Fukuoka 819-0395, Japan  International Center for Quantum-Field Measurement
Systems for Studies of the Universe and Particles (QUP), KEK, Oho 1-1,
Tsukuba, Ibaraki 305-0801, Japan

###### Abstract

In this study, we investigate the decoherence of a spatially superposed
electrically neutral spin-$\frac{1}{2}$ particle in the presence of a
relativistic quantum electromagnetic field in Minkowski spacetime. We
demonstrate that decoherence due to the spin-magnetic field coupling can be
categorized into two distinct factors: local decoherence, originating from the
two-point correlation functions along each branch of the superposed
trajectories, and nonlocal decoherence, which arises from the correlation
functions between the two superposed trajectories. These effects are linked to
phase damping and amplitude damping. We also show that if the quantum field is
prepared in a thermal state, decoherence monotonically increases with the
field temperature.

##  I Introduction

In recent years, the quantum superposition of the center-of-mass (COM) degree
of freedom of a particle has attracted considerable attention, particularly in
studies exploring the intersection between quantum theory and gravity. Seminal
works of Bose et al. [1] and Marletto and Vedral [2] proposed an experimental
scheme to test the quantumness of gravity, known as the Bose-Marletto-Vedral
(BMV) experiment—also referred to as the quantum gravity induced entanglement
of masses (QGEM) protocol. In this protocol, two spatially superposed masses
experiencing the Newtonian gravitational potential are expected to generate
entanglement if gravity is a quantum entity.

These proposals employ a Stern-Gerlach-type scheme to generate spatial
superposition states. Let $\ket{C}$ be a mass’ COM state. The mass is assumed
to have an embedded spin, which is initialized in a superposition state,
$\ket{\uparrow}+\ket{\downarrow}$ up to a normalization constant. It is then
exposed to an inhomogeneous classical magnetic field, and the mass becomes
spatially superposed as
$\ket{\downarrow,C_{\downarrow}}+\ket{\uparrow,C_{\uparrow}}$, where
$\ket{C_{\downarrow}}$ and $\ket{C_{\uparrow}}$ denote the trajectories
corresponding to the spin state (e.g., left and right trajectories). The BMV
experiment employs two copies of such a spatial superposition state to test
the quantum nature of gravity.

A primary focus of spatial superposition states, in both experimental and
theoretical contexts, is decoherence. In the BMV experiment, decoherence
hinders our ability to detect quantum gravity-induced entanglement. As a
result, developing methods to mitigate these decoherence effects is of
paramount importance. Related studies have been carried out such as the effect
of decoherence on the gravitationally induced entanglement [3, 4, 5, 6, 7] and
methods for overcoming decoherence [8, 9, 10].

From a theoretical perspective, the decoherence of spatially superposed
particles is also crucial. A thought experiment, originally proposed in [11],
identifies an inconsistency between the principles of causality and
complementarity. It suggests that an observer’s decision can affect the
coherence of a spatially superposed mass, even if they are causally
disconnected. This problem was further analyzed and resolved in [12, 13, 14]
by considering the decoherence of spatially superposed particles and vacuum
fluctuations. See [15, 16, 17] for further analyses on the effect of
decoherence on this thought experiment.

These studies on the decoherence of spatially superposed particles typically
focus on the COM degree of freedom. However, to our knowledge, little
investigation has been done on the decoherence of a spatially superposed spin
degree of freedom caused by spin-magnetic field coupling. This is particularly
interesting because the embedded spin in a mass is responsible for generating
spatial superposition, yet it cannot avoid the decoherence from the vacuum
fluctuations of a quantum electromagnetic field even if the mass is
electrically neutral. See [18, 19, 20, 21] for the decoherence of an electron
coupled to a quantum electromagnetic field, and [22] for gravity-induced
decoherence on a spin-$\frac{1}{2}$ system.

In this paper, we analyze the decoherence of a spin’s superposition state
caused by a quantum electromagnetic field. Assuming the spin particle, which
is a nonrelativistic first quantized system, is prepared in a spatial
superposition state, we allow the superposed spin particle to interact with a
relativistic quantum electromagnetic field via spin-magnetic field coupling
for a finite period. For simplicity, we consider a superposed trajectory at
rest (i.e., no acceleration or relative velocity) in $(3+1)$-dimensional
Minkowski spacetime. Nevertheless, our analysis can be readily extended to a
spin particle traveling along arbitrary trajectories and to quantum
electromagnetic fields in an $(n+1)$-dimensional curved spacetime. We carry
out a perturbative analysis to obtain the final density matrix for the spin
particle and compute a decoherence measure. We find that the spatially
superposed spin particle undergoes decoherence due to amplitude damping and
phase damping effects. Each of these effects contributes to the decoherence
measure both locally and nonlocally. The local contribution depends solely on
each branch of the superposed trajectory and is independent of the spatial
splitting configuration. Conversely, the nonlocal contribution depends on the
field correlation functions between the two branches of the superposed
trajectory. Depending on the spatial superposition configuration, the nonlocal
contribution may mitigate the decoherence caused by the local terms. Finally,
we consider the quantum electromagnetic field in a thermal state and show that
the spin particle monotonically decoheres with the field temperature.

The paper is organized as follows. In Sec. II, we review the quantum theory of
electromagnetic fields and the Hamiltonian describing the interaction between
a single spin and the field. Assuming a small coupling constant, we employ a
perturbative analysis. In Sec. III, we derive the final density matrix for the
spin system, which is initially entangled with the COM degree of freedom. We
provide concrete expressions of the elements of the final density matrix when
the field is initially in the Minkowski vacuum and thermal states. We then
introduce the $l_{1}$ norm of coherence as a coherence measure and define the
measure for decoherence in Sec. IV. The decoherence effect due to vacuum
fluctuations and thermal noise is examined. Our conclusion is given in Sec. V.
Appendix A provides detailed calculations for the elements of the final
density matrix, and Appendix B discusses the case of the scalar field.

Unless otherwise stated, we work in units where $\hbar=c=k_{\text{B}}=1$ and
use the mostly-plus signature convention, $(-,+,+,+)$.

##  II Setup

###  II.1 Quantum electromagnetic field

We begin by reviewing the quantized electromagnetic field. Consider a
classical Lagrangian density for the electromagnetic field in
$(3+1)$-dimensional Minkowski spacetime:

$\displaystyle\mathcal{L}$
$\displaystyle=-\dfrac{1}{4}F_{\mu\nu}F^{\mu\nu}\,,$ (1)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field
strength tensor with $A^{\mu}$ being the four-potential. In the radiation
gauge $A^{0}=0$, $\bm{\nabla}\cdot\bm{A}=0$, the equation of motion becomes

$\displaystyle\square\bm{A}(\mathsf{x})=0\,,$ (2)

where $\square$ is the d’Alembert operator and $\mathsf{x}$ is a spacetime
point.

Quantizing the field, the vector potential operator $\hat{\bm{A}}(\mathsf{x})$
can be written as a mode decomposition:

$\displaystyle\hat{\bm{A}}(\mathsf{x})=\int_{\mathbb{R}^{3}}\dfrac{\text{d}^{3}k}{\sqrt{(2\pi)^{3}2|\bm{k}|}}\sum_{\lambda=1}^{2}\left(\bm{e}_{({\bm{k}},\lambda)}e^{-\mathsf{i}|\bm{k}|t+\mathsf{i}{\bm{k}}\cdot{\bm{x}}}\hat{a}_{{\bm{k}},\lambda}+\text{h.c.}\right),$
(3)

where $\bm{e}_{({\bm{k}},\lambda)}$ is the polarization vector with $\lambda$
representing the two kinds of polarizations, and it obeys

$\displaystyle\bm{e}_{({\bm{k}},\lambda)}\cdot\bm{e}_{({\bm{k}},\lambda^{\prime})}^{*}=\delta_{\lambda,\lambda^{\prime}}\,,$
(4a)
$\displaystyle{\bm{k}}\cdot\bm{e}_{({\bm{k}},\lambda)}=0\quad\forall\lambda\,.$
(4b)

The Minkowski vacuum $\ket{0}$ is defined to be

$\displaystyle\hat{a}_{{\bm{k}},\lambda}\ket{0}=0\quad\forall{\bm{k}},\lambda\,,$
(5)

and the commutation relation for the creation and annihilation operators reads

$\displaystyle[\hat{a}_{{\bm{k}},\lambda},\hat{a}_{{\bm{k}}^{\prime},\lambda^{\prime}}^{\dagger}]$
$\displaystyle=\delta^{(3)}({\bm{k}}-{\bm{k}}^{\prime})\delta_{\lambda,\lambda^{\prime}}\,.$
(6)

The quantum magnetic field
$\hat{\bm{B}}(\mathsf{x})\coloneqq\bm{\nabla}\times\hat{\bm{A}}(\mathsf{x})$
reads

$\displaystyle\hat{\bm{B}}(\mathsf{x})$
$\displaystyle=\int_{\mathbb{R}^{3}}\text{d}^{3}k\sum_{\lambda=1}^{2}\left(\bm{B}_{({\bm{k}},\lambda)}(\mathsf{x})\hat{a}_{{\bm{k}},\lambda}+\bm{B}_{({\bm{k}},\lambda)}^{*}(\mathsf{x})\hat{a}_{{\bm{k}},\lambda}^{\dagger}\right)\,,$
(7)

where

$\displaystyle\bm{B}_{({\bm{k}},\lambda)}(\mathsf{x})$
$\displaystyle\equiv\dfrac{\mathsf{i}{\bm{k}}\times\bm{e}_{({\bm{k}},\lambda)}}{\sqrt{(2\pi)^{3}2|\bm{k}|}}e^{-\mathsf{i}|\bm{k}|t+\mathsf{i}{\bm{k}}\cdot{\bm{x}}}$
(8)

is the mode function for the magnetic field. In this paper, we will use the
following projector:

$\displaystyle\sum_{\lambda}\bm{e}_{({\bm{k}},\lambda)}^{(i)}\bm{e}_{({\bm{k}},\lambda)}^{(j)*}=\delta^{ij}-\dfrac{k^{i}k^{j}}{|\bm{k}|^{2}}\,,$
(9)

where $\bm{e}_{({\bm{k}},\lambda)}^{(i)}$ and $k^{i}$ are the components of
the vectors $\bm{e}_{({\bm{k}},\lambda)}$ and ${\bm{k}}$, respectively.

We comment on reparametrization of the time variable of the free Hamiltonian.
So far, we have quantized the electromagnetic field with respect to the
Minkowski time $t$. The free Hamiltonian for the field is therefore a
generator of time-translation with respect to $t$, and we indicate this by
writing

$\displaystyle\hat{H}_{\text{EM,0}}^{t}$
$\displaystyle=\int_{\mathbb{R}^{3}}\text{d}^{3}k\sum_{\lambda=1}^{2}|\bm{k}|\hat{a}_{{\bm{k}},\lambda}^{\dagger}\hat{a}_{{\bm{k}},\lambda}\,.$
(10)

In the following sections, we consider the interaction between the field and a
spin particle with proper time $\tau$, and we will express the full
Hamiltonian in terms of $\tau$. Hence, we will use the field’s free
Hamiltonian as a generator of time-translation with respect to the proper time
$\tau$, $\hat{H}_{\text{EM,0}}^{\tau}$, given by

$\displaystyle\hat{H}_{\text{EM,0}}^{\tau}$
$\displaystyle=\dfrac{\text{d}t(\tau)}{\text{d}\tau}\hat{H}_{\text{EM,0}}^{t}\,.$
(11)

This can be understood as follows [23, 24]. In general, the Schrödinger
equation with respect to $t$ is
$\mathsf{i}\frac{\text{d}}{\text{d}t}\ket{\psi(t)}=\hat{H}^{t}(t)\ket{\psi(t)}$.
By changing the time variable to $\tau$ using $t=t(\tau)$, we obtain the above
relation:

$\displaystyle\mathsf{i}\dfrac{\text{d}}{\text{d}\tau}\ket{\psi(t(\tau))}$
$\displaystyle=\dfrac{\text{d}t(\tau)}{\text{d}\tau}\hat{H}^{t}(t(\tau))\ket{\psi(t(\tau))}\equiv\hat{H}^{\tau}(\tau)\ket{\psi(\tau)}\,.$

###  II.2 Spin-magnetic field system

Consider a spin-$\frac{1}{2}$ particle interacting with a quantum
electromagnetic field. In this paper, we consider three degrees of freedom:
the spin, the COM, and the quantum field. The Hilbert space is given by
$\mathcal{H}_{\text{s}}\otimes\mathcal{H}_{\text{COM}}\otimes\mathcal{H}_{\text{EM}}$,
with ‘s’ standing for the spin system. However, since the COM degree of
freedom is only relevant for spatial superposition and is not involved in the
interaction, we focus only on
$\mathcal{H}_{\text{s}}\otimes\mathcal{H}_{\text{EM}}$ for now.

We first assume that the spin particle is exposed to a classical magnetic
field $\bm{B}({\bm{x}})$, which is directed in the $+z$-axis, i.e.,

$\displaystyle\bm{B}({\bm{x}})$
$\displaystyle=[0,0,B_{0}({\bm{x}})]^{\intercal}\,.$ (12)

Note that the magnetic field is not uniform in general, thereby it could
depend on ${\bm{x}}$. In the Schrödinger picture, the free Hamiltonian for the
spin system $\hat{H}_{\text{s,0}}^{\tau}$ as a generator of the time-
translation with respect to the spin’s proper time $\tau$ is given by

$\displaystyle\hat{H}_{\text{s,0}}^{\tau}$
$\displaystyle=\hat{\bm{\mu}}\cdot\bm{B}({\bm{x}})=\mu_{\text{B}}B_{0}({\bm{x}})\hat{\sigma}_{3}\in\mathcal{B}(\mathcal{H}_{\text{s}})\,,$
(13)

where $\mu_{\text{B}}$ is the Bohr magneton and
$\hat{\bm{\mu}}\equiv\mu_{\text{B}}\hat{\bm{\sigma}}$ is the magnetic moment
operator defined with the vector of the Pauli operators
$\hat{\bm{\sigma}}=[\hat{\sigma}_{1},\hat{\sigma}_{2},\hat{\sigma}_{3}]^{\intercal}$.
Here, $\mathcal{B}(\mathcal{H}_{\text{s}})$ denotes the space of bounded
operators on the spin’s Hilbert space. It is worth noting that the Bohr
magneton $\mu_{\text{B}}$ has the dimension of length in natural units. We
define $\ket{\uparrow}$ and $\ket{\downarrow}$ as the eigenstates of
$\hat{H}_{\text{s,0}}^{\tau}$ (or equivalently, eigenstates of
$\hat{\sigma}_{3}$):

$\displaystyle\hat{H}_{\text{s,0}}^{\tau}\ket{\uparrow}$
$\displaystyle=\mu_{\text{B}}B_{0}({\bm{x}})\ket{\uparrow}\,,\quad\hat{H}_{\text{s,0}}^{\tau}\ket{\downarrow}=-\mu_{\text{B}}B_{0}({\bm{x}})\ket{\downarrow}\,.$
(14)

Let us now consider the interaction between the spin system and a quantized
magnetic field $\hat{\bm{B}}(\mathsf{x})$. In the Schrödinger picture, the
total Hamiltonian $\hat{H}_{\text{S,tot}}$ is given by

$\displaystyle\hat{H}_{\text{S,tot}}^{\tau}$
$\displaystyle=\hat{H}_{\text{s,0}}^{\tau}\otimes\mathds{1}_{\text{EM}}+\mathds{1}_{\text{s}}\otimes\hat{H}_{\text{EM,0}}^{\tau}+\hat{H}_{\text{S,int}}^{\tau}(\tau)\,,$
(15)

where

$\displaystyle\hat{H}_{\text{S,int}}^{\tau}(\tau)$
$\displaystyle=\hat{\bm{\mu}}(\tau)\cdot\hat{\bm{B}}(\mathsf{x})=\mu_{\text{B}}\chi(\tau)\hat{\sigma}_{i}\otimes\hat{B}^{i}(\mathsf{x})\,.$
(16)

Here, $\tau$ is the proper time of the spin particle, $\chi(\tau)$ is the
switching function that governs the time dependence of the interaction, and
the Einstein summation convention is used for $i=1,2,3$. The switching
function $\chi(\tau)$ is typically assumed to be a compactly supported smooth
function, $\chi\in C_{0}^{\infty}(\mathbb{R})$. In the literature, it is also
common that $\chi(\tau)$ is chosen to be a smooth superpolynomial function
such as a Gaussian function.

