#  Thermal effect in a causal diamond: open quantum systems approach

A. Chakraborty  Department of Physics, University of Houston. Houston, Texas
77024-5005, USA  H. E. Camblong  Department of Physics and Astronomy,
University of San Francisco. San Francisco, California 94117-1080, USA  C.
Ordóñez  Department of Physics, University of Houston. Houston, Texas
77024-5005, USA

###### Abstract

A static observer with a finite lifetime has causal access to only a limited
region of spacetime known as the causal diamond. The presence of an apparent
horizon in the causal diamond, due to the observer’s finite lifetime, is the
origin of an Unruh-like thermal effect. Thus, even though the observer is
static and the background is flat, the finite-lifetime observer experiences a
thermal bath in the Minkowski vacuum. In this article, we provide an open
quantum systems approach that yields a complete thermal characterization via
the observer’s steady-state density matrix, which is shown to be thermal with
a temperature inversely proportional to its lifetime. This associated diamond
temperature agrees with the established result derived from other methods.
Moreover, our approach is particularly useful for designing entanglement
harvesting protocols in the causal diamond. In addition, we introduce an
insightful procedure that defines diamond coordinates using conformal
transformations, and which leads to a more direct derivation of the thermal
properties.

##  I Introduction

The concept of the vacuum state is observer dependent in quantum field theory
in curved spacetime. This was illustrated by Stephen Hawking in his seminal
papers hawking1 ; hawking2 ; hawking3  on thermal radiation emitted by black
holes. In another seminal paper unruh76 , William Unruh showed that the
Hawking effect is equivalent to a uniformly accelerated observer detecting
thermal particles in the Minkowski vacuum, and the temperature is proportional
to the acceleration of the observer. The temperature $T_{U}=a/2\pi$ (in
natural units $\hbar=G=c=k_{B}=1$) is known as the Unruh temperature and the
effect is known as the Unruh effect. The Unruh effect showed explicitly that
the notion of the presence or absence of particles in a quantum state is
related to the motion of a detector. This was corroborated further by Davies,
Fulling, and others in a series of papers wald75 ; parker75 ; ufd76 ; davies78
; df77 . The origin of this thermality is attributed to the presence of an
event horizon (in the case of a black hole) or an apparent horizon (in the
case of a uniformly accelerated observer) israel76 . Due to the presence of a
horizon, the observer can only access a certain region of the spacetime. Thus,
any measurement in the frame of the observer should involve an average over
the degrees of freedom associated with the inaccessible region. This averaging
yields a mixed state for the observer; and in the specific case of an
accelerated observer, it yields a thermal state. The contribution of the
horizon to determination of the thermality was further illustrated by the
conformal field theory approach, where the central charge of the Virasoro
algebra was shown to be related to the Hawking temperature carlip1 ; carlip2 ;
birminghamsen ; cardy . Another near-horizon approach used the conformal
symmetry of the fields near the horizon to show that it is possible to derive
the area-entropy formula for generic black holes by statistical mode counting
nhcamblong ; nhcamblong-sc ; nhcamblong-tightness . In a recent series of
papers, using a quantum optics approach, it was shown that the presence of the
horizon is responsible for creating the thermal atmosphere in static and
stationary black holes chakraborty1 ; chakraborty2 ; chakraborty3 ;
chakraborty4 .

In all the approaches mentioned above, the Unruh effect and the Hawking effect
are always associated with an accelerated system—either the observer or the
field. However, in 2003, Martinetti and Rovelli, in their seminal paper
martinetti-1 , showed that a finite-lifetime observer can detect thermal
particles in the Minkwoski vacuum despite being static. In essence, the birth
and death of the finite-lifetime observer restrict the observer’s causal
access to a finite region of the whole Minkowski spacetime. This is the region
causally connected with the observer with a given lifetime; thus, it is the
intersection between the future light cone of the birth event and the past
light cone of the death event. Because of its diamond-shaped appearance, this
region is called the causal diamond (Fig. 1). Due to the presence of the
apparent horizons in the causal diamond structure, the finite-lifetime
observer will not perceive the Minkowski vacuum as a vacuum state. Martinetti
and Rovelli showed that the static observer experiences a variable temperature
in the Minkowski vacuum. The physical meaning of this result was clarified in
a series of papers su-ralph-1 ; su-ralph-2 ; light-cone ; jacobson  in the
context of black holes, AdS spacetime, and Unruh-deWitt detectors. We
particularly focus on the detector approach su-ralph-1 ; su-ralph-2 , in which
the finite-lifetime observer can be treated as a two-level system with the
energy gap between the two-levels fixed in the system’s reference frame. These
two-level systems, known as the diamond observers, perceive a different vacuum
state than the Minkowski vacuum and the excitation rate of these observers in
the Minkowski vacuum is thermal in nature. This temperature
$T_{D}=2/\pi\mathcal{T}$ is known as the diamond temperature and is inversely
proportional to the lifetime $\mathcal{T}$ of the observer. One can also show
that the annihilation and creation operators of a diamond observer and a
Minkowski observer are related by a nontrivial Bogolyubov transformation su-
ralph-1 , which is a trademark of the inequivalence of the diamond and
Minkowski vacuum. Moreover, the expression for the diamond temperature
satisfies the requirement that the static observer becomes a Minkowski
observer as the lifetime increases to infinity.

Figure 1: Here, A is the point of birth of the observer and B is the point of
death. So the length of AB is the lifetime of the observer
$\mathcal{T}=2\alpha$. The intersection of the future light cone of A and the
past light cone of B is the causal diamond (shaded region in green). The
boundary of this diamond region acts as a horizon for a diamond observer.

Given the physical relevance of this result, it is worthwhile to explore other
derivations of the thermality of the state of the finite-lifetime observer. In
this paper, we focus on a derivation using the open quantum systems approach,
where the diamond observers are treated as the system and the scalar field
configuration is treated as the environment. This derivation is important for
the following three purposes: (i) it illustrates the role of decoherence in
determining the diamond temperature, (ii) it paves the path for designing
entanglement harvesting protocols between two diamond observers, and (iii) it
can be helpful in simulating the evolution of the system in a quantum
computer. The role of decoherence in determining the thermality of a system
has been previously illustrated in the literature for accelerated observers
and black holes benatti ; scully ; yu-zhang-1 . In our case, we show that the
system-environment interaction can lead to thermality for a static observer in
a causal diamond. This approach also shows the importance of the role played
by tracing out the environment degrees of freedom to generate a reduced
thermal density matrix for the diamond observer. On the other hand, the
interaction of two observers with the same environment can lead to enhancement
or destruction of entanglement between the two systems. The open quantum
systems approach offers a more direct way to evaluate the evolution of the
entanglement between the two particles, as has been illustrated in a number of
papers benatti ; huyu-1 ; huyu-2 ; huyu-3 ; yu-zhang-2 ; yu-zhang-3 . Open
quantum systems have also been simulated successfully in quantum computers,
and some efficient quantum algorithms have been used to solve specific cases
of the Lindblad equation oqs-qc-1 ; oqs-qc-2 ; oqs-qc-3 ; oqs-qc-4 .
Therefore, it is worth studying the open quantum system approach for causal
diamonds as it relates to the feasibility of simulating it in a quantum
computer in the near future.

For an accelerated system, the open quantum system approach has been used for
a broad class of Hamiltonians and interactions benatti ; yu-zhang-1 . For the
purposes of our paper, we consider a similar approach for causal diamonds and
illustrate the mechanism of decoherence for the case where the Hamiltonian is
directed along the $z$-direction and the interaction between the diamond
observer and the field is a monopole interaction with a single scalar field.
In addition, we define a coordinate system suitable for the diamond observer.
There are a few variants of the diamond coordinates that have appeared in the
literature over the years martinetti-2 ; jacobson ; su-ralph-1 . In this
article, we provide a general method of defining the diamond coordinates using
conformal transformations, which is more direct than the one used in the
original derivation by Martinetti and Rovelli martinetti-1 ; martinetti-2 ,
and can be traced back to similar forms in the literature jacobson ; casini .

This paper is organized as follows. In Sec. II, we discuss the generic
procedure for defining the diamond coordinates, as well as particular cases,
including the choice used by Martinetti and Rovelli; and we consider the
trajectory of a static observer in the diamond coordinates. In Sec. III, we
define our model of an Unruh-DeWitt detector. In Sec. IV, we briefly outline
the open quantum systems approach, apply the Lindblad equation to our system,
and find the evolution of the density matrix of the system. In Sec. V, we
determine the steady-state density matrix of the system and discuss its
thermal nature. We conclude the paper with a discussion of our findings and
directions for future work in Sec. VI. The appendices show how to evaluate the
integral critical to thermality and the details of a generalized comparative
framework for the Bloch-vector form of the Lindblad equation.

##  II Diamond Coordinates

In the introduction, we defined the causal diamond to be the region bounded by
the intersection of the past light cone of the death event and the future
light cone of the birth event of a finite-lifetime observer (Fig. 1). In this
section, we are going to introduce a generic procedure to define diamond
coordinates, namely, coordinate systems suitable for the associated diamond
observer with a finite lifetime. Let $(\xi,\eta)$ denote diamond coordinates
that cover the whole range of the causal diamond region. Such coordinates are
required for the quantization of the field and to describe any causally
accessible event from the observer’s perspective. However, this coordinate
chart is not unique, as will be obvious from the procedure outlined below; in
effect, in the literature, there exist a few alternative versions of diamond
coordinates. Thus, we propose an underlying principle for defining such a
coordinate chart: the existence of a nonunique conformal mapping between the
right Rindler wedge $R\coloneqq\\{(x,t):|t|\leq x\text{ and }x\geq 0\\}$ and
the diamond region $D\coloneqq\\{(x,t):|t|+|x|\leq\alpha\\}$, where $2\alpha$
is the lifetime of the observer. In what follows, we will refer to $2\alpha$
as the size of the diamond; and will denote the $(x,t)$ points in Minkowski
space restricted to region $R$ as $(x_{\\!{}_{R}},t_{\\!{}_{R}})$, and the
$(x,t)$ points restricted to the diamond region as
$(x_{\\!{}_{D}},t_{\\!{}_{D}})$. A visual representation of these regions can
be seen in Fig. 2.

Figure 2: (a) The shaded region of the (1+1)-dimensional Minkowski spacetime
shows the right Rindler wedge, which we call $R$. (b) The diamond-shaped
shaded region $D$ centered at the origin is the causal diamond for a static
observer at the origin with finite lifetime from $-\alpha$ to $\alpha$.

Once such a mapping between
$(x_{\\!{}_{R}},t_{\\!{}_{R}})\leftrightarrow(x_{\\!{}_{D}},t_{\\!{}_{D}})$ is
established, we can use the relation between the Rindler wedge $R$ and Rindler
coordinates to define a coordinate chart $(\xi,\tau)$ suitable for the diamond
observer,

$\tau=\alpha\tanh^{-1}\left(\frac{t_{\\!{}_{R}}}{x_{\\!{}_{R}}}\right)\qquad\qquad\xi=\frac{\alpha}{2}\ln\left[\alpha^{-2}(x_{\\!{}_{R}}^{2}-t_{\\!{}_{R}}^{2})\right]\;,$
(1)

where $\xi,\tau\in(-\infty,\infty)$, i.e. they cover the whole
$\mathbb{R}^{2}$ plane. Equation (1) gives the inverse transformation of the
relation between the Rindler coordinates and Minkowski coordinates,

$t_{\\!{}_{R}}=\alpha\,e^{\xi/\alpha}\sinh(\tau/\alpha)\;,\qquad\qquad
x_{\\!{}_{R}}=\alpha\,e^{\xi/\alpha}\cosh(\tau/\alpha)\;.$ (2)

Here, $\xi=$ constant denotes a uniformly accelerated observer with
acceleration $(\alpha\,e^{\xi/\alpha})^{-1}$. The sequence of steps above
establishes a mapping from the diamond region to the $\mathbb{R}^{2}$ plane
via the path
$(x_{\\!{}_{D}},t_{\\!{}_{D}})\rightarrow(x_{\\!{}_{R}},t_{\\!{}_{R}})\rightarrow(\xi,\tau)$.
The $(\xi,\tau)$ coordinate system is suitable for the diamond observer
because each possible value of $(\xi,\tau)$ is connected to a unique spacetime
point in the diamond via this mapping. For future convenience in Sec. III, we
rescale the time coordinate $\tau$ in $(\xi,\tau)$ by a factor of 1/2, i.e.,
we define $\eta=\tau/2$. This rescaling is one-to-one and the new coordinate
chart $(\xi,\eta)$ also covers the $\mathbb{R}^{2}$ plane. In the new
coordinate system we have

$\eta=\frac{\alpha}{2}\tanh^{-1}\left(\frac{t_{\\!{}_{R}}}{x_{\\!{}_{R}}}\right)\qquad\qquad\xi=\frac{\alpha}{2}\ln\left[\alpha^{-2}(x_{\\!{}_{R}}^{2}-t_{\\!{}_{R}}^{2})\right]\;,$
(3)

The critical step in the definition of the diamond coordinates is the
conformal mapping between the regions $R$ and $D$. The nonuniqueness of this
mapping is the reason for the appearance of different diamond coordinate
charts for $D$ in the literature. In the following, we first provide a generic
procedure to generate such diamond coordinate charts, and we consider the
simplest possible choice that we will later use for the open quantum systems
approach. In addition, we will show that this general procedure also yields
the coordinate chart used by Martinetti and Rovelli martinetti-1 ;
martinetti-2 .

###  II.1 General procedure for the conformal mapping

The building blocks for defining the mapping between $D$ and $R$ are the three
following conformal transformations:

  1. 1.

Special conformal transformation $K(\rho)$ defined as

$K(\rho)\;x^{\mu}=\frac{x^{\mu}-b^{\mu}(x\cdot x)}{1-2(b\cdot x)+(b\cdot
b)(x\cdot x)}$ (4)

where $b^{\mu}=(0,-\rho,0,0)$ is directed along the $x$-axis in four-
dimensional spacetime. The dot product represents the inner product in the
Minkowski space with metric signature $(-,+,+,+)$. The special conformal
transformation $K(\rho)$ maps the right Rindler wedge $R$ to a diamond of size
$1/\rho$ centered at $(1/2\rho,0)$ in the $(x,t)$ plane. In general, this type
of special conformal transformation leaves the structure of the light cone
invariant and maps a diamond region into another diamond region.

  2. 2.

Scaling transformation $\Lambda(l)$ defined as:

$\Lambda(l)\;x^{\mu}=lx^{\mu}\;.$ (5)

This transformation is useful to rescale the size of the diamond. Applying
this transformation on a diamond of size $1/\rho$ would yield a diamond of
size $l/\rho$.

  3. 3.

Spatial translation $T(a)$ defined as

$T(a)\;x=x+a\;.$ (6)

This transformation can be used to bring the center of the diamond to the
origin.

The general procedure is a composition of these building blocks. The first
step is to map the right wedge $R$, which can be thought as an infinite-sized
diamond, to a finite-sized diamond using the special conformal transformation
$K(\rho)$. Once the diamond is brought to a finite size, one can use rescaling
and translation as intermediate steps to bring the diamond to the desired size
and center it at the origin. One can choose any finite value of $\rho$ for the
special conformal transformation and use multiple rescaling and translation to
get the desired diamond size. This gives rise to different coordinate charts.
Here, one should note that the generators of the three transformations
$K(\rho)$, $\Lambda(l)$, and $T(a)$ do not commute. In fact, the generators
satisfy the $sl(2,\mathbb{R})$ algebra. So, changing the order of the
transformations will lead to a different mapping.