Let us write the sum of free Hamiltonians as
$\hat{H}_{\text{0}}^{\tau}\equiv\hat{H}_{\text{s,0}}^{\tau}+\hat{H}_{\text{EM,0}}^{\tau}$.
The interaction Hamiltonian $\hat{H}_{\text{I}}^{\tau}$ in the interaction
picture is then [25, 26]

$\displaystyle\hat{H}_{\text{I}}^{\tau}$
$\displaystyle=e^{\mathsf{i}\hat{H}_{\text{0}}^{\tau}\tau}\hat{H}_{\text{S,int}}^{\tau}(\tau)e^{-\mathsf{i}\hat{H}_{\text{0}}^{\tau}\tau}$
$\displaystyle=\mu_{\text{B}}\chi(\tau)\Big{[}e^{\mathsf{i}\Omega({\bm{x}})\tau}\hat{\sigma}_{+}\otimes\hat{B}_{-}(\mathsf{x}(\tau))$
$\displaystyle\quad+e^{-\mathsf{i}\Omega({\bm{x}})\tau}\hat{\sigma}_{-}\otimes\hat{B}_{+}(\mathsf{x}(\tau))+\hat{\sigma}_{3}\otimes\hat{B}_{3}(\mathsf{x}(\tau))\Big{]}\,,$
(17)

where $\Omega({\bm{x}})\coloneqq 2\mu_{\text{B}}B_{0}({\bm{x}})$ is the energy
difference between the two spin states, $\hat{\sigma}_{\pm}$ are the raising
and lowering operators
$\hat{\sigma}_{\pm}\coloneqq(\hat{\sigma}_{1}\pm\mathsf{i}\hat{\sigma}_{2})/2$
(or equivalently, $\hat{\sigma}_{+}=\ket{\uparrow}\bra{\downarrow}$ and
$\hat{\sigma}_{-}=\ket{\downarrow}\bra{\uparrow}$), and
$\hat{B}_{\pm}(\mathsf{x})\coloneqq\hat{B}_{1}(\mathsf{x})\pm\mathsf{i}\hat{B}_{2}(\mathsf{x})$.
The first two terms involving $\hat{\sigma}_{\pm}$ in (17) correspond to
interactions with a spin flip, while the third term does not. Hence, the first
two terms are analogous to the Unruh-DeWitt particle detector model [27, 28],
which is linearly coupled to a quantum scalar field.

The time-evolution operator $\hat{U}_{\text{I}}$ is given by

$\displaystyle\hat{U}_{\text{I}}$
$\displaystyle=\mathcal{T}_{t}\exp\left[-\mathsf{i}\int_{\mathbb{R}}\text{d}t\,\dfrac{\text{d}\tau}{\text{d}t}\hat{H}_{\text{I}}^{\tau}(\tau(t))\right]\,,$
(18)

where $\mathcal{T}_{t}$ is a time-ordering symbol with respect to time $t$. In
what follows, we will assume that the energy difference $\Omega$ is constant
during the interaction for simplicity: $\Omega({\bm{x}})=\Omega$. This energy
$\Omega$ can be thought of as a typical energy scale of our system. To use a
perturbative analysis, we assume that the dimensionless quantity
$\mu_{\text{B}}\Omega$ is much smaller than 1, i.e., $\mu_{\text{B}}\Omega\ll
1$.111 One needs to consider a dimensionless coupling constant to compare it
to unity.  By performing the Dyson series expansion, we obtain

$\displaystyle\hat{U}_{\text{I}}$
$\displaystyle=\mathds{1}+\hat{U}_{\text{I}}^{(1)}+\hat{U}_{\text{I}}^{(2)}+\cdots\,,$
(19)

where

$\displaystyle\hat{U}_{\text{I}}^{(1)}$
$\displaystyle\coloneqq-\mathsf{i}\int_{\mathbb{R}}\text{d}t\,\dfrac{\text{d}\tau}{\text{d}t}\hat{H}_{\text{I}}^{\tau}(\tau(t))\,,$
(20a) $\displaystyle\hat{U}_{\text{I}}^{(2)}$
$\displaystyle\coloneqq-\int_{\mathbb{R}}\text{d}t\int_{\mathbb{R}}\text{d}t^{\prime}\,\Theta(t-t^{\prime})\dfrac{\text{d}\tau}{\text{d}t}\dfrac{\text{d}\tau}{\text{d}t^{\prime}}\hat{H}_{\text{I}}^{\tau}(\tau(t))\hat{H}_{\text{I}}^{\tau}(\tau(t^{\prime}))\,,$
(20b)

with $\Theta(t)$ being the Heaviside step function. Here,
$\hat{U}_{\text{I}}^{(j)}$ indicates that it contains the $j$th power of the
Bohr magneton: $\mu_{\text{B}}^{j}$.

##  III Final density matrix

Using perturbation theory, we derive the final density matrix for the spin
particle. Here, we explicitly include the COM degree of freedom to consider
the spin’s spatial superposition state.

###  III.1 Initial state

Let us assume that in the infinite past, the initial state of the total system
is given by

$\displaystyle\ket{\psi_{\text{s,0}}}\otimes\ket{C}\otimes\ket{\text{EM}}\,,$

where $\ket{C}$ is the COM state, $\ket{\psi_{\text{s,0}}}$ is the spin degree
of freedom’s initial pure state given by

$\displaystyle\ket{\psi_{\text{s,0}}}$
$\displaystyle=\alpha\ket{\uparrow}+e^{\mathsf{i}\vartheta}\sqrt{1-\alpha^{2}}\ket{\downarrow}\,,$
(21)

where $\alpha\in[0,1]$ and $\vartheta\in[0,2\pi)$ is the relative phase, and
$\ket{\text{EM}}$ corresponds to the initial state of the field. We note that
the field state $\ket{\text{EM}}$ is written in this way for the sake of
convenience.

Before the spin particle interacts with the quantum field, we assume the
classical magnetic field $B_{0}({\bm{x}})$ causes the particle’s trajectory to
split based on its spin. To be precise, a Stern-Gerlach-type protocol
entangles the COM state $\ket{C}$ with the spin degree of freedom:

$\displaystyle\ket{\psi_{\text{s,0}}}\otimes\ket{C}\mapsto\alpha\ket{\uparrow,C_{\uparrow}}+e^{\mathsf{i}\vartheta}\sqrt{1-\alpha^{2}}\ket{\downarrow,C_{\downarrow}}\,.$
(22)

Here $\ket{C_{\uparrow}}$ and $\ket{C_{\downarrow}}$ represent the
trajectories corresponding to the up-spin and down-spin, respectively.
Furthermore, for simplicity, we assume that each wavepacket is very narrow so
that $\\{\ket{C_{\uparrow}},\ket{C_{\downarrow}}\\}$ is orthonormal. As a
result, the dimension of the joint Hilbert space of the spin and the COM after
splitting is $2\times 2$. In the basis
$\\{\ket{\uparrow,C_{\uparrow}},\ket{\uparrow,C_{\downarrow}},\ket{\downarrow,C_{\uparrow}},\ket{\downarrow,C_{\downarrow}}\\}$,
the initial joint density matrix for the spin particle is

$\displaystyle\rho_{\text{s,0}}$
$\displaystyle=\begin{bmatrix}\alpha^{2}&0&0&\alpha\sqrt{1-\alpha^{2}}e^{-\mathsf{i}\vartheta}\\\
0&0&0&0\\\ 0&0&0&0\\\
\alpha\sqrt{1-\alpha^{2}}e^{\mathsf{i}\vartheta}&0&0&1-\alpha^{2}\end{bmatrix}\,,$
(23)

and we will denote the total initial density matrix $\rho_{\text{tot,0}}$ by

$\displaystyle\rho_{\text{tot,0}}$
$\displaystyle=\rho_{\text{s,0}}\otimes\rho_{\text{EM,0}}\,,$ (24)

where $\rho_{\text{EM,0}}$ is the field’s initial state.

The spatial dependence of the classical magnetic field vector
$B_{0}({\bm{x}})$ allows for the possibility of manipulating the spin’s
trajectory based on its spin state. By ingeniously engineering this position
dependence of $B_{0}({\bm{x}})$, it becomes feasible to direct the spin’s
trajectory in such a way that the displacement vector, which characterizes the
direction of split between trajectories, can either be aligned parallel to the
$z$-axis or lie perpendicular to it, within the $(x,y)$-plane. This concept is
illustrated in Fig. 1.

Figure 1: Comparing the two cases. (a) The two superposed trajectories split
in the $z$-direction and remain static relative to Minkowski time $t$. The red
and blue lines indicate the duration of interaction. (b) The scenario where
the trajectories split in the $(x,y)$-plane, i.e., their displacement vector
is perpendicular to the $z$-axis.

Finally, let us specify the field’s initial state $\rho_{\text{EM,0}}$. In
this paper, we assume that the initial state of the field,
$\rho_{\text{EM,0}}$, is a quasifree state. This is a special type of state
where any connected $n$-point correlation function can be expressed as
products of the Wightman function (i.e., two-point correlation function). To
be precise, let us define the $n$-point correlation function as

$\displaystyle\braket{\hat{B}_{i}(\mathsf{x}_{1})\hat{B}_{j}(\mathsf{x}_{2})\cdots\hat{B}_{k}(\mathsf{x}_{n})}_{\rho_{\text{EM,0}}}$
$\displaystyle\coloneqq\operatorname{Tr}[\hat{B}_{i}(\mathsf{x}_{1})\hat{B}_{j}(\mathsf{x}_{2})\cdots\hat{B}_{k}(\mathsf{x}_{n})\rho_{\text{EM,0}}]\,,$
(25)

where $\hat{B}_{i}(\mathsf{x})$ is the $i$th component of the quantum magnetic
field $\hat{\bm{B}}(\mathsf{x})$ (7). Then, $\rho_{\text{EM,0}}$ is said to be
a quasifree state if any $n$-point correlation function can be written as
products of the Wightman function
$\braket{\hat{B}_{i}(\mathsf{x})\hat{B}_{j}(\mathsf{x}^{\prime})}_{\rho_{\text{EM,0}}}$.
Examples of quasifree states include, but are not limited to, the Minkowski
vacuum state, the squeezed state, and the thermal state known as the Kubo-
Martin-Schwinger (KMS) state [29, 30].

###  III.2 Final state

Up to this point, the interaction between the spin and the field has not been
turned on. We assume that, after we split the trajectories, the classical
magnetic field is uniform and time-independent so that $\Omega$ can be treated
as a constant throughout the interaction. We then smoothly turn on the
interaction for duration $\sim\sigma$. After the interaction, we perform a
reverse Stern-Gerlach-type procedure to converge the trajectories by admitting
the spacetime dependence of $B_{0}({\bm{x}})$.

Let us employ a perturbative method to obtain the final density matrix of the
spin particle. Starting from the initial state (24) with (23), the entire
system evolves under the unitary operator $\hat{U}_{\text{I}}$ given by (19)
with (20a) and (20b). The final density matrix of the spin particle
$\rho_{\text{s}}$ (i.e., the spin and COM degrees of freedom) after the
interaction can be obtained by tracing out the field degree of freedom from
the final total state as follows.

$\displaystyle\rho_{\text{s}}$
$\displaystyle=\operatorname{Tr}_{\text{EM}}[\hat{U}_{\text{I}}\rho_{\text{tot,0}}\hat{U}_{\text{I}}^{\dagger}]$
$\displaystyle=\rho_{\text{s}}^{(0,0)}+\rho_{\text{s}}^{(1,1)}+\rho_{\text{s}}^{(2,0)}+\rho_{\text{s}}^{(0,2)}+\mathcal{O}(\mu_{\text{B}}^{4})\,,$
(26)

where $\operatorname{Tr}_{\text{EM}}$ is understood as a partial trace of the
field degree of freedom, and

$\displaystyle\rho_{\text{s}}^{(i,j)}$
$\displaystyle\coloneqq\operatorname{Tr}_{\text{EM}}[\hat{U}_{\text{I}}^{(i)}\rho_{\text{tot,0}}\hat{U}_{\text{I}}^{(j)\dagger}]$
(27)

with $\rho_{\text{s}}^{(0,0)}\equiv\rho_{\text{s,0}}$. Note that
$\rho_{\text{s}}$ includes only even powers of $\mu_{\text{B}}$ because the
odd-point correlation functions vanish for quasifree states.