The simplest way jacobson  to map the region $R$ to a diamond $D$ of size
$2\alpha$ is to use the composite map

$T(-\alpha)\circ K\left(\frac{1}{2\alpha}\right)\;.$ (7)

Here, one maps the region $R$ to a diamond of the desired size $2\alpha$ and
then use translation to bring the center of the diamond to the origin. The
scaling transformation is not used in this case as we directly map $R$ to a
diamond of the desired size. One can use the expression for $K(\rho)$ and
$T(a)$ to obtain the mapping between
$(x_{\\!{}_{R}},t_{\\!{}_{R}})\rightarrow(x_{\\!{}_{D}},t_{\\!{}_{D}})$
explicitly:

$\displaystyle t_{\\!{}_{D}}$
$\displaystyle=\frac{t_{\\!{}_{R}}}{1+(x_{\\!{}_{R}}/\alpha)+(x_{\\!{}_{R}}^{2}-t_{\\!{}_{R}}^{2})/4\alpha^{2}}\;,$
(8) $\displaystyle x_{\\!{}_{D}}$
$\displaystyle=\frac{(x_{\\!{}_{R}}^{2}-t_{\\!{}_{R}}^{2})/4\alpha-\alpha}{1+(x_{\\!{}_{R}}/\alpha)+(x_{\\!{}_{R}}^{2}-t_{\\!{}_{R}}^{2})/4\alpha^{2}}\;.$
(9)

However, one needs the inverse transformation
$(x_{\\!{}_{D}},t_{\\!{}_{D}})\rightarrow(x_{\\!{}_{R}},t_{\\!{}_{R}})$ to
define the diamond coordinates in Eq. (3). The inverse transformation can be
obtained easily using the inverted composite map $K^{-1}(1/2\alpha)\circ
T(\alpha)$, where $K^{-1}(1/2\alpha)=K(-1/2\alpha)$, which yields the
following coordinate chart,

$\displaystyle t_{\\!{}_{R}}$
$\displaystyle=\frac{4\alpha^{2}t_{\\!{}_{D}}}{(x_{\\!{}_{D}}+\alpha)^{2}-t_{\\!{}_{D}}^{2}-4\alpha
x_{\\!{}_{D}}}$ (10) $\displaystyle x_{\\!{}_{R}}$
$\displaystyle=\frac{4\alpha^{2}(x_{\\!{}_{D}}+\alpha)-2\alpha[(x_{\\!{}_{D}}+\alpha)^{2}-t_{\\!{}_{D}}^{2}]}{(x_{\\!{}_{D}}+\alpha)^{2}-t_{\\!{}_{D}}^{2}-4\alpha
x_{\\!{}_{D}}}\;.$ (11)

In the remainder of the paper, we are going to use this particular mapping to
define the diamond coordinates due to its geometric simplicity. However, it is
obvious that this is only one particular way to define the diamond
coordinates. For completeness, we will also rederive the original diamond
coordinates by Martinetti and Rovelli below.

Figure 3: This figure shows the composite map, Eq. (7). The right Rindler
wedge $R$ can be thought of as a diamond of infinite size, i.e., a diamond
whose edges are taken to infinity.

###  II.2 Diamond coordinates by Martinetti and Rovelli

For comparison purposes, in Refs. martinetti-1 ; martinetti-2  the authors
obtained the conformal mapping between $R$ and $D$ using a composition of
scaling, relativistic inversion, translation, and reflection. However, the
transformations used in Refs. martinetti-1 ; martinetti-2  can be
reconstructed by using the building blocks described in Sec. II.1. In effect,
the composite map we need is

$\Lambda(\alpha)\circ T(-1)\circ\Lambda(2)\circ K(1)\;.$ (12)

One can easily verify that the transformation acting on coordinates
$(x_{\\!{}_{R}},t_{\\!{}_{R}})$ yields

$\displaystyle t_{\\!{}_{D}}$
$\displaystyle=\frac{2t_{\\!{}_{R}}\alpha}{1+2x_{\\!{}_{R}}+(x_{\\!{}_{R}}^{2}-t_{\\!{}_{R}}^{2})}$
(13) $\displaystyle x_{\\!{}_{D}}$
$\displaystyle=\frac{\left[-1+(x_{\\!{}_{R}}^{2}-t_{\\!{}_{R}}^{2})\right]\alpha}{1+2x_{\\!{}_{R}}+(x_{\\!{}_{R}}^{2}-t_{\\!{}_{R}}^{2})}\;,$
(14)

which agrees with the transformation obtained by Martinetti and Rovelli. In
this case, we see that the region $R$ is first mapped into a diamond of size
$1$, and then scaling and translation are used to bring the diamond to the
desired size and centered at origin (Fig. 4). The inverse transformation can
be obtained by inverting the composite map. This inverted mapping is the most
common form used in the literature su-ralph-1 ; arzano . However, we do not
need to go into detail about the inverse map, as we are going to use Eqs.
(10)–(11) for this article.

Figure 4: This figure illustrates the composite map Eq. (12) that yields the
transformation obtained in Refs. martinetti-1 ; martinetti-2 .

###  II.3 Static observer with a finite lifetime

Using Eqs. (3) and (10)–(11), we can now define the diamond coordinates as
follows

$\displaystyle\eta$
$\displaystyle=\frac{\alpha}{2}\tanh^{-1}\left(\frac{4\alpha^{2}t_{\\!{}_{D}}}{4\alpha^{2}(x_{\\!{}_{D}}+\alpha)-2\alpha[(x_{\\!{}_{D}}+\alpha)^{2}-t_{\\!{}_{D}}^{2}]}\right)$
(15) $\displaystyle\xi$
$\displaystyle=\frac{\alpha}{2}\ln\left(\frac{16\alpha^{4}t_{\\!{}_{D}}^{2}-\\{4\alpha^{2}(x_{\\!{}_{D}}+\alpha)-2\alpha[(x_{\\!{}_{D}}+\alpha)^{2}-t_{\\!{}_{D}}^{2}]\\}^{2}}{\alpha^{2}[(x_{\\!{}_{D}}+\alpha)^{2}-t_{\\!{}_{D}}^{2}-4\alpha
x_{\\!{}_{D}}]^{2}}\right)\;.$ (16)

While this transformation looks complicated, we only need the trajectory for
the static observer in the diamond coordinates for our discussion. This is
because, in our open quantum systems model, the system is the finite-lifetime
static detector sitting at the origin $(x_{\\!{}_{D}}=0)$. Using the mapping
in Eq. (15), one can find the relation between the diamond proper time $\eta$
for the diamond observer and the Minkowski time $t_{\\!{}_{D}}$.

$\eta=\frac{\alpha}{2}\tanh^{-1}\left(\frac{2\alpha
t_{\\!{}_{D}}}{\alpha^{2}+t_{D}^{2}}\right)\;.$ (17)

So, the trajectory of a diamond observer in terms of the diamond coordinates
can be written as:

$\displaystyle t_{\\!{}_{D}}$
$\displaystyle=\alpha\tanh\left(\frac{\eta}{\alpha}\right)\;,$ (18a)
$\displaystyle x_{\\!{}_{D}}$ $\displaystyle=0\;.$ (18b)

In this section, we have clarified the procedure of defining the diamond
coordinates. We have also obtained the trajectory of the diamond observer in
these coordinates, and established the relation between the diamond proper
time $\eta$ and the Minkowski time $t_{\\!{}_{D}}$. Our objective now is to
treat the diamond observer and the scalar field as a composite system. This
requires specific details about the physics of the diamond observer, which we
model as a finite-lifetime two-level system interacting with a scalar field.
This is the subject of the next section.

##  III Unruh-DeWitt detector with scaled energy

We can now define a finite-lifetime two-level detector as the diamond observer
which interacts with a scalar field. A similar construction was used
previously by Unruh and DeWitt for an accelerated detector in the region $R$
to find its thermal excitation probability birrell-davies . These detectors
are therefore called Unruh-DeWitt detectors. We are going to use a similar
two-level detector as the diamond observer, but with a slight modification. In
this case, the Hamiltonian of the two-level system is scaled by a factor su-
ralph-1 ; olson-ralph  such that the energy gap between the two levels is
fixed with respect to the proper time of the detector, which is the diamond
time $\eta$. For this purpose, we define the Hamiltonian of the two-level
system as

$H_{0}^{\prime}=\frac{H_{0}}{1-\frac{t^{2}}{\alpha^{2}}}\;,$ (19)

where

$H_{0}=\frac{\omega_{0}}{2}\sigma_{z}$ (20)

is the usual Hamiltonian of a two-level system, with energy gap $\omega_{0}$.
A remark about notation is in order. The time $t$ appearing in the scale
factor in Eq. (19) is strictly the Minkowski time $t_{D}$, but we will use $t$
instead of $t_{D}$ for the sake of simplicity. This is mainly because $t_{D}$
is still the Minkowski time, albeit restricted to $[-\alpha,\alpha]$, and we
are not going to use $t_{R}$ in the remainder of this paper—thus eliminating
the source of any confusion.
Now we introduce the interaction Hamiltonian $H_{I}^{\prime}$, which is
usually taken to be a monopole or dipole interaction with the scalar field. In
this case, to keep things simple, we will only consider the monopole
interaction. The interaction Hamiltonian is linear in both the field and
detector operator,

$H_{I}^{\prime}=m\sigma_{x}\otimes\Phi\;.$ (21)

Here, $m$ stands for the coupling constant; and we take the monopole moment
operator to be the Pauli matrix $\sigma_{x}$, which can be written as the
addition of the lowering and raising operators flipping the state of the two-
state system. The scalar field $\Phi$ can be decomposed into field modes
satisfying the Klein-Gordon equation. In general,

$\Phi(x,t)=\sum_{k}\left[u_{k}(x,t)\,a_{k}+u^{*}_{k}(x,t)\,a_{k}^{\dagger}\right]\;,$
(22)

where $u_{k}$ are the field modes obtained by solving the wave equation in a
particular spacetime background. It now can be seen that the state
$|\Psi\rangle$ of the detector evolves according to the Schrödinger equation

$\frac{d|\Psi\rangle}{dt}=\left[\frac{H_{0}}{1-\frac{t^{2}}{\alpha^{2}}}+H_{I}^{\prime}\right]|\Psi\rangle\;,$
(23)

in the Minkowski time $t$. However, for the detector, the proper time is the
diamond time $\eta$. Thus, in its own reference frame, the Schrödinger
equation takes the form

$\frac{d|\Psi\rangle}{d\eta}=\bigl{(}H_{0}+H_{I}\bigr{)}|\Psi\rangle\;,$ (24)

where

$H_{I}=\frac{H_{I}^{\prime}}{\cosh^{2}(\eta/\alpha)}$ (25)

is the new scaled interaction Hamiltonian. We now see the effect of the scale
factor introduced in the original Hamiltonian $H_{0}^{\prime}$: it reduces to
the Hamiltonian of a two-level system in the reference frame of the diamond
observer. We will now use this modified interaction Hamiltonian to find the
evolution of the finite-lifetime detector.

##  IV Lindblad equation: open quantum systems approach

The state of the detector can be described by the density matrix of the two-
level system. However, the system (the detector) here is not isolated as it
interacts with the scalar field in a flat spacetime. Due to the interaction,
we can model the evolution of the detector’s state under the paradigm of open
quantum systems. We can treat the field configuration here as the environment.
The field is assumed to be in the Minkowski vacuum which is the natural choice
of vacuum in this case as the background geometry is flat. We consider the
system-environment interaction to be weak, so that the Markovian approximation
holds. Under the Markovian approximation, the evolution of the reduced density
matrix of the system ($\rho_{\\!{}_{S}}$) is governed by the Lindblad equation
breuer-book

$\frac{d\rho_{\\!{}_{S}}}{d\eta}=-i[H_{\rm
eff},\rho_{\\!{}_{S}}(\eta)]+\mathcal{L}[\rho_{\\!{}_{S}}(\eta)]\;.$ (26)

Here, we describe the evolution in terms of the diamond proper time $\eta$.

The right-hand side of the Lindblad equation (26) consists of two distinctly
different terms: the unitary evolution with respect to a effective Hamiltonian
$H_{\rm eff}$ and the dissipative part $\mathcal{L}[\rho_{\\!{}_{S}}(\eta)]$,
which is known as the Lindbladian operator. The effective Hamiltonian is of
the form

$H_{\rm eff}=\frac{\Omega}{2}\sigma_{z}\;,$ (27)

where $\Omega$ is a renormalized shifted frequency that has the same origin as
the Lamb shift benatti ; breuer-book . In subsequent discussions, we do not
need the explicit expression for $\Omega$. The Lindbladian operator
$\mathcal{L}[\rho_{\\!{}_{S}}(\eta)]$ is defined for a general interaction of
the form $H_{I}=\sum_{\alpha}S_{\alpha}\otimes B_{\alpha}$, where $S_{\alpha}$
are operators in the Hilbert space of the system and $B_{\alpha}$ are
operators in the Hilbert space of the environment (bath). Then,
$\mathcal{L}[\rho_{\\!{}_{S}}(\eta)]$ can be written the following form,

$\mathcal{L}[\rho_{\\!{}_{S}}(\eta)]=\sum_{\omega,\alpha,\beta}\gamma_{\alpha\beta}(\omega)\left(S_{\beta}(\omega)\rho_{\\!{}_{S}}S_{\alpha}^{\dagger}(\omega)-\frac{1}{2}\left\\{S_{\alpha}^{\dagger}(\omega)S_{\beta}(\omega),\rho_{\\!{}_{S}}\right\\}\right)\;,$
(28)

where $\\{,\\}$ stands for the anticommutator and the factors
$\gamma_{\alpha\beta}(\omega)$ encode the contribution of the environment. The
reduced density matrix $\rho_{\\!{}_{S}}$ of the system can be derived from
the system-environment density matrix by averaging over the degrees of freedom
of the field. This averaging leads to the factors
$\gamma_{\alpha\beta}(\omega)$.

In our case, the interaction Hamiltonian only has one term: $\displaystyle
H_{I}=m\sigma_{x}\otimes\frac{\Phi}{\cosh^{2}(\eta/\alpha)}$. Therefore, the
double sums over $\alpha$ and $\beta$ are superfluous in the expression of the
Lindblad operator, and Eq. (28) simplifies to

$\mathcal{L}[\rho_{\\!{}_{S}}(\eta)]=\sum_{\omega}\gamma(\omega)\left(S(\omega)\rho_{\\!{}_{S}}S^{\dagger}(\omega)-\frac{1}{2}\left\\{S^{\dagger}(\omega)S(\omega),\rho_{\\!{}_{S}}\right\\}\right)\;.$
(29)

Now, the field is in the Minkowski vacuum, so that its density matrix is
$\rho_{\\!{}_{B}}=|0_{M}\rangle\langle 0_{M}|$. In addition, $\gamma(\omega)$
is the Fourier transform

$\gamma(\omega)=\int_{-\infty}^{\infty}d\Delta\eta\;e^{i\omega\Delta\eta}\;\langle
0_{M}|B(\eta)B(\eta^{\prime})|0_{M}\rangle$ (30)

of the two-point correlation function

$G^{+}(\Delta\eta)\equiv\langle
0_{M}|B(\eta)B(\eta^{\prime})|0_{M}\rangle=\langle
0_{M}|\frac{\Phi(\eta)\Phi(\eta^{\prime})}{\cosh^{2}(\eta/\alpha)\cosh^{2}{(\eta^{\prime}/\alpha)}}|0_{M}\rangle\;,$
(31)

which has an argument $\Delta\eta=\eta-\eta^{\prime}$ that only depends on the
difference of the values of the two proper times involved in the Wightman
function. This effectively amounts to using $\eta$ as the argument of $G^{+}$
and setting $\eta^{\prime}=0$ without any loss of generality. This symmetry
property will be verified in the next section.

In order to find the solutions to the Lindblad equation, we still need to
define the factors $S_{\alpha}(\omega)$. For the case being analyzed, and
omitting the subscripts, $S(\omega)$ is defined using projection operators
$\Pi(\epsilon)=|\epsilon\rangle\langle\epsilon|$, where
$\\{|\epsilon\rangle\\}$ defines an eigenbasis of the Hamiltonian $H_{0}$ of
the system; thus,

$S(\omega)=\sum_{\epsilon^{\prime}-\epsilon=\omega}\Pi(\epsilon)\;S\;\Pi(\epsilon^{\prime})\;.$
(32)

For the two-level system, the two energy eigenvalues are
$\epsilon=\pm\omega_{0}/2$ and the eigenbasis of $H_{0}$ consists of the
eigenvectors $|\pm\rangle_{z}$ of the $\sigma_{z}$ operator; with this
notation, $|\epsilon\rangle=|\operatorname{sgn}(\epsilon)\rangle_{z}$, but we
will use the standard shorthand $|\pm\rangle$ instead of $|\pm\rangle_{z}$ or
$|\epsilon\rangle$. In addition, when writing the frequency components
$S(\omega)$, one considers all possible transitions in the two-level system,
according to Eq. (32), i.e.,
$\omega=\epsilon^{\prime}-\epsilon=\pm\omega_{0},0$. Moreover, for the
interaction Hamiltonian (21), the operator $S$ is just $\sigma_{x}$. (For a
generalization of this choice, see Appendix B.) Therefore, the only three
relevant operators $S(\omega)$ are given by