Since we are considering the spatial superposition state, it is necessary to
take into account the spacetime point at which the field observables
$\hat{B}_{i}(\mathsf{x})$ interact. Recall that each
$\hat{U}_{\text{I}}^{(i)}$ contains the interaction Hamiltonians
$\hat{H}_{\text{I}}^{\tau}(\tau)$ (17), each of which takes the form
$\hat{\sigma}_{i}\otimes\hat{B}_{i}(\mathsf{x})$. We impose that the field
observable is defined on $\mathsf{x}_{\uparrow}$ (resp.
$\mathsf{x}_{\downarrow}$) when
$\hat{\sigma}_{i}\otimes\hat{B}_{i}(\mathsf{x})$ acts on the states associated
with $\ket{C_{\uparrow}}$ (resp. $\ket{C_{\downarrow}}$), that is, for any
$s\in\\{\uparrow,\downarrow\\}$,

$\displaystyle[\hat{\sigma}_{i}\otimes\hat{B}_{i}(\mathsf{x})](\ket{s,C_{\uparrow}}\otimes\ket{\text{EM}})\mapsto\hat{\sigma}_{i}\ket{s,C_{\uparrow}}\otimes\hat{B}_{i}(\mathsf{x}_{\uparrow})\ket{\text{EM}}\,,$
$\displaystyle[\hat{\sigma}_{i}\otimes\hat{B}_{i}(\mathsf{x})](\ket{s,C_{\downarrow}}\otimes\ket{\text{EM}})\mapsto\hat{\sigma}_{i}\ket{s,C_{\downarrow}}\otimes\hat{B}_{i}(\mathsf{x}_{\downarrow})\ket{\text{EM}}\,.$

The resulting density matrix $\rho_{\text{s}}$ reads

$\displaystyle\rho_{\text{s}}$
$\displaystyle=\begin{bmatrix}\rho_{11}&0&0&\rho_{14}\\\
0&\rho_{22}&\rho_{23}&0\\\ 0&\rho_{23}^{*}&\rho_{33}&0\\\
\rho_{14}^{*}&0&0&\rho_{44}\end{bmatrix}+\mathcal{O}(\mu_{\text{B}}^{4})\,,$
(28)

where

$\displaystyle\rho_{11}$
$\displaystyle=\rho_{11}^{(0)}[1-P_{\uparrow\to\downarrow}^{C_{\uparrow}}(\Omega)]\,,$
(29a) $\displaystyle\rho_{22}$
$\displaystyle=\rho_{44}^{(0)}P_{\downarrow\to\uparrow}^{C_{\downarrow}}(\Omega)\,,$
(29b) $\displaystyle\rho_{33}$
$\displaystyle=\rho_{11}^{(0)}P_{\uparrow\to\downarrow}^{C_{\uparrow}}(\Omega)\,,$
(29c) $\displaystyle\rho_{44}$
$\displaystyle=\rho_{44}^{(0)}[1-P_{\downarrow\to\uparrow}^{C_{\downarrow}}(\Omega)]\,,$
(29d) $\displaystyle\rho_{14}$
$\displaystyle=\rho_{14}^{(0)}(1-\mathcal{D}_{\text{nl}}-\mathcal{A}_{\text{loc}}-\mathcal{D}_{\text{loc}})\,,$
(29e) $\displaystyle\rho_{23}$
$\displaystyle=\rho_{14}^{(0)*}\mathcal{M}_{\text{nl}}\,,$ (29f)

where $\rho_{11}^{(0)}=\alpha^{2},\rho_{44}^{(0)}=1-\alpha^{2}$, and
$\rho_{14}^{(0)}=\alpha\sqrt{1-\alpha^{2}}e^{-\mathsf{i}\vartheta}$, and

$\displaystyle
P_{\downarrow\to\uparrow}^{C_{\downarrow}}(\Omega)\coloneqq\int_{\mathbb{R}}\text{d}t\int_{\mathbb{R}}\text{d}t^{\prime}\,\braket{\mathscr{B}_{+}(\mathsf{x}_{\downarrow})\mathscr{B}_{-}(\mathsf{x}^{\prime}_{\downarrow})}_{\rho_{\text{EM,0}}}\,,$
(30a) $\displaystyle
P_{\uparrow\to\downarrow}^{C_{\uparrow}}(\Omega)\coloneqq\int_{\mathbb{R}}\text{d}t\int_{\mathbb{R}}\text{d}t^{\prime}\,\braket{\mathscr{B}_{-}(\mathsf{x}_{\uparrow})\mathscr{B}_{+}(\mathsf{x}^{\prime}_{\uparrow})}_{\rho_{\text{EM,0}}}\,,$
(30b)
$\displaystyle\mathcal{M}_{\text{nl}}\coloneqq\int_{\mathbb{R}}\text{d}t\int_{\mathbb{R}}\text{d}t^{\prime}\,\braket{\mathscr{B}_{-}(\mathsf{x}_{\uparrow})\mathscr{B}_{-}(\mathsf{x}^{\prime}_{\downarrow})}_{\rho_{\text{EM,0}}}\,,$
(30c)
$\displaystyle\mathcal{D}_{\text{nl}}\coloneqq\int_{\mathbb{R}}\text{d}t\int_{\mathbb{R}}\text{d}t^{\prime}\,\braket{\mathscr{B}_{3}(\mathsf{x}_{\downarrow})\mathscr{B}_{3}(\mathsf{x}^{\prime}_{\uparrow})}_{\rho_{\text{EM,0}}}\,,$
(30d)
$\displaystyle\mathcal{A}_{\text{loc}}\coloneqq\int_{\mathbb{R}}\text{d}t\int_{\mathbb{R}}\text{d}t^{\prime}\,\Theta(t-t^{\prime})\braket{\mathscr{B}_{-}(\mathsf{x}_{\uparrow})\mathscr{B}_{+}(\mathsf{x}^{\prime}_{\uparrow})}_{\rho_{\text{EM,0}}}$
$\displaystyle\quad+\int_{\mathbb{R}}\text{d}t\int_{\mathbb{R}}\text{d}t^{\prime}\,\Theta(t^{\prime}-t)\braket{\mathscr{B}_{+}(\mathsf{x}_{\downarrow})\mathscr{B}_{-}(\mathsf{x}^{\prime}_{\downarrow})}_{\rho_{\text{EM,0}}}\,,$
(30e)
$\displaystyle\mathcal{D}_{\text{loc}}\coloneqq\int_{\mathbb{R}}\text{d}t\int_{\mathbb{R}}\text{d}t^{\prime}\,\Theta(t-t^{\prime})\braket{\mathscr{B}_{3}(\mathsf{x}_{\uparrow})\mathscr{B}_{3}(\mathsf{x}^{\prime}_{\uparrow})}_{\rho_{\text{EM,0}}}$
$\displaystyle\quad+\int_{\mathbb{R}}\text{d}t\int_{\mathbb{R}}\text{d}t^{\prime}\,\Theta(t^{\prime}-t)\braket{\mathscr{B}_{3}(\mathsf{x}_{\downarrow})\mathscr{B}_{3}(\mathsf{x}^{\prime}_{\downarrow})}_{\rho_{\text{EM,0}}}\,,$
(30f)

and

$\displaystyle\hat{\mathscr{B}}_{\mp}(\mathsf{x}(t))$
$\displaystyle\coloneqq\mu_{\text{B}}\dfrac{\text{d}\tau}{\text{d}t}\chi(\tau(t))e^{\pm\mathsf{i}\Omega\tau(t)}\hat{B}_{\mp}(\mathsf{x}(\tau(t)))\,,$
(31a) $\displaystyle\hat{\mathscr{B}}_{3}(\mathsf{x}(t))$
$\displaystyle\coloneqq\mu_{\text{B}}\dfrac{\text{d}\tau}{\text{d}t}\chi(\tau(t))\hat{B}_{3}(\mathsf{x}(\tau(t)))\,.$
(31b)

We note that the terms with
$\braket{\hat{B}_{3}\hat{B}_{\pm}}_{\rho_{\text{EM,0}}}$ vanish.

Here, $P^{C_{\downarrow}}_{\downarrow\to\uparrow}(\Omega)$ [resp.
$P^{C_{\uparrow}}_{\uparrow\to\downarrow}(\Omega)$] is the transition
probability of the spin degree of freedom from $\ket{\downarrow}$ to
$\ket{\uparrow}$ (resp. from $\ket{\uparrow}$ to $\ket{\downarrow}$). The
superscript $C_{s}$ with $s\in\\{\uparrow,\downarrow\\}$ indicates the path
along which it is evaluated. In other words, the transition probabilities are
obtained by evaluating the two-point correlation functions along each
corresponding trajectory:
$\braket{\hat{B}_{+}(\mathsf{x}_{\downarrow})\hat{B}_{-}(\mathsf{x}^{\prime}_{\downarrow})}_{\rho_{\text{EM,0}}}$
or
$\braket{\hat{B}_{-}(\mathsf{x}_{\uparrow})\hat{B}_{+}(\mathsf{x}^{\prime}_{\uparrow})}_{\rho_{\text{EM,0}}}$.
Henceforth, we will refer to such quantities that depend on a single
trajectory as “local contributions.” Other local contributions are
$\mathcal{A}_{\text{loc}}$ and $\mathcal{D}_{\text{loc}}$. We will see in the
next section that the real parts of these local contributions,
$\text{Re}[\mathcal{A}_{\text{loc}}]$ and
$\text{Re}[\mathcal{D}_{\text{loc}}]$, correspond to decoherence terms. They
can be written as

$\displaystyle\text{Re}[\mathcal{A}_{\text{loc}}]$
$\displaystyle=\dfrac{1}{2}[P_{\uparrow\to\downarrow}^{C_{\uparrow}}(\Omega)+P_{\downarrow\to\uparrow}^{C_{\downarrow}}(\Omega)]\,,$
(32) $\displaystyle\text{Re}[\mathcal{D}_{\text{loc}}]$
$\displaystyle=\dfrac{1}{2}\bigg{(}\int_{\mathbb{R}}\text{d}t\int_{\mathbb{R}}\text{d}t^{\prime}\,\braket{\mathscr{B}_{3}(\mathsf{x}_{\uparrow})\mathscr{B}_{3}(\mathsf{x}^{\prime}_{\uparrow})}$
$\displaystyle\quad+\int_{\mathbb{R}}\text{d}t\int_{\mathbb{R}}\text{d}t^{\prime}\,\braket{\mathscr{B}_{3}(\mathsf{x}_{\downarrow})\mathscr{B}_{3}(\mathsf{x}^{\prime}_{\downarrow})}\bigg{)}\,.$
(33)

We will call them the “local decoherence terms.”

On the other hand, $\mathcal{M}_{\text{nl}}$ and $\mathcal{D}_{\text{nl}}$ are
“nonlocal” since they depend on the two-point correlation functions across two
different trajectories:
$\braket{\hat{B}_{-}(\mathsf{x}_{\uparrow})\hat{B}_{-}(\mathsf{x}_{\downarrow}^{\prime})}_{\rho_{\text{EM,0}}}$
and
$\braket{\hat{B}_{3}(\mathsf{x}_{\downarrow})\hat{B}_{3}(\mathsf{x}_{\uparrow}^{\prime})}_{\rho_{\text{EM,0}}}$.
In particular, $\mathcal{D}_{\text{nl}}$ only depends on the anticommutator
(i.e., the Hadamard distribution)
$\braket{\\{\hat{B}_{3}(\mathsf{x}_{\downarrow}),\hat{B}_{3}(\mathsf{x}_{\uparrow}^{\prime})\\}}_{\rho_{\text{EM,0}}}$.

Let us introduce the types of decoherence: amplitude damping and phase damping
(dephasing). Amplitude damping is a process corresponding to the spontaneous
emission. In our case, this is governed by the first two terms of the form
$e^{\mathsf{i}\Omega\tau}\hat{\sigma}_{+}\hat{B}_{-}+e^{-\mathsf{i}\Omega\tau}\hat{\sigma}_{-}\hat{B}_{+}$
in the interaction Hamiltonian (17). The spin particle exchanges energy with
the field and a spin flip occurs. In this way, the spin and the emitted photon
become entangled, leading to the spin’s decoherence. Amplitude damping, in
general, alters both diagonal and off-diagonal elements in a density matrix,
which is also the case for our density matrix (28). The elements of the
density matrix with $\hat{B}_{\pm}(\mathsf{x})$, such as the transition
probabilities, the nonlocal term $\mathcal{M}_{\text{nl}}$, and the local
decoherence term $\mathcal{A}_{\text{loc}}$ are the consequence of amplitude
damping.

Phase damping, on the other hand, is governed by the term
$\hat{\sigma}_{3}\hat{B}_{3}$ in the interaction Hamiltonian (17). Since
$\hat{\sigma}_{3}$ does not flip the states $\ket{\uparrow}$ and
$\ket{\downarrow}$, phase damping corresponds to the loss of the relative
phase [31] in the spin particle without exchanging energy with the field.
Thus, a density matrix loses the off-diagonal elements only. In our final
density matrix (28), the elements $\mathcal{D}_{\text{nl}}$ and
$\mathcal{D}_{\text{loc}}$ are the results of phase damping.

In the next section, we compute a decoherence measure. As we will see, the
decoherence measure can be written with $\text{Re}[\mathcal{A}_{\text{loc}}]$,
$\text{Re}[\mathcal{D}_{\text{loc}}]$, $\text{Re}[\mathcal{D}_{\text{nl}}]$,
and $|\mathcal{M}_{\text{nl}}|$. Thus, let us first evaluate each of these by
specifying the field’s initial state $\rho_{\text{EM,0}}$ and the spin’s
trajectory $\mathsf{x}(\tau)$.

####  III.2.1 The Minkowski vacuum case

Here, we assume that the field is initially in the Minkowski vacuum $\ket{0}$,
and that the spin particle is at rest with respect to the Minkowski time $t$
so that the trajectory can be written as
$\mathsf{x}_{s}(\tau)=(\tau,{\bm{x}}_{s})$, where ${\bm{x}}_{s}$ with
$s\in\\{\uparrow,\downarrow\\}$ is a fixed spatial vector describing the
spatial position of the spin with the COM state $\ket{C_{s}}$, and $\tau$ is
now understood as $\tau=t$.

We first evaluate the excitation transition probability
$P^{C_{\downarrow}}_{\downarrow\to\uparrow}(\Omega)$. The same transition
probability was examined in [25], but they encountered an ultraviolet (UV)
divergence. We will comment on their results below.

Inserting the mode expansion given in (7), one finds

$\displaystyle\braket{\hat{B}_{+}(\mathsf{x}(\tau))\hat{B}_{-}(\mathsf{x}(\tau^{\prime}))}_{\rho_{\text{EM,0}}}=\braket{0}{\hat{B}_{+}(\mathsf{x}(\tau))\hat{B}_{-}(\mathsf{x}(\tau^{\prime}))}{0}$
$\displaystyle=\int_{\mathbb{R}^{3}}\text{d}^{3}k\sum_{\lambda}B_{({\bm{k}},\lambda)}^{(+)}(\mathsf{x}(\tau))B_{({\bm{k}},\lambda)}^{(+)*}(\mathsf{x}(\tau^{\prime}))\,,$

where $B_{({\bm{k}},\lambda)}^{(+)}(\mathsf{x})\coloneqq
B_{({\bm{k}},\lambda)}^{(1)}(\mathsf{x})+\mathsf{i}B_{({\bm{k}},\lambda)}^{(2)}(\mathsf{x})$,
and so the transition probability reduces to

$\displaystyle P^{C_{\downarrow}}_{\downarrow\to\uparrow}(\Omega)$
$\displaystyle=\mu_{\text{B}}^{2}\int_{\mathbb{R}^{3}}\dfrac{\text{d}^{3}k}{(2\pi)^{3}2|\bm{k}|}(|\bm{k}|^{2}+k_{3}^{2})|\tilde{\chi}(|\bm{k}|+\Omega)|^{2}$
$\displaystyle=\dfrac{\mu_{\text{B}}^{2}}{3\pi^{2}}\int_{0}^{\infty}\text{d}|\bm{k}|\,|\bm{k}|^{3}|\tilde{\chi}(|\bm{k}|+\Omega)|^{2}\,,$
(34)

where we employed the Fourier transformation

$\displaystyle\tilde{\chi}(\omega)$
$\displaystyle\coloneqq\int_{\mathbb{R}}\text{d}\tau\,\chi(\tau)e^{\mathsf{i}\omega\tau}\,,$
(35)

and used the identity

$\displaystyle\sum_{\lambda}({\bm{k}}\times\bm{e}_{\lambda}({\bm{k}}))_{+}({\bm{k}}\times\bm{e}_{\lambda}^{*}({\bm{k}}))_{-}=|\bm{k}|^{2}+k_{3}^{2}\,,$
(36)

where
$({\bm{k}}\times\bm{e}_{({\bm{k}},\lambda)})_{\pm}\equiv({\bm{k}}\times\bm{e}_{({\bm{k}},\lambda)})_{1}\pm\mathsf{i}({\bm{k}}\times\bm{e}_{({\bm{k}},\lambda)})_{2}$.
This identity can be directly proven by using the projector (9). Note that we
have not specified the switching function $\chi(\tau)$ yet in (34).