$\displaystyle S(+\omega_{0})$
$\displaystyle=|-\rangle\langle-|\sigma_{x}|+\rangle\langle+|=|-\rangle\langle+|$
(33) $\displaystyle S(-\omega_{0})$
$\displaystyle=|+\rangle\langle+|\sigma_{x}|-\rangle\langle-|=|+\rangle\langle-|$
(34) $\displaystyle S(0)$ $\displaystyle=0\;.$ (35)

In short, we now have all the ingredients to solve the Lindblad equation. One
method for its solution, which is ideally tailored for a two-level system,
consists in representing the reduced density matrix $\rho_{S}(\eta)$ as a
Bloch vector $|\rho(\eta)\rangle$ with components $\rho_{i}(\eta)$ ($i=1,2,3$)
defined via the generic expansion for operators in terms of the basis of the
three Pauli matrices and the unit matrix,

$\rho_{\\!{}_{S}}=\frac{1}{2}\left(\mathit{I}+\sum_{i=1}^{3}\rho_{i}\sigma_{i}\right)\;.$
(36)

The prefactor of one half guarantees the unit-trace condition of the density
matrix, and otherwise defines conventionally the components $\rho_{i}(\eta)$.
The equation for the components $\rho_{i}(\eta)$ of the Bloch vector can be
found by expanding all the operators from Eqs. (26), (27), and (36) in this
basis. Then, the Lindblad equation reduces a Schrödinger-like vector equation
for the three-component Bloch vector $|\rho\rangle$ of the form

$\frac{d|\rho\rangle}{d\eta}=-2\mathcal{H}|\rho\rangle+|n\rangle\;,$ (37)

where the operator $\mathcal{H}$ has the $3\times 3$ matrix representation

$\mathcal{H}=\begin{pmatrix}A/2&\Omega/2&0\\\ -\Omega/2&A/2&0\\\
0&0&A\end{pmatrix}\;,$ (38)

and $|n\rangle$ is the constant vector $(0,0,-2B)$. Thus, in matrix form,

$\begin{pmatrix}\dot{\rho}_{1}\\\ \dot{\rho}_{2}\\\
\dot{\rho}_{3}\end{pmatrix}=-2\mathcal{H}\begin{pmatrix}\rho_{1}\\\
\rho_{2}\\\ \rho_{3}\end{pmatrix}+\begin{pmatrix}0\\\ 0\\\
-2B\end{pmatrix}\;.$ (39)

In these equations,

$A=\frac{1}{2}[\gamma(\omega_{0})+\gamma(-\omega_{0})]\;\;;\;\;B=\frac{1}{2}[\gamma(\omega_{0})-\gamma(-\omega_{0})]\;.$
(40)

In the next section, we are going to find the system’s steady-state density
matrix and investigate its properties, which determine the thermal character
of the system.

##  V Steady-state reduced density matrix and diamond temperature

In the evolution of the system interacting with the environment, a steady
state is achieved asymptotically as $\eta\rightarrow\infty$. At the steady
state, the left-hand side of Eq. (37) becomes zero. This leads to the
equilibrium values of the coefficients $\rho_{i}$ which we denote
$\rho_{i}(\infty)$; the corresponding Bloch vector
$|\rho_{S}\rangle_{\\!{}_{\infty}}$ is

$|\rho_{\\!{}_{S}}\rangle_{\infty}=\frac{1}{2}\mathcal{H}^{-1}|n\rangle=-\frac{B}{A}\begin{pmatrix}0\\\
0\\\ 1\end{pmatrix}\;.$ (41)

Then, there is only one nonzero component of the density matrix in the Bloch
representation (corresponding to the $z$ axis), which is due to the form of
the initial Hamiltonian $H_{0}$ with a similar decomposition.

Moreover, one can further analyze the evolution of the density matrix by
explicitly solving the linear equation (37) by direct integration,

$|\rho(\eta)\rangle=e^{-2\mathcal{H}\eta}\,|\rho(0)\rangle+(1-e^{-2\mathcal{H}\eta})\,|\rho_{\\!{}_{\infty}}\rangle\;,$
(42)

which verifies its asymptotic approach to the form (41). This analysis can be
further extended by evaluating the exponential $e^{-2\mathcal{H}\eta}$ of the
matrix (38) of degree three by using the Cayley-Hamilton theorem, which
displays the nonunitary dissipation and decoherence behavior governed by the
thermal Boltzmann factor benatti ; yu-zhang-1 . The coefficients that govern
both the steady-state density matrix and its evolution with dissipation and
decoherence, can be found by evaluating $\gamma(\omega)$, as can be seen from
Eq. (40); this is given by the Fourier transform of the two-point correlation
function, as in Eqs. (30) and (31). The Wightman function $G^{+}$ only depends
on the difference of the values of the two proper times, i.e., it is function
of the simple argument $\Delta\eta=\eta-\eta^{\prime}$. This is explicitly
verified below. In flat spacetime, the two-point correlation function $\langle
0_{M}|\Phi({\bf x},t)\Phi({\bf x}^{\prime},t^{\prime})|0_{M}\rangle$ has the
expression

$\langle 0_{M}|\Phi({\bf x},t)\Phi({\bf
x}^{\prime},t^{\prime})|0_{M}\rangle=-\frac{1}{4\pi^{2}}\frac{1}{|t-t^{\prime}-i\epsilon|^{2}-|{\bf
x}-{\bf x}^{\prime}|^{2}}\;.$ (43)

For a finite-lifetime diamond detector, the trajectory of the detector is
defined by the relation between Minkowski time $t$ and diamond time $\eta$, as
the detector is located at the origin (${\bf x}=0$) for the entirety of its
existence. In Eq. (18), we found that the two times are related by
$t=\alpha\tanh\left(\eta/\alpha\right)$. Replacing this in Eq. (43), we get

$G^{+}(\eta-\eta^{\prime})=\langle
0_{M}|\frac{\Phi(\eta)\Phi(\eta^{\prime})}{\cosh^{2}(\eta/\alpha)\cosh^{2}(\eta^{\prime}/\alpha)}|0_{M}\rangle=-\frac{1}{4\pi^{2}\alpha^{2}}\,\frac{1}{\sinh^{2}\left[(\eta-\eta^{\prime})/\alpha-i\epsilon\right]}\;.$
(44)

where we have absorbed a positive function of $\eta,\,\eta^{\prime},\,\alpha$
into $\epsilon$. This verifies, as expected, that the scaled two-point
function above is dependent only on the proper time interval
$\eta-\eta^{\prime}$. The value of $\gamma(\omega)$ can be computed from the
Fourier transform (30)

$\gamma(\omega)=-\frac{1}{4\pi^{2}\alpha^{2}}\int_{-\infty}^{\infty}d\eta\;e^{i\omega\eta}\;\frac{1}{\sinh^{2}\left(\eta/\alpha-i\epsilon\right)}=\frac{\omega}{2\pi}\frac{1}{1-e^{-\pi\alpha\omega}}\;.$
(45)

This important integral is the key that yields a thermal density matrix, as
shown below; its evaluation requires the $i\epsilon$ prescription that places
the pole corresponding to $\eta=0$ in the upper half plane, as is easily shown
by contour integration: see Appendix A for the evaluation of the integral.