We first comment on the results obtained by [25]. The authors in [25]
considered a special case of our result (34) when a rectangular switching
function is chosen. In general, the Fourier transform of the rectangular
function with width $2\sigma$ is known to be $2\sin(\omega\sigma)/\omega$. The
integral in (34) takes the form

$\displaystyle\int_{0}^{\infty}\text{d}|\bm{k}|\,\dfrac{|\bm{k}|^{3}\sin^{2}[(|\bm{k}|+\Omega)\sigma]}{(|\bm{k}|+\Omega)^{2}}\,,$
(37)

which is UV divergent, even though the switching function is compactly
supported. This divergence is attributed to the poor choice of a switching
function, which does not converge at large $|\bm{k}|$. One can instead choose
a more appropriate switching function, such as a Gaussian function, which
ensures convergence:

$\displaystyle\chi(\tau)$ $\displaystyle=e^{-\tau^{2}/2\sigma^{2}}\,,$ (38)

where $\sigma$ is the typical time scale. The Fourier transform of such a
Gaussian function reads

$\displaystyle\tilde{\chi}(\omega)$ $\displaystyle=\sqrt{2\pi}\sigma
e^{-\omega^{2}\sigma^{2}/2}\,,$ (39)

which is also a Gaussian, and the integrand in the transition probability (34)
converges at large $|\bm{k}|$. The resulting transition probability reads

$\displaystyle P^{C_{\downarrow}}_{\downarrow\to\uparrow}(\Omega)$
$\displaystyle=\dfrac{\mu_{\text{B}}^{2}}{6\pi\sigma^{2}}\Big{[}2e^{-\Omega^{2}\sigma^{2}}(1+\Omega^{2}\sigma^{2})$
$\displaystyle\quad-\sqrt{\pi}\Omega\sigma(3+2\Omega^{2}\sigma^{2})\text{erfc}(\Omega\sigma)\Big{]}\,,$
(40)

where $\text{erfc}(x)$ is the complementary error function. The deexcitation
transition probability $P^{C_{\uparrow}}_{\uparrow\to\downarrow}(\Omega)$ is
obtained as

$\displaystyle P^{C_{\uparrow}}_{\uparrow\to\downarrow}(\Omega)$
$\displaystyle=\dfrac{\mu_{\text{B}}^{2}}{6\pi\sigma^{2}}\Big{[}2e^{-\Omega^{2}\sigma^{2}}(1+\Omega^{2}\sigma^{2})$
$\displaystyle\quad+\sqrt{\pi}\Omega\sigma(3+2\Omega^{2}\sigma^{2})\text{erfc}(-\Omega\sigma)\Big{]}\,,$
(41)

which is just $P^{C_{\downarrow}}_{\downarrow\to\uparrow}(-\Omega)$. Adding
these transition probabilities yields the (vacuum) local amplitude damping
term $\text{Re}[\mathcal{A}_{\text{loc}}]$ in (32).

In the same manner, by using the identity

$\displaystyle\sum_{\lambda}({\bm{k}}\times\bm{e}_{({\bm{k}},\lambda)})_{3}({\bm{k}}\times\bm{e}_{({\bm{k}},\lambda)}^{*})_{3}=|\bm{k}|^{2}-k_{3}^{2}\,,$
(42)

the local phase damping term $\text{Re}[\mathcal{D}_{\text{loc}}]$ reads

$\displaystyle\text{Re}[\mathcal{D}_{\text{loc}}]$
$\displaystyle=\dfrac{\mu_{\text{B}}^{2}}{6\pi\sigma^{2}}\,.$ (43)

The nonlocal terms $\text{Re}[\mathcal{D}_{\text{nl}}]$ and
$|\mathcal{M}_{\text{nl}}|$ depend on the configuration of the spin’s spatial
splitting. Let ${\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow}$ be a displacement
vector pointing from the point on $\mathsf{x}_{\uparrow}(\tau)$ to the point
on $\mathsf{x}_{\downarrow}(\tau)$ at a fixed time $\tau$. Since we are
assuming static superposed trajectories, the displacement vector is time-
independent. The length of the displacement vector,
$L\equiv|{\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow}|$, represents the spatial
distance between the two trajectories. Then, as we will show below, one can
consider two scenarios: (i) when the displacement vector is parallel to the
$z$-axis [see Fig. 1(a)]; and (ii) when the displacement vector is
perpendicular to the $z$-axis [see Fig. 1(b)]. We provide a detailed
calculation in Appendix A.

For the nonlocal phase damping term $\text{Re}[\mathcal{D}_{\text{nl}}]$, a
straightforward calculation leads to:
(i) When the spin is split along the $z$-axis:
${\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow}=[0,0,L]^{\intercal}$,

$\displaystyle\text{Re}[\mathcal{D}_{\text{nl}}]$
$\displaystyle=\dfrac{\mu_{\text{B}}^{2}\sigma}{2\pi
L^{3}}\left[-\dfrac{L}{\sigma}+\left(\dfrac{L^{2}}{\sigma^{2}}+2\right)D^{+}\left(\dfrac{L}{2\sigma}\right)\right]\,,$
(44)

where $D^{+}(x)\coloneqq\sqrt{\pi}e^{-x^{2}}\text{erfi}(x)/2$ is the Dawson
function with $\text{erfi}(x)$ being the imaginary error function. In
particular, if the spatial splitting is small, it reduces to

$\displaystyle\text{Re}[\mathcal{D}_{\text{nl}}]$
$\displaystyle\simeq\dfrac{\mu_{\text{B}}^{2}}{6\pi\sigma^{2}}\quad({\rm
for}~{}L\ll\sigma)\,.$ (45)

(ii) When the spin is split along the $x$-axis:
${\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow}=[L,0,0]^{\intercal}$,

$\displaystyle\text{Re}[\mathcal{D}_{\text{nl}}]$
$\displaystyle=\dfrac{\mu_{\text{B}}^{2}\sigma}{8\pi
L^{3}}\left[\dfrac{L^{3}}{\sigma^{3}}+2\dfrac{L}{\sigma}-\left(\dfrac{L^{4}}{\sigma^{4}}+4\right)D^{+}\left(\dfrac{L}{2\sigma}\right)\right]\,,$
(46)

which also reduces to

$\displaystyle\text{Re}[\mathcal{D}_{\text{nl}}]$
$\displaystyle\simeq\dfrac{\mu_{\text{B}}^{2}}{6\pi\sigma^{2}}\quad({\rm
for}~{}L\ll\sigma)\,.$ (47)

While $\text{Re}[\mathcal{D}_{\text{nl}}]$ is nonnegative for case (i), it can
be negative for case (ii) in some regions in a parameter space. Nevertheless,
the amount of decoherence coincides with the local phase damping (43) when the
spatial displacement is small.

For the other nonlocal term $|\mathcal{M}_{\text{nl}}|$, we find:
(i) When the spin is split along the $z$-axis:
${\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow}=[0,0,L]^{\intercal}$,

$\displaystyle|\mathcal{M}_{\text{nl}}|$ $\displaystyle=0\,;$ (48)

(ii) When the spin is split along the $x$-axis:
${\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow}=[L,0,0]^{\intercal}$,

$\displaystyle|\mathcal{M}_{\text{nl}}|=\dfrac{-3\mu_{\text{B}}^{2}\sigma
e^{-\Omega^{2}\sigma^{2}}}{4\pi L^{3}}$
$\displaystyle\times\left[\dfrac{L}{\sigma}\left(\dfrac{L^{2}}{6\sigma^{2}}+1\right)-2\left(\dfrac{L^{4}}{12\sigma^{4}}+\dfrac{L^{2}}{3\sigma^{2}}+1\right)D^{+}\left(\dfrac{L}{2\sigma}\right)\right]\,,$
(49)

which reduces to

$\displaystyle|\mathcal{M}_{\text{nl}}|\simeq\dfrac{\mu_{\text{B}}^{2}L^{2}e^{-\Omega^{2}\sigma^{2}}}{30\pi\sigma^{4}}\quad({\rm
for}~{}L\ll\sigma)\,.$ (50)

Let us comment on the rotation symmetry of the system. Although we considered
the perpendicular scenario (ii) with a specific displacement vector
${\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow}=[L,0,0]^{\intercal}$, it is worth
pointing out that the whole composite system is invariant under rotation about
the $z$-axis. Therefore, the result does not change for
${\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow}=[L\cos\theta_{3},L\sin\theta_{3},0]^{\intercal}$
as long as the quantum field is also rotated. To see this, it is
straightforward to show that
$[\hat{H}_{\text{S,tot}},\hat{U}_{\text{s}}(\theta_{3})\otimes\hat{U}_{\text{EM}}(\theta_{3})]$=0,
where $\hat{H}_{\text{S,tot}}$ is the full Hamiltonian in the Schrödinger
picture, and
$\hat{U}_{\text{s}}(\theta_{3})(=e^{-\mathsf{i}\theta_{3}\hat{\sigma}_{3}/2})$
and $\hat{U}_{\text{EM}}(\theta_{3})$ are the unitary representations of
rotation about the $z$-axis for the spin particle and the quantum field,
respectively. This commutation relation can be computed by realizing that
$\hat{U}_{\text{s}}^{\dagger}(\theta_{3})\hat{\bm{\sigma}}\hat{U}_{\text{s}}(\theta_{3})=D(\theta_{3})\hat{\bm{\sigma}}$
as well as
$\hat{U}_{\text{EM}}^{\dagger}(\theta_{3})\hat{\bm{B}}(\mathsf{x})\hat{U}_{\text{EM}}(\theta_{3})=D(\theta_{3})\hat{\bm{B}}(R^{-1}(\theta_{3})\mathsf{x})$,
where

$\displaystyle D(\theta_{3})$
$\displaystyle=\begin{bmatrix}\cos\theta_{3}&-\sin\theta_{3}&0\\\
\sin\theta_{3}&\cos\theta_{3}&0\\\ 0&0&1\\\ \end{bmatrix}$ (51)

is the rotation matrix and $R(\theta_{3})$ is the rotation matrix for a
spacetime point. These relations allow us to show that the
$\hat{\bm{\sigma}}\cdot\hat{\bm{B}}(\mathsf{x})$ term in the interaction
Hamiltonian is invariant under the joint unitary transformation
$\hat{U}_{\text{s}}(\theta_{3})\otimes\hat{U}_{\text{EM}}(\theta_{3})$. On the
other hand, if one rotates only the spin particle, i.e.,
$\hat{U}_{\text{s}}(\theta_{3})\otimes\mathds{1}_{\text{EM}}$, the entire
system is not invariant. This leads to $\theta_{3}$-dependent results for
${\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow}=[L\cos\theta_{3},L\sin\theta_{3},0]^{\intercal}$.

####  III.2.2 Thermal state

We are also interested in the case where the quantum field is in a thermal
state. In quantum field theory, the thermal state is identified as the Kubo-
Martin-Schwinger (KMS) state [29, 30]. Technically, the Gibbs thermal state at
inverse temperature $\beta$,

$\displaystyle\rho_{\beta}$
$\displaystyle=\dfrac{1}{Z}e^{-\beta\hat{H}_{\text{EM,0}}^{t}}\,,$ (52)

where $Z\coloneqq\operatorname{Tr}[e^{-\beta\hat{H}_{\text{EM,0}}^{t}}]$ is
the partition function, is not well-defined for quantum fields in free space.
This stems from the fact that the field’s Hilbert space has uncountably many
dimensions. Nevertheless, as discussed in Ref. [32], we can consider a large
finite-sized box to make the Hilbert space countable and then take the
infinite-size limit. This allows us to practically compute the correlation
functions using the Gibbs state (52), and the final result will agree with the
rigorous definition of the KMS thermal state.

It is straightforward to see that one-point correlation functions vanish for
the thermal state, and using Eqs. (52) and (10), we obtain [32]

$\displaystyle\operatorname{Tr}[\rho_{\beta}\hat{a}_{{\bm{k}},\lambda}\hat{a}_{{\bm{k}}^{\prime},\lambda^{\prime}}^{\dagger}]=\dfrac{e^{\beta|\bm{k}|}}{e^{\beta|\bm{k}|}-1}\delta^{(3)}({\bm{k}}-{\bm{k}}^{\prime})\delta_{\lambda,\lambda^{\prime}}\,,$
(53a)
$\displaystyle\operatorname{Tr}[\rho_{\beta}\hat{a}_{{\bm{k}},\lambda}^{\dagger}\hat{a}_{{\bm{k}}^{\prime},\lambda^{\prime}}]=\dfrac{1}{e^{\beta|\bm{k}|}-1}\delta^{(3)}({\bm{k}}-{\bm{k}}^{\prime})\delta_{\lambda,\lambda^{\prime}}\,,$
(53b)
$\displaystyle\operatorname{Tr}[\rho_{\beta}\hat{a}_{{\bm{k}},\lambda}\hat{a}_{{\bm{k}}^{\prime},\lambda^{\prime}}]=\operatorname{Tr}[\rho_{\beta}\hat{a}_{{\bm{k}},\lambda}^{\dagger}\hat{a}_{{\bm{k}}^{\prime},\lambda^{\prime}}^{\dagger}]=0\,.$
(53c)

From these properties, any thermal two-point correlation functions
$\braket{\hat{B}_{i}\hat{B}_{j}}_{\text{th}}$ can be written as

$\displaystyle\braket{\hat{B}_{i}\hat{B}_{j}}_{\text{th}}$
$\displaystyle\equiv\braket{\hat{B}_{i}\hat{B}_{j}}_{\rho_{\beta}}\coloneqq\operatorname{Tr}[\rho_{\beta}\hat{B}_{i}\hat{B}_{j}]$
$\displaystyle=\braket{\hat{B}_{i}\hat{B}_{j}}_{\text{vac}}+\braket{\hat{B}_{i}\hat{B}_{j}}_{\beta}\,,$
(54)

where
$\braket{\hat{B}_{i}\hat{B}_{j}}_{\text{vac}}\coloneqq\braket{0}{\hat{B}_{i}\hat{B}_{j}}{0}$
is the vacuum two-point correlation function and

$\displaystyle\braket{\hat{B}_{i}\hat{B}_{j}}_{\beta}$
$\displaystyle\coloneqq\int_{\mathbb{R}^{3}}\text{d}^{3}k\sum_{\lambda}\dfrac{B^{(i)}_{({\bm{k}},\lambda)}B^{(j)*}_{({\bm{k}},\lambda)}+\text{c.c.}}{e^{\beta|\bm{k}|}-1}$
(55)

is the contribution coming from the thermal state. Therefore, any quantities
that we have obtained so far, such as transition probabilities, can be
decomposed into the vacuum and thermal contributions:

$\displaystyle
P_{\downarrow\to\uparrow}^{(\text{th})}=P_{\downarrow\to\uparrow}^{(\text{vac})}+P_{\downarrow\to\uparrow}^{(\beta)}\,.$
(56)

Here, $P_{\downarrow\to\uparrow}^{(\text{vac})}$ and
$P_{\downarrow\to\uparrow}^{(\beta)}$ come from
$\braket{\hat{B}_{+}\hat{B}_{-}}_{\text{vac}}$ and
$\braket{\hat{B}_{+}\hat{B}_{-}}_{\beta}$ in (54), respectively. The
excitation probability in the vacuum,
$P_{\downarrow\to\uparrow}^{(\text{vac})}$, is given by (40) and

$\displaystyle P_{\downarrow\to\uparrow}^{(\beta)}$
$\displaystyle=P_{\uparrow\to\downarrow}^{(\beta)}=\dfrac{\mu_{\text{B}}^{2}}{2(2\pi)^{3}}\int_{\mathbb{R}^{3}}\dfrac{\text{d}^{3}k}{|\bm{k}|(e^{\beta|\bm{k}|}-1)}(|\bm{k}|^{2}+k_{3}^{2})$
$\displaystyle\quad\times\left(|\tilde{\chi}(\Omega+|\bm{k}|)|^{2}+|\tilde{\chi}(\Omega-|\bm{k}|)|^{2}\right)$
(57)
$\displaystyle=\dfrac{\mu_{\text{B}}^{2}}{3\pi^{2}}\int_{0}^{\infty}\text{d}|\bm{k}|\dfrac{|\bm{k}|^{3}}{e^{\beta|\bm{k}|}-1}$
$\displaystyle\quad\times\left(|\tilde{\chi}(\Omega+|\bm{k}|)|^{2}+|\tilde{\chi}(\Omega-|\bm{k}|)|^{2}\right)\,.$

Employing the Gaussian switching function (38),

$\displaystyle
P_{\downarrow\to\uparrow}^{(\beta)}=P_{\uparrow\to\downarrow}^{(\beta)}$
$\displaystyle=\dfrac{4\mu_{\text{B}}^{2}\sigma^{2}e^{-\Omega^{2}\sigma^{2}}}{3\pi}\int_{0}^{\infty}\text{d}|\bm{k}|\dfrac{|\bm{k}|^{3}e^{-|\bm{k}|^{2}\sigma^{2}}}{e^{\beta|\bm{k}|}-1}\cosh(2\Omega|\bm{k}|\sigma^{2})\,.$
(58)

Therefore, the local decoherence due to amplitude damping
$\text{Re}[\mathcal{A}_{\text{loc}}^{(\text{th})}]$ in (32) becomes

$\displaystyle\text{Re}[\mathcal{A}_{\text{loc}}^{(\text{th})}]$
$\displaystyle=\text{Re}[\mathcal{A}_{\text{loc}}^{(\text{vac})}]+\text{Re}[\mathcal{A}_{\text{loc}}^{(\beta)}]\,,$
(59)

where

$\displaystyle\text{Re}[\mathcal{A}_{\text{loc}}^{(\beta)}]=(P_{\downarrow\to\uparrow}^{(\beta)}+P_{\uparrow\to\downarrow}^{(\beta)})/2$
$\displaystyle=\dfrac{4\mu_{\text{B}}^{2}\sigma^{2}e^{-\Omega^{2}\sigma^{2}}}{3\pi}\int_{0}^{\infty}\text{d}|\bm{k}|\dfrac{|\bm{k}|^{3}e^{-|\bm{k}|^{2}\sigma^{2}}}{e^{\beta|\bm{k}|}-1}\cosh(2\Omega|\bm{k}|\sigma^{2})\,.$
(60)

Similarly, the local decoherence due to phase damping
$\text{Re}[\mathcal{D}_{\text{loc}}^{(\text{th})}]$ in (33) reads

$\displaystyle\text{Re}[\mathcal{D}_{\text{loc}}^{(\text{th})}]$
$\displaystyle=\text{Re}[\mathcal{D}_{\text{loc}}^{(\text{vac})}]+\text{Re}[\mathcal{D}_{\text{loc}}^{(\beta)}]\,,$
(61)

where $\text{Re}[\mathcal{D}_{\text{loc}}^{(\text{vac})}]$ is given in (43)
and

$\displaystyle\text{Re}[\mathcal{D}_{\text{loc}}^{(\beta)}]$
$\displaystyle=\dfrac{\mu_{\text{B}}^{2}}{(2\pi)^{3}}\int_{\mathbb{R}^{3}}\text{d}^{3}k\dfrac{|\bm{k}|^{2}-k_{3}^{2}}{|\bm{k}|(e^{\beta|\bm{k}|}-1)}|\tilde{\chi}(|\bm{k}|)|^{2}$
(62)
$\displaystyle=\dfrac{2\mu_{\text{B}}^{2}\sigma^{2}}{3\pi}\int_{0}^{\infty}\text{d}|\bm{k}|\dfrac{|\bm{k}|^{3}e^{-|\bm{k}|^{2}\sigma^{2}}}{e^{\beta|\bm{k}|}-1}\,.$
(63)

The nonlocal contributions,

$\displaystyle\text{Re}[\mathcal{D}_{\text{nl}}^{(\text{th})}]$
$\displaystyle=\text{Re}[\mathcal{D}_{\text{nl}}^{(\text{vac})}]+\text{Re}[\mathcal{D}_{\text{nl}}^{(\beta)}]\,,$
(64) $\displaystyle\mathcal{M}_{\text{nl}}^{(\text{th})}$
$\displaystyle=\mathcal{M}_{\text{nl}}^{(\text{vac})}+\mathcal{M}_{\text{nl}}^{(\beta)}\,,$
(65)

where

$\displaystyle\text{Re}[\mathcal{D}_{\text{nl}}^{(\beta)}]$
$\displaystyle=\dfrac{\mu_{\text{B}}^{2}}{(2\pi)^{3}}\int_{\mathbb{R}^{3}}\text{d}^{3}k\dfrac{|\bm{k}|^{2}-k_{3}^{2}}{|\bm{k}|(e^{\beta|\bm{k}|}-1)}|\tilde{\chi}(|\bm{k}|)|^{2}$
$\displaystyle\quad\times\cos[{\bm{k}}\cdot({\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow})]\,,$
(66) $\displaystyle\mathcal{M}_{\text{nl}}^{(\beta)}$
$\displaystyle=\dfrac{-\mu_{\text{B}}^{2}}{(2\pi)^{3}}\int_{\mathbb{R}^{3}}\text{d}^{3}k\dfrac{\tilde{\chi}(\Omega-|\bm{k}|)\tilde{\chi}(\Omega+|\bm{k}|)}{|\bm{k}|(e^{\beta|\bm{k}|}-1)}$
$\displaystyle\quad\times\text{Re}[k_{-}^{2}e^{-\mathsf{i}{\bm{k}}\cdot({\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow})}]\,,$
(67)

depend on the splitting configuration:
(i) When the spin is split along the $z$-axis:
${\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow}=[0,0,L]^{\intercal}$,

$\displaystyle\text{Re}[\mathcal{D}_{\text{nl}}^{(\beta)}]$
$\displaystyle=\dfrac{2\mu_{\text{B}}^{2}\sigma^{2}}{\pi
L^{3}}\int_{0}^{\infty}\text{d}|\bm{k}|\dfrac{e^{-|\bm{k}|^{2}\sigma^{2}}}{e^{\beta|\bm{k}|}-1}$
$\displaystyle\quad\times[-|\bm{k}|L\cos(|\bm{k}|L)+\sin(|\bm{k}|L)]\,,$ (68)
$\displaystyle\mathcal{M}_{\text{nl}}^{(\beta)}$ $\displaystyle=0\,.$ (69)

This means $\mathcal{M}_{\text{nl}}^{(\text{th})}=0$ because both the vacuum
and thermal contributions are zero.
(ii) When the spin is split along the $x$-axis:
${\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow}=[L,0,0]^{\intercal}$,

$\displaystyle\text{Re}[\mathcal{D}_{\text{nl}}^{(\beta)}]$
$\displaystyle=\dfrac{\mu_{\text{B}}^{2}\sigma^{2}}{\pi
L^{3}}\int_{0}^{\infty}\text{d}|\bm{k}|\dfrac{e^{-|\bm{k}|^{2}\sigma^{2}}}{e^{\beta|\bm{k}|}-1}$
$\displaystyle\quad\times[|\bm{k}|L\cos(|\bm{k}|L)+(|\bm{k}|^{2}L^{2}-1)\sin(|\bm{k}|L)]\,,$
(70) $\displaystyle\mathcal{M}_{\text{nl}}^{(\beta)}$
$\displaystyle=\dfrac{-\mu_{\text{B}}^{2}\sigma^{2}e^{-\Omega^{2}\sigma^{2}}}{\pi
L^{3}}\int_{0}^{\infty}\text{d}|\bm{k}|\dfrac{e^{-|\bm{k}|^{2}\sigma^{2}}}{e^{\beta|\bm{k}|}-1}$
$\displaystyle\quad\times\Big{[}3|\bm{k}|L\cos(|\bm{k}|L)+(|\bm{k}|^{2}L^{2}-3)\sin(|\bm{k}|L)\Big{]}\,.$
(71)

Figure 2: Decoherence $\mathscr{D}/(\mu_{\text{B}}^{2}\sigma^{-2})$ as a
function of the splitting separation $L$ in units of typical interaction scale
$\sigma$. The solid and dashed lines correspond to the splitting in the $z$\-
and $x$-directions, respectively. (a) When the field is initially prepared in
the Minkowski vacuum; and (b) when the field is in the thermal state at
temperature $T\sigma=1$.

##  IV Decoherence of spin

###  IV.1 (De)coherence measure

We employ the coherence measure known as the $l_{1}$ norm of coherence
$\mathcal{C}_{l_{1}}$ [33, 34] defined as

$\displaystyle\mathcal{C}_{l_{1}}$ $\displaystyle\coloneqq\sum_{i\neq
j}|\rho_{ij}|\,.$ (72)

We are particularly interested in a $4\times 4$ density matrix given in (28).
The corresponding $l_{1}$ norm of coherence is

$\displaystyle\mathcal{C}_{l_{1}}$
$\displaystyle=2(|\rho_{14}|+|\rho_{23}|)\,.$

Based on (29), we have

$\displaystyle\mathcal{C}_{l_{1}}$
$\displaystyle=2|\rho_{14}^{(0)}|\Big{(}1-\text{Re}[\mathcal{D}_{\text{nl}}+\mathcal{D}_{\text{loc}}+\mathcal{A}_{\text{loc}}]+|\mathcal{M}_{\text{nl}}|\Big{)}$
$\displaystyle\quad+\mathcal{O}(\mu_{\text{B}}^{4})\,,$ (73)

where $\text{Re}[\mathcal{A}_{\text{loc}}]$ is identical to the “average” of
transition probabilities: Eq. (32). From this expression, we define the
measure for decoherence $\mathscr{D}$ as

$\displaystyle\mathscr{D}$
$\displaystyle\coloneqq\text{Re}[\mathcal{D}_{\text{nl}}+\mathcal{D}_{\text{loc}}+\mathcal{A}_{\text{loc}}]-|\mathcal{M}_{\text{nl}}|~{}(\geq
0)\,,$ (74)

which is independent of the initial amount of coherence. The explicit form for
each term in (74) is given in the previous section.

The local decoherences from amplitude damping
$\text{Re}[\mathcal{A}_{\text{loc}}]$ and phase damping
$\text{Re}[\mathcal{D}_{\text{loc}}]$ in $\mathscr{D}$ are always present
regardless of the spin’s superposition configuration. However, the nonlocal
terms, $\text{Re}[\mathcal{D}_{\text{nl}}]$ and $|\mathcal{M}_{\text{nl}}|$,
depend on the superposition configuration because they rely on the two-point
correlation functions evaluated along the two trajectories,
$\mathsf{x}_{\uparrow}(\tau)$ and $\mathsf{x}_{\downarrow}(\tau)$. The
nonlocal term $|\mathcal{M}_{\text{nl}}|$ from amplitude damping is always
nonnegative, and thus “mitigates” decoherence. In contrast, the nonlocal term
$\text{Re}[\mathcal{D}_{\text{nl}}]$ from phase damping can be either positive
or negative. Therefore, it can either worsen or reduce decoherence depending
on the spatial superposition configuration and the chosen parameters.

In the following sections, we compute decoherence $\mathscr{D}$ divided by the
dimensionless coupling constant $(\mu_{\text{B}}/\sigma)^{2}$.

###  IV.2 Splitting dependence

In Fig. 2, we show the decoherence
$\mathscr{D}/(\mu_{\text{B}}^{2}\sigma^{-2})$ as a function of the logarithm
of the splitting separation $L$ in units of the typical interaction time scale
$\sigma$. The graph includes both the Minkowski vacuum case (a) and the
thermal state case (at temperature $T\sigma=1$) (b), each depicting
$z$-splitting (solid lines) and $x$-splitting (dashed lines). From these
figures, we can see that decoherence worsens as the energy gap increases due
to amplitude damping, and higher temperatures cause more decoherence as
expected.

Recall that the local decoherence terms are all independent of $L$. In Fig. 2,
the values of the local decoherence correspond to those at large $L/\sigma$
because the nonlocal contributions vanish in the limit $L/\sigma\to\infty$. In
contrast, decoherence $\mathscr{D}$ with smaller splitting separation $L$
includes both local and nonlocal contributions. Note that
$\mathcal{M}_{\text{nl}}\to 0$ as $L\to 0$, thus the nonlocal phase damping
$\mathcal{D}_{\text{nl}}$ accounts for the increase in decoherence near
$L\approx 0$. The nonlocal amplitude damping $\mathcal{M}_{\text{nl}}$
mitigates decoherence, which is evident in the dashed lines around
$\log_{10}L/\sigma\approx 0.5$. Note that the nonlocal terms depend on the
splitting configuration because of the electromagnetic field’s polarization.
We compare this to the scalar field case in Appendix B.

###  IV.3 Temperature dependence

Figure 3: Decoherence $\mathscr{D}/(\mu_{\text{B}}^{2}\sigma^{-2})$ as a
function of the field temperature $T\sigma$. The solid lines indicate
splitting in the $z$-direction, and the dashed lines indicate splitting in the
$x$-direction. Here, we have chosen $L/\sigma=1$.

Here, we examine the temperature dependence of decoherence $\mathscr{D}$. In
Fig. 3, we show decoherence $\mathscr{D}$ as a function of the field
temperature $T$ in units of $\sigma$. We observe that decoherence increases
monotonically with $T$ for all energy gaps $\Omega$.