Using Eq. (45), the only non-zero coefficient in the expansion of the density
matrix in Eq. (36) reduces to

$\rho_{\\!{}_{3}}(\infty)=-\frac{B}{A}=\frac{\gamma(-\omega_{0})-\gamma(\omega_{0})}{\gamma(\omega_{0})+\gamma(-\omega_{0})}=\frac{1-e^{\pi\alpha\omega_{0}}}{1+e^{\pi\alpha\omega_{0}}}\;.$
(46)

The steady-state density matrix then can be written using the expansion of Eq.
(36),

$\rho_{\\!{}_{S}}(\infty)=\begin{pmatrix}\displaystyle\frac{1}{1+e^{\pi\alpha\omega_{0}}}&0\\\
0&\displaystyle\frac{e^{\pi\alpha\omega_{0}}}{1+e^{\pi\alpha\omega_{0}}}\end{pmatrix}=\frac{e^{-\pi\alpha
H_{0}}}{{\rm Tr}\left[e^{-\pi\alpha H_{0}}\right]}\;.$ (47)

This expression has the same form as the density matrix of the two-level
system kept in a thermal bath with temperature $T=1/\beta$,

$\rho_{\\!{}_{S}}=\frac{e^{-\beta H_{0}}}{{\rm Tr}\left[e^{-\beta
H_{0}}\right]}\;,$ (48)

if one identifies the temperature with the quantity $T=1/(\pi\alpha)$.
Clearly, the steady-state density operator of the diamond observer has the
form of a thermal density matrix. Thus, the diamond observer experiences a
thermal bath in the Minkowski vacuum with temperature inversely proportional
to its lifetime,

$T_{D}=\frac{1}{\pi\alpha}=\frac{2}{\pi\mathcal{T}}\;.$ (49)

In conclusion, Eq. (49) gives the diamond temperature $T_{D}$ su-ralph-1  in
terms of $\mathcal{T}=2\alpha$, the lifetime of the particle or the size of
the diamond (according to Minkowski coordinates).

##  VI Conclusions and outlook

In this article, we have introduced a systematic, rigorous method for defining
the diamond coordinates, which extends existing work in the literature.
Furthermore, we have used the Lindblad equation to identify the state of the
diamond observer in an open quantum systems approach. This approach provides
another meaning for the temperature of the diamond: it is the temperature of
the thermal steady-state reduced density matrix of a two-state finite-lifetime
detector with a scaled energy gap. Such methodology has obvious advantages, as
the reduced density matrix provides a full characterization of the state of
the system. Moreover, the open quantum systems approach opens up new
directions for future work. First, it leads to the quantum information
theoretic description of the system where one can cast the Lindblad equation
in terms of Kraus operators nielsen-chuang ; kraus-lindblad . Second, it
suggests the feasibility of simulating the system-environment interaction in
the diamond in a quantum computer. Third, it provides a methodology for
modeling the evolution of the correlation between two-level finite-lifetime
detectors, possibly leading to an entanglement harvesting protocol. Finally,
this framework has great potential for revisiting a broad range of problems at
the intersection of general relativity and quantum field theory.

###### Acknowledgements.

H.E.C. acknowledges support by the University of San Francisco Faculty
Development Fund. This material is based upon work supported by the Air Force
Office of Scientific Research under Grant No. FA9550-21-1-0017 (C. R. O. and
A.C.).

##  Appendix A Evaluation of integral in Eq. (45)

In this appendix, we show the steps to evaluate the Fourier transform of the
Wightman function given in Eq. (45):

$\mathcal{I}=\int_{-\infty}^{\infty}d\eta\;\;\frac{e^{i\omega\eta}}{\sinh^{2}(\eta/\alpha-i\epsilon)}\;.$
(50)

We start by a redefinition of variables: $z=\eta/\alpha$ and $q=\omega\alpha$.
So, we can now recast Eq. (50) in the following way:

$\hat{\mathcal{I}}\equiv\frac{\mathcal{I}}{\alpha}=\int_{-\infty}^{\infty}dz\;\;\frac{e^{iqz}}{\sinh^{2}(z-i\epsilon)}\equiv\int_{-\infty}^{\infty}dz\;\;e^{iqz}f(z)\;.$
(51)

The integral $\hat{\mathcal{I}}$ can now be evaluated using complex contour
integration. Using Jordan’s Lemma, we can close the contour in the upper-half
complex z-plane since $q>0$,

$\int_{-\infty}^{\infty}dz\;\;e^{iqz}f(z)=\oint_{C}\,dz\;\;e^{iqz}f(z)=2\pi
i\sum_{k}{\rm Res}\,(z_{k})\;,$ (52)

where $C$ is the semicircular contour with infinite radius closed in the
upper-half $z$ plane (see Fig. 5). Res $(z_{k})$ is the residue of the
function $e^{iqz}f(z)$ for the pole at $z=z_{k}$.

Figure 5: The semicircular contour C, used to evaluate Eq. (51), with radius
$R\rightarrow\infty$, closed in the upper-half plane.

So, to evaluate the integral in Eq. (51), we need to find the poles of the
integrand in the upper-half z-plane, and calculate their corresponding
residues. The integrand has second-order poles at the zeros of the function
$\sinh^{2}(z-i\epsilon)$, i.e.,

$\sinh^{2}(z_{k}-i\epsilon)=0\qquad\implies\qquad z_{k}=i\pi k+i\epsilon\;,$
(53)

where $k\in\mathbb{Z}$ (all integers). Only the poles with integers $k\geq 0$
are enclosed by the contour. The residues are

${\rm Res}\,(z_{k})=iqe^{-k\pi q}\;.$ (54)

Using Eq. (52), we can now evaluate the integral in Eq. (51):

$\hat{\mathcal{I}}=2\pi i(iq)\sum_{k=0}^{\infty}e^{-k\pi q}=-\frac{2\pi
q}{1-e^{-\pi q}}\;.$ (55)

Restoring the original variables back, we get

$\mathcal{I}=-2\pi\omega\alpha^{2}(1-e^{\pi\omega\alpha})^{-1}\;,$ (56)

which gives us the final result of Eq. (45).

##  Appendix B Generalized comparative framework for the Bloch-Vector Form of
the Lindblad Equation

The calculation shown in the main text, Sec. III, leading to the Schrödinger-
like Bloch-vector form of the Lindblad equation (37), specializes to an
interaction Hamiltonian (21). This Hamiltonian involves a system’s operator
that is proportional to $\sigma_{x}$ alone, i.e. $S=\sigma_{x}$; or, more
generally, $S=m\sigma_{x}$, including a coupling strength $m$. In this
Appendix, we present a larger framework that is a modified (and simplified)
alternative to the one in Ref. benatti , adapted for a comparison of different
literature results using a uniform notation. This should help clarify some
possible confusion regarding the selection of numerical factors in the final
equations for $\mathcal{H}$ and $[|n\rangle]$ as needed for Eq. (37),
including those found in published references—see Eqs. (69) and (70).
Additional generalizations are possible, following the techniques of Ref.
benatti —but unlike this reference, we are only considering the case where the
effective Hamiltonian (27) of the two-level system has a quantization axis
along $z$.

The generic system’s operators can be written in the form

$S=\sum_{\mu}m_{\mu}\,\sigma_{\mu}\;,$ (57)

with a set of coupling strengths $m_{\mu}$ ($\mu=0,1,2,3$), and with the usual
conventions for the Pauli matrices:
$\sigma_{j}\equiv\sigma_{x},\sigma_{y},\sigma_{z}$, for $j=1,2,3$, and
$\sigma_{0}\equiv\mathit{I}$ for the identity matrix. In this basis, the
general resolution of the density matrix is given by Eq. (36). As discussed in
the explanation for Eqs. (32)–(35), there are only three possible transitions
in the two-level system, with $\omega=\epsilon^{\prime}-\epsilon=\pm\omega,0$,
which give the only three relevant components of $S(\omega)$:

$S_{\pm}\equiv S(\pm\omega_{0})=\left(m_{1}\pm
im_{2}\right)\,P_{\mp\pm}\;\;,\;\;\;\;\;S_{0}\equiv
S(0)=m_{0}\mathit{I}+m_{3}\sigma_{3}\;,$ (58)

with the operators $P_{\mp\pm}=|\mp\rangle\langle\pm|=\left(\sigma_{x}\mp
i\sigma_{y}\right)\\!/2$.