Let us mathematically show that, for an arbitrary switching function
$\chi(\tau)$, decoherence increases monotonically with temperature
$\beta^{-1}$, based on the analysis in [35]. Consider two inverse temperatures
$\beta_{\text{H}}$ and $\beta_{\text{C}}$ with
$\beta_{\text{H}}^{-1}>\beta_{\text{C}}^{-1}$. Here, the subscripts ‘H’ and
‘C’ refer to hot and cold temperatures, respectively. We explicitly indicate
the $\beta$-dependence as
$\mathcal{D}_{\text{nl}}^{(\beta)}(\beta_{\text{H}})$. For convenience, we
first show that the phase damping and amplitude damping terms satisfy

$\displaystyle\text{Re}[\mathcal{D}_{\text{loc}}^{(\beta)}(\beta_{\text{H}})]+\mathcal{D}_{\text{nl}}^{(\beta)}(\beta_{\text{H}})\geq\text{Re}[\mathcal{D}_{\text{loc}}^{(\beta)}(\beta_{\text{C}})]+\mathcal{D}_{\text{nl}}^{(\beta)}(\beta_{\text{C}})\,,$
(75)
$\displaystyle\text{Re}[\mathcal{A}_{\text{loc}}^{(\beta)}(\beta_{\text{H}})]-|\mathcal{M}_{\text{nl}}^{(\beta)}(\beta_{\text{H}})|\geq\text{Re}[\mathcal{A}_{\text{loc}}^{(\beta)}(\beta_{\text{C}})]-|\mathcal{M}_{\text{nl}}^{(\beta)}(\beta_{\text{C}})|\,,$
(76)

which lead to our conclusion

$\displaystyle\mathscr{D}^{\text{(th)}}(\beta_{\text{H}})\geq\mathscr{D}^{\text{(th)}}(\beta_{\text{C}})\,,$
(77)

where $\mathscr{D}^{\text{(th)}}$ is understood as

$\displaystyle\mathscr{D}^{\text{(th)}}(\beta)$
$\displaystyle=\mathscr{D}^{\text{(vac)}}+\mathscr{D}^{(\beta)}(\beta)\,.$
(78)

First, let us focus on the phase damping terms and prove Eq. (75). From (62),
it is straightforward to show that

$\displaystyle\text{Re}[\mathcal{D}_{\text{loc}}^{(\beta)}(\beta_{\text{H}})]$
$\displaystyle=\text{Re}[\mathcal{D}_{\text{loc}}^{(\beta)}(\beta_{\text{C}})]+\dfrac{\mu_{\text{B}}^{2}}{(2\pi)^{3}}\int_{\mathbb{R}^{3}}\text{d}^{3}k\dfrac{|\bm{k}|^{2}-k_{3}^{2}}{|\bm{k}|}|\tilde{\chi}(|\bm{k}|)|^{2}h(|\bm{k}|)\,,$
(79)

where

$\displaystyle h(|\bm{k}|)$
$\displaystyle\coloneqq\dfrac{1}{e^{\beta_{\text{H}}|\bm{k}|}-1}-\dfrac{1}{e^{\beta_{\text{C}}|\bm{k}|}-1}\,(\geq
0)\,.$ (80)

Likewise, the nonlocal phase damping term
$\mathcal{D}_{\text{nl}}^{(\beta)}(\beta)$ in (66) results in

$\displaystyle\mathcal{D}_{\text{nl}}^{(\beta)}(\beta_{\text{H}})$
$\displaystyle=\mathcal{D}_{\text{nl}}^{(\beta)}(\beta_{\text{C}})+\dfrac{\mu_{\text{B}}^{2}}{(2\pi)^{3}}\int_{\mathbb{R}^{3}}\text{d}^{3}k\dfrac{|\bm{k}|^{2}-k_{3}^{2}}{|\bm{k}|}$
$\displaystyle\quad\times|\tilde{\chi}(|\bm{k}|)|^{2}h(|\bm{k}|)\cos[{\bm{k}}\cdot({\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow})]$
$\displaystyle\geq\mathcal{D}_{\text{nl}}^{(\beta)}(\beta_{\text{C}})$
$\displaystyle\quad-\dfrac{\mu_{\text{B}}^{2}}{(2\pi)^{3}}\int_{\mathbb{R}^{3}}\text{d}^{3}k\dfrac{|\bm{k}|^{2}-k_{3}^{2}}{|\bm{k}|}|\tilde{\chi}(|\bm{k}|)|^{2}h(|\bm{k}|)\,,$
(81)

and therefore, adding the two results yields Eq. (75).

We can also obtain Eq. (76) through a similar calculation. Since
$k_{-}^{2}=|\bm{k}|^{2}\sin^{2}\theta e^{-\mathsf{i}2\varphi}$ in spherical
coordinates $(|\bm{k}|,\theta,\varphi)$, Eq. (67) becomes

$\displaystyle|\mathcal{M}_{\text{nl}}^{(\beta)}(\beta_{\text{H}})|\leq|\mathcal{M}_{\text{nl}}^{(\beta)}(\beta_{\text{C}})|$
$\displaystyle\quad+\dfrac{\mu_{\text{B}}^{2}}{(2\pi)^{3}}\int_{\mathbb{R}^{3}}\text{d}^{3}k\,|\bm{k}|h(|\bm{k}|)|\tilde{\chi}(\Omega-|\bm{k}|)\tilde{\chi}(\Omega+|\bm{k}|)|\,.$
(82)

We also recall that
$P_{\downarrow\to\uparrow}^{(\beta)}=P_{\uparrow\to\downarrow}^{(\beta)}$, and
Eq. (57) gives us

$\displaystyle\text{Re}[\mathcal{A}_{\text{loc}}^{(\beta)}(\beta)]$
$\displaystyle=\dfrac{\mu_{\text{B}}^{2}}{2(2\pi)^{3}}\int_{\mathbb{R}^{3}}\dfrac{\text{d}^{3}k}{|\bm{k}|(e^{\beta|\bm{k}|}-1)}(|\bm{k}|^{2}+k_{3}^{2})$
$\displaystyle\quad\times\left(|\tilde{\chi}(\Omega+|\bm{k}|)|^{2}+|\tilde{\chi}(\Omega-|\bm{k}|)|^{2}\right)\,.$
(83)

Since $|\bm{k}|^{2}+k_{3}^{2}=|\bm{k}|^{2}(1+\cos^{2}\theta)$ in spherical
coordinates, we obtain

$\displaystyle\text{Re}[\mathcal{A}_{\text{loc}}^{(\beta)}(\beta_{\text{H}})]$
$\displaystyle\geq\text{Re}[\mathcal{A}_{\text{loc}}^{(\beta)}(\beta_{\text{C}})]+\dfrac{\mu_{\text{B}}^{2}}{2(2\pi)^{3}}\int_{\mathbb{R}^{3}}\text{d}^{3}k\,|\bm{k}|h(|\bm{k}|)$
$\displaystyle\quad\times\left(|\tilde{\chi}(\Omega+|\bm{k}|)|^{2}+|\tilde{\chi}(\Omega-|\bm{k}|)|^{2}\right)\,.$
(84)

Therefore,

$\displaystyle\text{Re}[\mathcal{A}_{\text{loc}}^{(\beta)}(\beta_{\text{H}})]-|\mathcal{M}_{\text{nl}}^{(\beta)}(\beta_{\text{H}})|\geq\text{Re}[\mathcal{A}_{\text{loc}}^{(\beta)}(\beta_{\text{C}})]-|\mathcal{M}_{\text{nl}}^{(\beta)}(\beta_{\text{C}})|$
$\displaystyle+\dfrac{\mu_{\text{B}}^{2}}{2(2\pi)^{3}}\int_{\mathbb{R}^{3}}\text{d}^{3}k\,|\bm{k}|h(|\bm{k}|)\left(|\tilde{\chi}(\Omega-|\bm{k}|)|-|\tilde{\chi}(\Omega+|\bm{k}|)|\right)^{2}$
$\displaystyle\geq\text{Re}[\mathcal{A}_{\text{loc}}^{(\beta)}(\beta_{\text{C}})]-|\mathcal{M}_{\text{nl}}^{(\beta)}(\beta_{\text{C}})|\,,$
(85)

which is Eq. (76). Hence, we have shown that the spatially superposed spin
particle experiences monotonic decoherence with increasing temperature as
described in Eq. (77).

###  IV.4 Order estimation

In this section, we bring back the fundamental constants $c,\hbar$, and
$k_{\text{B}}$ and estimate the decoherence in a table-top experiment. First,
let us discuss the interaction time scale. While our Gaussian switching
function is not exactly a compactly supported function, we use $10\sigma$ as
the effective total interaction time. This is because the range
$\tau\in[-5\sigma,5\sigma]$ is sufficient to provide accurate results, as
shown by the normalized Gaussian integral:

$\displaystyle\int_{-5\sigma}^{5\sigma}\text{d}\tau\,\dfrac{1}{\sqrt{2\pi}\sigma}e^{-\tau^{2}/2\sigma^{2}}=0.999999\,.$

Thus, if the interaction time is 1 s then $10\sigma=1\,\text{s}$.

####  IV.4.1 Vacuum fluctuations

Let us estimate the amount of decoherence caused by vacuum fluctuations. We
can estimate the local decoherence terms
$\text{Re}[\mathcal{A}_{\text{loc}}^{(\text{vac})}]$ and
$\text{Re}[\mathcal{D}_{\text{loc}}^{(\text{vac})}]$ by expressing
$\mu_{\text{B}}/\sigma$ and $\Omega\sigma$ in SI units [see Eqs. (40), (41),
and (43)]. Given that $\mu_{\text{B}}=e/(2m_{e})$ in natural units, where $e$
and $m_{e}$ are the electron’s charge and mass, we can use the fine-structure
constant $\alpha_{\text{f}}=e/(4\pi)$ to convert $\mu_{\text{B}}/\sigma$ to SI
units as follows.

$\displaystyle\dfrac{\mu_{\text{B}}}{\sigma}$
$\displaystyle=\dfrac{\sqrt{4\pi\alpha_{\text{f}}}\hbar}{2m_{e}c^{2}}\dfrac{1}{\sigma}\simeq
1.2\times 10^{-20}\left(\dfrac{1\,\text{s}}{10\sigma}\right)\,.$ (86)

From this value, the local phase damping decoherence term in (43) can be
estimated as follows.

$\displaystyle\text{Re}[\mathcal{D}_{\text{loc}}^{(\text{vac})}]$
$\displaystyle\simeq 8.0\times
10^{-42}\left(\dfrac{1\,\text{s}}{10\sigma}\right)^{2}\,.$ (87)

We recall that the energy gap in natural units is
$\Omega=2\mu_{\text{B}}B_{0}$, thereby

$\displaystyle\Omega\sigma$
$\displaystyle=\dfrac{2\mu_{\text{B}}B_{0}}{\hbar}\sigma\simeq 2.8\times
10^{9}\left(\dfrac{B_{0}}{1\,\text{T}}\right)\left(\dfrac{10\sigma}{1\,\text{s}}\right)\,.$
(88)

Since this value is large, deexcitation primarily occurs. The local amplitude
damping decoherence in (32), using (40) and (41), can be approximated as

$\displaystyle\text{Re}[\mathcal{A}_{\text{loc}}^{(\text{vac})}]$
$\displaystyle\approx\dfrac{1}{3\sqrt{\pi}}\dfrac{\mu_{\text{B}}^{2}}{\sigma^{2}}(\Omega\sigma)^{3}\quad({\rm
for}~{}\Omega\sigma\gg 1)$ $\displaystyle\simeq 6.2\times
10^{-13}\left(\dfrac{10\sigma}{1\,\text{s}}\right)\left(\dfrac{B_{0}}{1\,\text{T}}\right)^{3}\,.$
(89)

Therefore, the dominant factor in local decoherence is amplitude damping.

We now estimate the nonlocal terms. For a splitting separation of $L\ll\sigma$
(e.g., $L\sim 100\,\mu\text{m}$), we saw in Eqs. (47) and (45) that the
nonlocal phase damping term
$\text{Re}[\mathcal{D}_{\text{nl}}^{(\text{vac})}]$ is

$\displaystyle\text{Re}[\mathcal{D}_{\text{nl}}^{(\text{vac})}]\approx\dfrac{\mu_{\text{B}}^{2}}{6\pi\sigma^{2}}=8.0\times
10^{-42}\left(\dfrac{1\,\text{s}}{10\sigma}\right)^{2}\,.$ (90)

Although $\mathcal{D}_{\text{nl}}^{(\text{vac})}$ depends on the splitting
configuration [see Eqs. (44) and (46)], they both coincide with the local
phase damping decoherence in this experimental setup. Moreover, the nonlocal
mitigation term $|\mathcal{M}_{\text{nl}}^{(\text{vac})}|$ is exactly zero for
the $z$-splitting [Eq. (48)], and
$|\mathcal{M}_{\text{nl}}^{(\text{vac})}|\approx 0$ for the $x$-splitting [Eq.
(50)].

Overall, local amplitude damping is the main source of decoherence from vacuum
fluctuations, but it is still a small amount.

####  IV.4.2 Thermal noise

Here, we estimate the decoherence from local amplitude damping caused by
thermal noise (60). To this end, let us introduce

$\displaystyle\overline{\mu}_{\text{B}}$
$\displaystyle\coloneqq\dfrac{\sqrt{4\pi\alpha_{\text{f}}}k_{\text{B}}}{2m_{e}c^{2}}\simeq
2.6\times 10^{-11}\,,$ (91a) $\displaystyle\overline{\Omega}$
$\displaystyle\coloneqq\dfrac{\hbar\Omega}{k_{\text{B}}}=\dfrac{2\mu_{\text{B}}B_{0}}{k_{\text{B}}}\simeq
1.3\left(\dfrac{B_{0}}{1\,\text{T}}\right)\,,$ (91b)
$\displaystyle\overline{\sigma}$
$\displaystyle\coloneqq\dfrac{k_{\text{B}}\sigma}{\hbar}\simeq 2.1\times
10^{9}\left(\dfrac{10\sigma}{1\,\text{s}}\right)\,.$ (91c)

We change the variable from $|\bm{k}|$ to $r\equiv\beta|\bm{k}|$ in (60) and
use the approximation
$\cosh[2(\overline{\Omega}/T)(\overline{\sigma}T)^{2}r]\approx\exp[2(\overline{\Omega}/T)(\overline{\sigma}T)^{2}r]/2$
since $\overline{\sigma}$ is large. Then, changing the variable to
$r^{\prime}\equiv(r-\overline{\Omega}/T)\overline{\sigma}T$ and using
$r^{\prime}/(\overline{\sigma}T)\ll 1$, we get

$\displaystyle\text{Re}[\mathcal{A}_{\text{loc}}^{(\beta)}]$
$\displaystyle\approx\dfrac{\overline{\mu}_{\text{B}}^{2}\overline{\sigma}}{3\sqrt{\pi}}\dfrac{\overline{\Omega}^{3}}{e^{\overline{\Omega}/T}-1}[1+\text{erf}(\overline{\Omega}\overline{\sigma})]\,,$
(92)

and at higher temperatures ($T\gg 1\,\text{K}$),

$\displaystyle\text{Re}[\mathcal{A}_{\text{loc}}^{(\beta)}]$
$\displaystyle\simeq\dfrac{2\overline{\mu}_{\text{B}}^{2}\overline{\sigma}}{3\sqrt{\pi}}\overline{\Omega}^{2}T$
$\displaystyle=9.2\times
10^{-11}\left(\dfrac{10\sigma}{1\,\text{s}}\right)\left(\dfrac{B_{0}}{1\,\text{T}}\right)^{2}\left(\dfrac{T}{100\,\text{K}}\right)\,.$
(93)

Thus, local amplitude damping due to thermal noise is more significant than
that caused by vacuum fluctuations.