For the field operators, we will keep the parameters defined in Eq. (40), and
further expand them to include the transition of zero frequency, i.e.,

$A=\frac{1}{2}[\gamma(\omega_{0})+\gamma(-\omega_{0})]\;\;;\;\;B=\frac{1}{2}[\gamma(\omega_{0})-\gamma(-\omega_{0})]\;\;;\;\;C=\gamma(0)-A\;.$
(59)

The intermediate steps in the calculation of the components of Eq. (37) are as
follows. They involve repeated application of the definitions of the Pauli
matrices and their product relations
$\sigma_{j}\sigma_{k}=\delta_{jk}\mathit{I}+i\epsilon_{jkl}\sigma_{l}$ (as
well as use of the Einstein summation convention for any pair of repeated
indices). First, for the products in the first term of Eq. (29), at each
transition frequency,

$\displaystyle
S_{\pm}\,\rho\,S_{\pm}^{\dagger}=\frac{1}{2}\left(m_{1}^{2}+m_{2}^{2}\right)\left(1\pm\rho_{3}\right)\Pi_{\mp}$
(60) $\displaystyle
S_{0}\,\rho\,S_{0}^{\dagger}=\left[\frac{1}{2}\left(m_{0}^{2}+m_{3}^{2}\right)+m_{0}m_{3}\rho_{3}\right]\mathit{I}+\left[m_{0}m_{3}+\frac{1}{2}\left(m_{0}^{2}+m_{3}^{2}\right)\rho_{3}\right]\sigma_{3}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{1}{2}\left(m_{0}^{2}-m_{3}^{2}\right)\,\left(\rho_{1}\sigma_{1}+\rho_{2}\sigma_{2}\right)\;,$
(61)

where $\Pi_{\pm}$ denotes the projectors
$\Pi_{\pm}=\left(\mathit{I}\pm\sigma_{3}\right)/2$. Second, for the
anticommutators in the second term of Eq. (29), the products
$S_{\pm}^{\dagger}S_{\pm}=\left(m_{1}^{2}+m_{1}^{2}\right)\,\Pi_{\pm}$ and
$S_{0}^{\dagger}S_{0}=\left(m_{0}^{2}+m_{3}^{2}\right)\,\mathit{I}+2m_{0}m_{3}\sigma_{3}$
give

$\displaystyle\frac{1}{2}\,\left\\{S_{\pm}^{\dagger}S_{\pm},\rho\right\\}=\left(m_{1}^{2}+m_{1}^{2}\right)\,\left(\frac{1}{2}\Pi_{\pm}\pm\frac{1}{4}\rho_{3}\mathit{I}+\frac{1}{4}\rho_{j}\sigma_{j}\right)$
(62)
$\displaystyle\frac{1}{2}\,\left\\{S_{0}^{\dagger}S_{0},\rho\right\\}=\frac{1}{2}S_{0}^{\dagger}S_{0}+m_{0}m_{3}\rho_{3}\mathit{I}+\frac{1}{2}\left(m_{0}^{2}+m_{3}^{2}\right)\,\left(\rho_{1}\sigma_{1}+\rho_{2}\sigma_{2}+\rho_{3}\sigma_{3}\right)\;.$
(63)

Then, the building blocks of the Lindbladian include

$\displaystyle L_{\pm}=S_{\pm}\rho
S_{\pm}^{\dagger}-\frac{1}{2}\left\\{S_{\pm}^{\dagger}S_{\pm},\rho\right\\}=\left(m_{1}^{2}+m_{2}^{2}\right)\,\left[\mp\frac{1}{2}\sigma_{3}-\frac{1}{4}\left(\rho_{1}\sigma_{1}+\rho_{2}\sigma_{2}+2\rho_{3}\sigma_{3}\right)\right]$
(64) $\displaystyle L_{0}=S_{0}\rho
S_{0}^{\dagger}-\frac{1}{2}\left\\{S_{0}^{\dagger}S_{0},\rho\right\\}=-m_{3}^{2}\left(\rho_{1}\sigma_{1}+\rho_{2}\sigma_{2}\right)\;,$

so that

$\displaystyle{\mathcal{L}}=\gamma_{+}L_{+}+\gamma_{-}L_{-}+\gamma_{0}L_{0}$
(65)
$\displaystyle=\left(m_{1}^{2}+m_{2}^{2}\right)\,\left[\frac{1}{2}\sigma_{3}\left(-\gamma_{+}+\gamma_{-}\right)-\frac{1}{4}\left(\rho_{1}\sigma_{1}+\rho_{2}\sigma_{2}+2\rho_{3}\sigma_{3}\right)\left(\gamma_{+}+\gamma_{-}\right)\right]-\gamma_{0}m_{3}^{2}\left(\rho_{1}\sigma_{1}+\rho_{2}\sigma_{2}\right)\;,$
(66)

with the notation $\gamma_{\pm}=\gamma(\pm\omega_{0})$ and
$\gamma_{0}=\gamma(0)$. Rearranging the terms in Eq. (66),

${\mathcal{L}}=-B^{\prime}\sigma_{3}-\frac{1}{2}A^{\prime}\left(\rho_{1}\sigma_{1}+\rho_{2}\sigma_{2}+2\rho_{3}\sigma_{3}\right)-\gamma_{0}m_{3}^{2}\left(\rho_{1}\sigma_{1}+\rho_{2}\sigma_{2}\right)\;,$
(67)

where $A^{\prime}=\left(m_{1}^{2}+m_{2}^{2}\right)A$ and
$B^{\prime}=\left(m_{1}^{2}+m_{2}^{2}\right)B$.

As a result, with the resolution in terms of Pauli operators, the Lindblad
equation (26) takes the form

$\frac{1}{2}\dot{\rho}_{k}\sigma_{k}=-\frac{\Omega}{2}\,\epsilon_{3kj}\rho_{j}\sigma_{k}-B^{\prime}\sigma_{3}-\left(\frac{1}{2}A^{\prime}+m_{3}^{2}\gamma_{0}\right)\left(\rho_{1}\sigma_{1}+\rho_{2}\sigma_{2}\right)-\frac{1}{2}(2A^{\prime})\rho_{3}\sigma_{3}\;,$
(68)

where the first term on the right-hand side arises from the unitary-evolution
commutator $[H_{\rm
eff},\rho]=\bigl{[}(\Omega/2)\sigma_{3},\left(\mathit{I}+\rho_{j}\sigma_{j}\right)/2\bigr{]}=-i(\Omega/2)\,\epsilon_{3kj}\rho_{j}\sigma_{k}$.
Also,
$\frac{1}{2}A^{\prime}+m_{3}^{2}\gamma_{0}=\left[\frac{1}{2}\left(m_{1}^{2}+m_{2}^{2}\right)+m_{3}^{2}\right]A+m_{3}^{2}C$.
Then, the linear combinations of Eq. (68) in the Pauli-matrix basis lead to
the components satisfying the Bloch-vector form of the Lindblad equation

$\begin{pmatrix}\dot{\rho}_{1}\\\ \dot{\rho}_{2}\\\
\dot{\rho}_{3}\end{pmatrix}=-2\mathcal{H}\begin{pmatrix}\rho_{1}\\\
\rho_{2}\\\ \rho_{3}\end{pmatrix}+\begin{pmatrix}0\\\ 0\\\
-2\left(m_{1}^{2}+m_{2}^{2}\right)\,B\end{pmatrix}\;,$ (69)

where the last term, $|n\rangle$, is the constant vector
$\bigl{(}0,0,-2\left(m_{1}^{2}+m_{2}^{2}\right)\,B\bigr{)}$, and the operator
$\mathcal{H}$ has the $3\times 3$ matrix representation

$\\!\mathcal{H}=\begin{pmatrix}\\!\left[\frac{1}{2}\left(m_{1}^{2}+m_{2}^{2}\right)+m_{3}^{2}\right]A+m_{3}^{2}C&\Omega/2&0\\\
\\!\\!-\Omega/2&\left[\frac{1}{2}\left(m_{1}^{2}+m_{2}^{2}\right)+m_{3}^{2}\right]A+m_{3}^{2}C&0\\\
0&0&\\!\\!\left(m_{1}^{2}+m_{2}^{2}\right)A\end{pmatrix}\,.$ (70)

In conclusion, if one considers a nonzero interaction with contributions from
all the Pauli matrices, and with all the couplings set to the same value
$m_{\mu}=m=1$, then the components of the Bloch-vector equation reduce to

$\mathcal{H}=\begin{pmatrix}2A+C&\Omega/2&0\\\ -\Omega/2&2A+C&0\\\
0&0&2A\end{pmatrix}=\begin{pmatrix}A+\gamma(0)&\Omega/2&0\\\
-\Omega/2&A+\gamma(0)&0\\\ 0&0&2A\end{pmatrix}\;,$ (71)

and $|n\rangle=(0,0,-4B)$; this agrees with the results in the original paper
of Ref. benatti  (for the particular case of an effective Hamiltonian (27)
with quantization axis along $z$).