##  V Conclusion

Maintaining coherence in a spin superposition state is crucial, particularly
for generating a spatial superposition state of mass. However, the spin degree
of freedom is inevitably influenced by the vacuum fluctuations of the
electromagnetic field, leading to unavoidable decoherence. In this paper, we
have investigated the decoherence of a spatially superposed spin particle
caused by the spin-magnetic field coupling. To isolate the pure quantum field
effect, we assumed that the spatially superposed spin trajectories are static
(i.e., without acceleration and relative velocity) in both Minkowski vacuum
and thermal states, and that the interaction occurs very smoothly.

The general expression for the coherence measure (the $l_{1}$ norm of
coherence $\mathcal{C}_{l_{1}}$) in the leading order of the coupling constant
is given by (73), which is valid as long as the quantum field is initially in
a quasifree state. We note that this expression can be straightforwardly
generalized to an arbitrary globally hyperbolic spacetime. From this
expression, we defined the amount of decoherence $\mathscr{D}$ in (74).
Decoherence originates from amplitude damping and phase damping effects, and
these come into $\mathscr{D}$ as local and nonlocal terms. The local terms
correspond to the decoherence that solely depends on each branch of spatially
superposed trajectories. In contrast, the nonlocal terms depend on the field
correlation functions between the two branches, and it may mitigate
decoherence depending on the configuration of the spatial superposition.

We then specified the field’s state to the Minkowski vacuum and thermal state,
and chose a Gaussian switching function to obtain the analytic form of
$\mathscr{D}$. The result depends on how the spin’s spatial superposition is
configured: (i) when the spin’s trajectory is split along the direction of the
classical magnetic field in the spin’s free Hamiltonian, and (ii) when it is
split perpendicular to it. Such configuration dependence comes from the
electromagnetic field’s polarization, but it is invisible in a table-top
experiment because the separation distance is very small. We note that a qubit
coupled to a quantum scalar field also exhibits such a nonlocal contribution,
which is independent of the splitting configuration. Unlike the
spin-$\frac{1}{2}$ particle case, the nonlocal term for a qubit mitigates the
local decoherence when the splitting distance is small. See Appendix B for
details.

We also analytically showed that a spatially superposed spin-$\frac{1}{2}$
particle decoheres monotonically with the field’s temperature. While the
thermal contribution dominates over the decoherence due to vacuum
fluctuations, the amount of decoherence is very small in a table-top
experiment.

Although our analysis focused on static superposition trajectories, it can be
applied to any trajectory in spacetime. One interesting example is a uniformly
accelerating trajectory. In this case, the spin particle experiences
thermality due to the Unruh effect [27] since the pullback of the Wightman
function along a single acceleration trajectory satisfies the KMS condition at
the Unruh temperature $T_{\text{U}}=a/(2\pi)$, where $a$ is the proper
acceleration. Consequently, any local decoherence terms (i.e.,
$\text{Re}[\mathcal{A}_{\text{loc}}]$ and
$\text{Re}[\mathcal{D}_{\text{loc}}]$) for the uniformly accelerating spin
particle are equivalent to those for the static particle in a thermal bath.
However, as pointed out in [36, 37], the uniformly accelerating spin particle
is expected to decohere more by emitting soft photons in the long interaction
limit, which does not occur in our thermal bath case. Such decoherence is
characteristic when Killing horizons exist. We suspect that this soft photon-
induced decoherence effect is encoded in the nonlocal terms
$\text{Re}[\mathcal{D}_{\text{nl}}]$ and $|\mathcal{M}_{\text{nl}}|$ in
$\mathscr{D}$.

###### Acknowledgements.

This work was supported by JSPS KAKENHI Grant Numbers JP3K25871.

##  Appendix A Elements in coherence

In this Appendix, we provide an explicit calculation for
$\mathcal{M}_{\text{nl}}$ in (48) and (49). The other nonlocal term
$\text{Re}[\mathcal{D}_{\text{nl}}]$ can be obtained in a similar way.

The nonlocal term $\mathcal{M}_{\text{nl}}$ is defined as

$\displaystyle\mathcal{M}_{\text{nl}}\coloneqq$
$\displaystyle\mu_{\text{B}}^{2}\int_{\mathbb{R}}\text{d}\tau\int_{\mathbb{R}}\text{d}\tau^{\prime}\,\chi(\tau)\chi(\tau^{\prime})e^{\mathsf{i}\Omega(\tau+\tau^{\prime})}\braket{\hat{B}_{-}(\mathsf{x}_{\uparrow})\hat{B}_{-}(\mathsf{x}^{\prime}_{\downarrow})}_{\rho_{\text{EM,0}}}\,.$
(94)

Recall that the quantum magnetic field $\hat{\bm{B}}(\mathsf{x})$ can be
expanded as [see Eqs. (7) and (8)]

$\displaystyle\hat{\bm{B}}(\mathsf{x})$
$\displaystyle=\int_{\mathbb{R}^{3}}\text{d}^{3}k\sum_{\lambda=1}^{2}\left(\bm{B}_{({\bm{k}},\lambda)}(\mathsf{x})\hat{a}_{{\bm{k}},\lambda}+\bm{B}_{({\bm{k}},\lambda)}^{*}(\mathsf{x})\hat{a}_{{\bm{k}},\lambda}^{\dagger}\right)\,,$

where

$\displaystyle\bm{B}_{({\bm{k}},\lambda)}(\mathsf{x})$
$\displaystyle\equiv\dfrac{\mathsf{i}{\bm{k}}\times\bm{e}_{({\bm{k}},\lambda)}}{\sqrt{(2\pi)^{3}2|\bm{k}|}}e^{-\mathsf{i}|\bm{k}|t+\mathsf{i}{\bm{k}}\cdot{\bm{x}}}\,.$

Denoting the components of the vector mode function
$\bm{B}_{({\bm{k}},\lambda)}(\mathsf{x})$ as

$\displaystyle\bm{B}_{({\bm{k}},\lambda)}(\mathsf{x})=[B_{({\bm{k}},\lambda)}^{(1)}(\mathsf{x}),B_{({\bm{k}},\lambda)}^{(2)}(\mathsf{x}),B_{({\bm{k}},\lambda)}^{(3)}(\mathsf{x})]^{\intercal}\,,$
(95)

each component of the quantum magnetic field can be explicitly written as

$\displaystyle\hat{\bm{B}}(\mathsf{x})$
$\displaystyle=\begin{bmatrix}\hat{B}_{1}(\mathsf{x})\\\
\hat{B}_{2}(\mathsf{x})\\\ \hat{B}_{3}(\mathsf{x})\end{bmatrix}$
$\displaystyle=\int_{\mathbb{R}^{3}}\text{d}^{3}k\sum_{\lambda=1}^{2}\Big{(}\begin{bmatrix}B_{({\bm{k}},\lambda)}^{(1)}\\\
B_{({\bm{k}},\lambda)}^{(2)}\\\
B_{({\bm{k}},\lambda)}^{(3)}\end{bmatrix}\hat{a}_{{\bm{k}},\lambda}+\begin{bmatrix}B_{({\bm{k}},\lambda)}^{(1)*}\\\
B_{({\bm{k}},\lambda)}^{(2)*}\\\
B_{({\bm{k}},\lambda)}^{(3)*}\end{bmatrix}\hat{a}_{{\bm{k}},\lambda}^{\dagger}\Big{)}\,,$

where we have abbreviated the spacetime dependence in the second equation for
simplicity.

Assuming that the field is initially in the Minkowski vacuum $\ket{0}$, the
vacuum two-point correlation function of
$\hat{B}_{-}(\equiv\hat{B}_{1}-\mathsf{i}\hat{B}_{2})$ reads

$\displaystyle\braket{0}{\hat{B}_{-}(\mathsf{x}_{\uparrow})\hat{B}_{-}(\mathsf{x}^{\prime}_{\downarrow})}{0}=\int_{\mathbb{R}^{3}}\text{d}^{3}k\sum_{\lambda=1}^{2}$
$\displaystyle\times\left(B_{({\bm{k}},\lambda)}^{(1)}(\mathsf{x}_{\uparrow})-\mathsf{i}B_{({\bm{k}},\lambda)}^{(2)}(\mathsf{x}_{\uparrow})\right)\left(B_{({\bm{k}},\lambda)}^{(1)*}(\mathsf{x}_{\downarrow}^{\prime})-\mathsf{i}B_{({\bm{k}},\lambda)}^{(2)*}(\mathsf{x}_{\downarrow}^{\prime})\right)$
$\displaystyle=\int_{\mathbb{R}^{3}}\dfrac{\text{d}^{3}k}{(2\pi)^{3}2|\bm{k}|}(-k_{-}^{2})e^{-\mathsf{i}|\bm{k}|t_{\uparrow}+\mathsf{i}{\bm{k}}\cdot{\bm{x}}_{\uparrow}}e^{\mathsf{i}|\bm{k}|t_{\downarrow}-\mathsf{i}{\bm{k}}\cdot{\bm{x}}_{\downarrow}}\,,$
(96)

where we have used the identity

$\displaystyle\sum_{\lambda}({\bm{k}}\times\bm{e}_{({\bm{k}},\lambda)})_{-}({\bm{k}}\times\bm{e}_{({\bm{k}},\lambda)}^{*})_{-}=-k_{-}^{2}\,,$
(97)

where
$({\bm{k}}\times\bm{e}_{({\bm{k}},\lambda)})_{-}\equiv({\bm{k}}\times\bm{e}_{({\bm{k}},\lambda)})_{1}-\mathsf{i}({\bm{k}}\times\bm{e}_{({\bm{k}},\lambda)})_{2}$
and $k_{-}\equiv k_{1}-\mathsf{i}k_{2}$.

We now assume that each superposed trajectory is at rest in a single reference
frame, so that $\mathsf{x}_{\uparrow}$ and $\mathsf{x}_{\downarrow}^{\prime}$
can be written as $\mathsf{x}_{\uparrow}(\tau)=(\tau,{\bm{x}}_{\uparrow})$ and
$\mathsf{x}_{\downarrow}(\tau^{\prime})=(\tau^{\prime},{\bm{x}}_{\downarrow})$.
This implies that the proper spatial separation
$L\equiv|{\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow}|$ is fixed throughout the
interaction. The nonlocal term $\mathcal{M}_{\text{nl}}$ reduces to

$\displaystyle\mathcal{M}_{\text{nl}}=\dfrac{\mu_{\text{B}}^{2}}{(2\pi)^{3}2}$
$\displaystyle\quad\times\int_{\mathbb{R}^{3}}\text{d}^{3}k\,\dfrac{(-k_{-}^{2})}{|\bm{k}|}e^{-\mathsf{i}{\bm{k}}\cdot({\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow})}\tilde{\chi}(\Omega-|\bm{k}|)\tilde{\chi}(\Omega+|\bm{k}|)\,,$
(98)

where $\tilde{\chi}(\omega)$ is the Fourier transformation (35) of the
switching function $\chi(\tau)$. We employ the spherical coordinates
$(|\bm{k}|,\theta,\varphi)$ for ${\bm{k}}$ so that
${\bm{k}}=[|\bm{k}|\sin\theta\cos\varphi,|\bm{k}|\sin\theta\sin\varphi,|\bm{k}|\cos\theta]^{\intercal}$
and the integration measure reads
$\text{d}^{3}k=|\bm{k}|^{2}\sin\theta\text{d}|\bm{k}|\text{d}\theta\text{d}\varphi$.
Consequently the nonlocal term becomes

$\displaystyle\mathcal{M}_{\text{nl}}=\dfrac{-\mu_{\text{B}}^{2}}{(2\pi)^{3}2}\int_{0}^{\infty}\text{d}|\bm{k}|\,|\bm{k}|^{3}\tilde{\chi}(\Omega-|\bm{k}|)\tilde{\chi}(\Omega+|\bm{k}|)$
$\displaystyle\times\int_{0}^{\pi}\text{d}\theta\,\sin^{3}\theta\int_{0}^{2\pi}\text{d}\varphi\,e^{-2\mathsf{i}\varphi}e^{-\mathsf{i}{\bm{k}}\cdot({\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow})}\,.$
(99)

This expression for $\mathcal{M}_{\text{nl}}$ tells us that the result depends
on the spin’s splitting configuration
${\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow}$. In what follows, we examine two
cases: (i) when the trajectory is split along the $z$-axis; and (ii) along the
$x$-axis.

(i) $z$-splitting:
${\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow}=[0,0,L]^{\intercal}$

In this case,
${\bm{k}}\cdot({\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow})=|\bm{k}|L\cos\theta$,
which is independent of $\varphi$. This means that the nonlocal term
$\mathcal{M}_{\text{nl}}=0$ since

$\displaystyle\int_{0}^{2\pi}\text{d}\varphi\,e^{-2\mathsf{i}\varphi}=0\,.$
(100)

(i) $x$-splitting:
${\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow}=[L,0,0]^{\intercal}$

Instead, if the spin trajectory is split along the $(x,y)$-plane, the inner
product ${\bm{k}}\cdot({\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow})$ depends on
$\varphi$. In the case of $x$-splitting, we have

$\displaystyle\int_{0}^{2\pi}\text{d}\varphi\,e^{-2\mathsf{i}\varphi}e^{-\mathsf{i}|\bm{k}|L\sin\theta\cos\varphi}=-2\pi
J_{2}(|\bm{k}|L\sin\theta)\,,$ (101)

where $J_{2}(z)$ is the Bessel function of the first kind. Further integrating
over $\theta$ yields

$\displaystyle\mathcal{M}_{\text{nl}}$
$\displaystyle=\dfrac{-\mu_{\text{B}}^{2}4\pi}{(2\pi)^{3}2L^{3}}\int_{0}^{\infty}\text{d}|\bm{k}|\,\tilde{\chi}(\Omega-|\bm{k}|)\tilde{\chi}(\Omega+|\bm{k}|)$
$\displaystyle\times\Big{[}3|\bm{k}|L\cos(|\bm{k}|L)+(|\bm{k}|^{2}L^{2}-3)\sin(|\bm{k}|L)\Big{]}\,.$
(102)

One obtains the final result (49) by specifying the Gaussian switching
function (38).

##  Appendix B UDW detector case

For a Unruh-DeWitt (UDW) detector, the interaction Hamiltonian only contains
the amplitude damping part. In the interaction picture, it is given by

$\displaystyle\hat{H}_{\text{I}}^{\tau}(\tau)$
$\displaystyle=\lambda\chi(\tau)(e^{\mathsf{i}\Omega\tau}\hat{\sigma}_{+}+e^{-\mathsf{i}\Omega\tau}\hat{\sigma}_{-})\otimes\hat{\phi}(\mathsf{x}(\tau))\,.$
(103)

In $(3+1)$ dimensions, the coupling constant $\lambda$ is dimensionless. We
assume the initial state

$\displaystyle\rho_{\text{tot,0}}$
$\displaystyle=\rho_{\text{UDW,0}}\otimes\rho_{\phi,0}\,,$ (104)

where $\rho_{\phi,0}$ is a quasifree state for the scalar field and
$\rho_{\text{UDW,0}}$ is the initial spatial superposition state of the
detector corresponding to

$\displaystyle\alpha\ket{\uparrow,C_{\uparrow}}+e^{\mathsf{i}\vartheta}\sqrt{1-\alpha^{2}}\ket{\downarrow,C_{\downarrow}}\,.$
(105)

The final density matrix $\rho_{\text{UDW}}$ of the detector reads

$\displaystyle\rho_{\text{UDW}}$
$\displaystyle=\begin{bmatrix}\rho_{11}&0&0&\rho_{14}\\\
0&\rho_{22}&\rho_{23}&0\\\ 0&\rho_{23}^{*}&\rho_{33}&0\\\
\rho_{14}^{*}&0&0&\rho_{44}\end{bmatrix}\,,$ (106)

where

$\displaystyle\rho_{11}$
$\displaystyle=\rho_{11}^{(0)}[1-P_{\uparrow\to\downarrow}^{C_{\uparrow}}(\Omega)]\,,$
(107a) $\displaystyle\rho_{22}$
$\displaystyle=\rho_{44}^{(0)}P_{\downarrow\to\uparrow}^{C_{\downarrow}}(\Omega)\,,$
(107b) $\displaystyle\rho_{33}$
$\displaystyle=\rho_{11}^{(0)}P_{\uparrow\to\downarrow}^{C_{\uparrow}}(\Omega)\,,$
(107c) $\displaystyle\rho_{44}$
$\displaystyle=\rho_{44}^{(0)}[1-P_{\downarrow\to\uparrow}^{C_{\downarrow}}(\Omega)]\,,$
(107d) $\displaystyle\rho_{14}$
$\displaystyle=\rho_{14}^{(0)}(1-\mathcal{A}_{\text{loc}})\,,$ (107e)
$\displaystyle\rho_{23}$
$\displaystyle=\rho_{14}^{(0)*}\mathcal{M}_{\text{nl}}\,,$ (107f)

with

$\displaystyle\mathcal{M}_{\text{nl}}$
$\displaystyle\coloneqq\lambda^{2}\int_{\mathbb{R}}\text{d}\tau\int_{\mathbb{R}}\text{d}\tau^{\prime}\,\chi(\tau)\chi(\tau^{\prime})e^{\mathsf{i}\Omega(\tau+\tau^{\prime})}$
$\displaystyle\quad\times\braket{\hat{\phi}(\mathsf{x}_{\uparrow})\hat{\phi}(\mathsf{x}_{\downarrow}^{\prime})}_{\rho_{\phi,0}}\,,$
(108a) $\displaystyle\mathcal{A}_{\text{loc}}$
$\displaystyle\coloneqq\lambda^{2}\int_{\mathbb{R}}\text{d}\tau\int_{\mathbb{R}}\text{d}\tau^{\prime}\,\Theta(t(\tau)-t(\tau^{\prime}))$
$\displaystyle\quad\times\chi(\tau)\chi(\tau^{\prime})e^{\mathsf{i}\Omega(\tau-\tau^{\prime})}\braket{\hat{\phi}(\mathsf{x}_{\uparrow})\hat{\phi}(\mathsf{x}_{\uparrow}^{\prime})}_{\rho_{\phi,0}}$
$\displaystyle\quad+\lambda^{2}\int_{\mathbb{R}}\text{d}\tau\int_{\mathbb{R}}\text{d}\tau^{\prime}\,\Theta(t(\tau^{\prime})-t(\tau))$
$\displaystyle\quad\times\chi(\tau)\chi(\tau^{\prime})e^{-\mathsf{i}\Omega(\tau-\tau^{\prime})}\braket{\hat{\phi}(\mathsf{x}_{\downarrow})\hat{\phi}(\mathsf{x}_{\downarrow}^{\prime})}_{\rho_{\phi,0}}\,.$
(108b)

Here, $\rho_{11}^{(0)}=\alpha^{2}$,
$\rho_{14}^{(0)}=\alpha\sqrt{1-\alpha^{2}}e^{-\mathsf{i}\vartheta}$, and
$\rho_{44}^{(0)}=1-\alpha^{2}$ [see Eq. (23)].

From the final density matrix, the $l_{1}$ norm of coherence can be computed
as

$\displaystyle\mathcal{C}_{l_{1}}$
$\displaystyle=2|\rho_{14}^{(0)}|(1-\text{Re}[\mathcal{A}_{\text{loc}}]+|\mathcal{M}_{\text{nl}}|)\,,$
(109)

and the amount of decoherence $\mathscr{D}$ is defined as

$\displaystyle\mathscr{D}$
$\displaystyle\coloneqq\text{Re}[\mathcal{A}_{\text{loc}}]-|\mathcal{M}_{\text{nl}}|\,.$
(110)

As in the case of the electromagnetic field (32), we have

$\displaystyle\text{Re}[\mathcal{A}_{\text{loc}}]$
$\displaystyle=\dfrac{1}{2}[P_{\uparrow\to\downarrow}^{C_{\uparrow}}(\Omega)+P_{\downarrow\to\uparrow}^{C_{\downarrow}}(\Omega)]\,.$
(111)

###  B.1 The Minkowski vacuum case

Consider a static detector coupled to the massless quantum scalar field in the
Minkowski vacuum in $(3+1)$-dimensional Minkowski spacetime. Employing the
Gaussian switching function (38), the excitation transition probability can be
computed as

$\displaystyle P_{\downarrow\to\uparrow}^{C_{\downarrow}}(\Omega)$
$\displaystyle=\dfrac{\lambda^{2}}{4\pi}\left[e^{-\Omega^{2}\sigma^{2}}-\sqrt{\pi}\Omega\sigma\text{erfc}(\Omega\sigma)\right]\,.$
(112)

Moreover, $\mathcal{M}_{\text{nl}}$ is independent of the direction of spatial
split and it reads

$\displaystyle\mathcal{M}_{\text{nl}}$
$\displaystyle=\dfrac{\lambda^{2}\sigma}{2\pi
L}e^{-\Omega^{2}\sigma^{2}}D^{+}(L/2\sigma)\,.$ (113)

Note that the Dawson function $D^{+}(x)$ has this property

$\displaystyle\lim_{x\to 0}\dfrac{D^{+}(x)}{x}=1\,,$ (114)

and therefore, $\mathcal{M}_{\text{nl}}$ takes the maximum value in the limit
$L\to 0$:

$\displaystyle\lim_{L/(2\sigma)\to 0}\mathcal{M}_{\text{nl}}$
$\displaystyle=\dfrac{\lambda^{2}}{4\pi}e^{-\Omega^{2}\sigma^{2}}\,.$ (115)

This is a very different behavior compared to the electromagnetic field case,
in which the nonlocal term $\mathcal{M}_{\text{nl}}$ vanishes at the
coincidence limit $L\to 0$. Then, one can easily check that, in the
coincidence limit $L\to 0$, a gapless detector ($\Omega=0$) does not decohere:
$\mathscr{D}=0$. Note that if we start the interaction without splitting the
trajectory, the density matrix for the UDW detector is a $2\times 2$ matrix.
In this case, the decoherence measure is
$\mathscr{D}=\text{Re}[\mathcal{A}_{\text{loc}}]-\text{Re}[\mathcal{M}_{\text{nl}}]$,
which is different from the splitting scenario. However, since
$\mathcal{M}_{\text{nl}}$ is real and positive,
$\text{Re}[\mathcal{M}_{\text{nl}}]=|\mathcal{M}_{\text{nl}}|$, and the
decoherence measure coincides.

On the other hand, the nonlocal term vanishes in the limit $L\to\infty$ and
therefore

$\displaystyle\lim_{L\to\infty}\mathscr{D}$
$\displaystyle=\text{Re}[\mathcal{A}_{\text{loc}}]\,.$

In contrast to the electromagnetic field case, decoherence reaches its maximum
at large distances. See Fig. 4(a) for the separation dependence.

Figure 4: The amount of decoherence $\mathscr{D}/\lambda^{2}$ as a function of
$\log_{10}L/\sigma$ for $\Omega\sigma=0,0.5$, and $1$ when (a) the scalar
field is in the Minkowski vacuum state, and (b) in the thermal state at the
temperature $T\sigma=1$.

###  B.2 The thermal state case

As we saw in the electromagnetic field case, it is known that the thermal
Wightman function $W_{\text{th}}(\mathsf{x},\mathsf{x}^{\prime})$ for a scalar
field is a sum of the vacuum and the pure-thermal Wightman functions [35]:

$\displaystyle W_{\text{th}}(\mathsf{x},\mathsf{x}^{\prime})$
$\displaystyle=W_{\text{vac}}(\mathsf{x},\mathsf{x}^{\prime})+W_{\beta}(\mathsf{x},\mathsf{x}^{\prime})\,,$
(116)

where

$\displaystyle W_{\beta}(\mathsf{x},\mathsf{x}^{\prime})$
$\displaystyle=\int_{\mathbb{R}^{3}}\dfrac{\text{d}^{3}k}{(2\pi)^{3}2|\bm{k}|}\dfrac{e^{-\mathsf{i}|\bm{k}|(t-t^{\prime})+\mathsf{i}{\bm{k}}\cdot({\bm{x}}-{\bm{x}}^{\prime})}+\text{c.c.}}{e^{\beta|\bm{k}|}-1}\,.$
(117)

Therefore, the amount of decoherence for the thermal state
$\mathscr{D}^{(\text{th})}$ can be decomposed as

$\displaystyle\mathscr{D}^{(\text{th})}$
$\displaystyle=\mathscr{D}^{(\text{vac})}+\mathscr{D}^{(\beta)}\,.$ (118)

where

$\displaystyle\mathscr{D}^{(\text{vac})}$
$\displaystyle=\text{Re}[\mathcal{A}_{\text{loc}}^{(\text{vac})}]-|\mathcal{M}_{\text{nl}}^{(\text{vac})}|\,,$
(119a) $\displaystyle\mathscr{D}^{(\beta)}$
$\displaystyle=\text{Re}[\mathcal{A}_{\text{loc}}^{(\beta)}]-|\mathcal{M}_{\text{nl}}^{(\beta)}|\,.$
(119b)

In this case,

$\displaystyle P_{\downarrow\to\uparrow}^{(\beta)}$
$\displaystyle=P_{\uparrow\to\downarrow}^{(\beta)}$
$\displaystyle=\lambda^{2}\int_{\mathbb{R}^{3}}\dfrac{\text{d}^{3}k}{(2\pi)^{3}2|\bm{k}|}\dfrac{|\tilde{\chi}(\Omega+|\bm{k}|)|^{2}+|\tilde{\chi}(\Omega-|\bm{k}|)|^{2}}{e^{\beta|\bm{k}|}-1}\,,$
(120) $\displaystyle\mathcal{M}_{\text{nl}}^{(\beta)}$
$\displaystyle=2\lambda^{2}\int_{\mathbb{R}^{3}}\dfrac{\text{d}^{3}k}{(2\pi)^{3}2|\bm{k}|}\dfrac{\tilde{\chi}(\Omega+|\bm{k}|)\tilde{\chi}(\Omega-|\bm{k}|)}{e^{\beta|\bm{k}|}-1}$
$\displaystyle\qquad\times\cos[{\bm{k}}\cdot({\bm{x}}_{\downarrow}-{\bm{x}}_{\uparrow})]\,.$
(121)

In particular, if we choose the Gaussian switching function (38), these become

$\displaystyle P_{\downarrow\to\uparrow}^{(\beta)}$
$\displaystyle=\dfrac{\lambda^{2}\sigma^{2}}{\pi}e^{-\Omega^{2}\sigma^{2}}\int_{0}^{\infty}\text{d}|\bm{k}|\,\dfrac{|\bm{k}|e^{-|\bm{k}|^{2}\sigma^{2}}}{e^{\beta|\bm{k}|}-1}\cosh(2\Omega|\bm{k}|\sigma^{2})\,,$
(122) $\displaystyle\mathcal{M}_{\text{nl}}^{(\beta)}$
$\displaystyle=\dfrac{\lambda^{2}\sigma^{2}}{\pi
L}e^{-\Omega^{2}\sigma^{2}}\int_{0}^{\infty}\text{d}|\bm{k}|\,\dfrac{e^{-|\bm{k}|^{2}\sigma^{2}}}{e^{\beta|\bm{k}|}-1}\sin(|\bm{k}|L)\,.$
(123)

The $L$-dependence is depicted in Fig. 4(b) when the field temperature is
$T\sigma=1$.

Following the same proof in Sec. IV.3, we can show that for an arbitrary
switching function $\chi(\tau)$, decoherence monotonically increases with
temperature $\beta^{-1}$. That is, for
$\beta_{\text{H}}^{-1}>\beta_{\text{C}}^{-1}$,

$\displaystyle\mathscr{D}^{(\text{th})}(\beta_{\text{H}})\geq\mathscr{D}^{(\text{th})}(\beta_{\text{C}})\,.$
(124)

A recent paper [38] discusses the decoherence of a uniformly accelerating UDW
detector. Let us compare our UDW setup with theirs. In our study, we consider
the qubit, the COM, and the field degrees of freedom:
$\mathcal{H}_{\text{UDW}}\otimes\mathcal{H}_{\text{COM}}\otimes\mathcal{H}_{\phi}$.
The COM degree of freedom is assumed to be fixed after splitting, so it does
not decohere. Specifically, the interaction Hamiltonian in the Schrödinger
picture takes the form
$\hat{\sigma}_{1}\otimes\mathds{1}_{\text{COM}}\otimes\hat{\phi}$, where
$\hat{\sigma}_{1}\in\mathcal{B}(\mathcal{H}_{\text{s}})$. Thus, this
Hamiltonian describes the decoherence of a UDW detector (spin degree of
freedom) caused by amplitude damping. In contrast, [38] considers
$\mathcal{H}_{\text{COM}}\otimes\mathcal{H}_{\phi}$ and examines the COM’s
decoherence caused by the quantum field. Similar to our case, each wavepacket
of the superposed system is assumed to be narrow and localized, making
$\mathcal{H}_{\text{COM}}$ a two-dimensional Hilbert space. Hence, they treat
the COM degree of freedom as a UDW detector. The interaction Hamiltonian is
derived from
$\ket{C_{\uparrow}}\bra{C_{\uparrow}}\otimes\hat{\phi}(\mathsf{x}_{\uparrow})+\ket{C_{\downarrow}}\bra{C_{\downarrow}}\otimes\hat{\phi}(\mathsf{x}_{\downarrow})$,
which can be reduced to
$\hat{\sigma}_{3}\otimes\varepsilon^{\mu}\partial_{\mu}\hat{\phi}(\mathsf{x})$,
where $\hat{\sigma}_{3}\in\mathcal{B}(\mathcal{H}_{\text{COM}})$ and
$\varepsilon^{\mu}$ describes the displacement between the two trajectories.
This describes the decoherence of the COM degree of freedom caused by phase
damping.

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