But when using only the Pauli matrix $\sigma_{x}$, as in the main text of our
paper, then Eqs. (38) and (39) are obtained. Clearly, the latter case cannot
be obtained from the former one by just enforcing the lack of transitions with
zero frequency, i.e., by setting $\gamma(0)=0$, because both Pauli matrices
$\sigma_{x}$ and $\sigma_{y}$ generate this outcome but contribute separately
to the final result in Eqs. (69) and (70)—this is a subtlety that can easily
lead to inconsistent results (e.g., the combination of Eqs. (4), (13), and
(15) in Ref. yu-zhang-1 ).

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# MSW effect with quark matter: Neutron Star as a case study

Hiranmaya Mishra<EMAIL_ADDRESS>Physical Research Laboratory, Navarangpura,
Ahmedabad, India  School of Physical Sciences, National Institute of Science
Education and Research Bhubanwswar, HBNI, Jatni 752050,Odisha, India  Prasanta
K. Panigrahi<EMAIL_ADDRESS>Indian Institute of Science Education
and Research Kolkata, India  Sudhanwa Patra<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology Bhilai, Raipur, India
Utpal Sarkar<EMAIL_ADDRESS>Indian Institute of Science Education
and Research Kolkata, India

###### Abstract

With the recent findings from various astrophysical results hinting towards
possible existence of strange quark matters with the baryonic resonances such
as $\Lambda^{0},\Sigma^{0},\Xi,\Omega$ in the core of neutron stars, we
investigate the MSW effect, in general, in quark matter. We find that the
resonance condition for the complete conversion of down-quark to strange quark
requires estremely large matter density ($\rho_{u}\simeq
10^{5}\,\mbox{fm}^{-3}$). Nonetheless the neutron stars provide a best
condition for the conversion to be statistically significant which is of the
same order as is expected from imposing charge neutrality condition. This has
a possibility of resolving the hyperon puzzle as well as the equation of state
for dense baryonic matter.

The paramount discovery of neutrino oscillation confirming that the neutrinos
have non-zero mass and they do change identities while propagating brought the
important difference to view of the universe Super-Kamiokande:2005wtt ;
SNO:2002tuh  and the physics beyond the Standard Model (SM) of particle
physics. With times, Mikheyev, Smirnov and Wolfenstein (MSW)
Wolfenstein:1977ue ; Mikheyev:1985zog ; Bethe:1986ej ; Smirnov:2004zv
mechanism is found to be very popular due to its potential to explain the
flavor conversion of solar neutrinos during their propagation in solar matter
even with the small vaccum mixing angle. The basic idea of MSW effect is that
neutrino while propagating in matter are subjected to a potential arising from
the coherent elastic scattering with the particles (protons, neutrons and
electrons) through charged current interaction. Even with small vaccum mixing
angle, the matter potential which acts as an index of refraction modifies the
in-medium mixing angle and can play an important role in flavor conversion of
neutrinos. It is well know that for solar neutrinos with energy around MeV,
the estimated mean free path of the neutrinos in normal matter ($\rho\sim
10^{-15}\,\mbox{fm}^{-3}$) is about $10^{14}$ km which at higher density
($\rho\sim 0.001\,\mbox{fm}^{-3}$) can be as small as about $1$ km
Giunti:2007ry . One can achieve such extreme high matter densities in neutron
stars and supernovae cores which has diameters of a few km.

Here, we investigate, for the first time, the MSW effect in dense quark matter
with possible conversion of down-quarks to strange quarks through flavor
oscillation. Our motivation to explore such a proposal is driven by the recent
observation of possibility of strange quark matter Annala:2019puf  considering
various astrophysical calculations or possible existence of hyperons in the
core of neutron star Vidana:2018bdi . Indeed, it is revealed that the interior
core of the neutron star indicates characteristics of deconfined phase which
can be interpreted as evidence of strange quark matter cores Lattimer:2004pg
and (or) existence of strange baryons in neutron star Schenke:2021nod ;
Tolos:2021lta .

Our focus in this letter is to put forward the MSW mechanism in quark matter.
The main theme of the proposal is the oscillation of quark flavor including
medium effects and explore the possibility of resonance oscillations of quark
flavors in dense quark matter.

We consider quark flavor oscillation for two generations using down (d) and
strange $s$ quarks. Within Standard Model (SM) of particle physics, the usual
quarks are classified in three generations where the left-handed ones are
structured as isospin doublets

$\displaystyle\begin{pmatrix}u\\\ d\end{pmatrix}_{L}\,,\begin{pmatrix}c\\\
s\end{pmatrix}_{L}\,,\begin{pmatrix}t\\\ b\end{pmatrix}_{L}\,$ (1)

while right-handed ones as isospin singlet fields. In general, the down-quark
mixes with strange and bottom quarks within three generation picture. As we
are limiting our discussion to two generations, the Cabibbo proposal
(Cabibbo:1963yz, ) of mixing among quarks is given as,

$\begin{pmatrix}u^{\prime}\\\ d^{\prime}\end{pmatrix}=\begin{pmatrix}u\\\
d\cos\theta_{C}+s\sin\theta_{C}\end{pmatrix}\,.$

where $\theta_{C}$ is Cabibbo mixing angle in vacuum. Here prime-indices
denote the weak eigenstate and the unprime quantity correspond to the mass
eigenstate. The basic idea of quark flavor oscillation is that quarks are
produced as flavor eigenstates. Since flavor states can not propagate, they
are expressed in mass basis using a unitary transformation. The vaccum effect
can be understood by relating mass eigenstates $(d,s)$ by their weak
eigenstates $(d^{\prime},s^{\prime})$ as

$\displaystyle\begin{pmatrix}d^{\prime}\\\ s^{\prime}\end{pmatrix}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}\cos\theta_{C}&\sin\theta_{C}\\\
-\sin\theta_{C}&\cos\theta_{C}\end{pmatrix}\begin{pmatrix}d\\\ s\end{pmatrix}$
(2)

where $\theta_{C}$ is the quark mixing angle (Cabibbo angle) and $d^{\prime}$,
$s^{\prime}$ are the flavour (weak) eigenstates.

The evolution of $d$ and $s$ mass eigenstates with masses $m_{d}$ and $m_{s}$
is governed by Schrodinger equation given as,

$i\frac{\partial}{\partial
t}|\psi(t)\rangle=E|\psi(t)\rangle\,,\quad|\psi(t)\rangle=\left[\begin{array}[]{c}d\\\
s\end{array}\right]$ (3)

Here, $|\psi(t)\rangle$ is denoted for two-component down-type quark state
vector as down-quark and strange-quark mass eigenstates. In the limit of
relativistic energy approximation $E_{i}\simeq p+m_{i}^{2}/2p\approx
E+m_{i}^{2}/2E$, we have

$i\frac{\partial}{\partial t}\left[\begin{array}[]{c}d\\\
s\end{array}\right]=\left[\left(\begin{array}[]{cc}E&0\\\
0&E\end{array}\right)+\frac{1}{2E}\left(\begin{array}[]{cc}m^{2}_{d}&0\\\
0&m^{2}_{s}\end{array}\right)\right]\left[\begin{array}[]{c}d\\\
s\end{array}\right]$ (4)

The term proportional to identity has no effect to d-s quark oscillation and
hence, can be dropped from the analysis. The final mass eigen states are
related to corresponding flavor eigenstates by another unitary mixing matrix.
The propagation and time evolution of down-quarks (part of $SU(2)$ doublets in
weak eigenstates) can be expressed in terms of weak eigenstates
$d^{\prime},s^{\prime}$ involving Cabibbo mixing angle as,

$i\frac{\partial}{\partial t}\left[\begin{array}[]{c}d^{\prime}\\\
s^{\prime}\end{array}\right]=H_{\rm vac}\left[\begin{array}[]{c}d^{\prime}\\\
s^{\prime}\end{array}\right]$ (5)

with vaccum part of effective Hamiltonian as,

$\displaystyle H_{\rm vac}$ $\displaystyle=$
$\displaystyle\frac{1}{2E}\left[\begin{array}[]{cc}m^{2}_{d}\cos^{2}\theta_{C}+m^{2}_{s}\sin^{2}\theta_{C}&-\cos\theta_{C}\sin\theta_{C}\Delta
m^{2}_{ds}\\\ -\cos\theta_{C}\sin\theta_{C}\Delta
m^{2}_{ds}&m^{2}_{d}\sin^{2}\theta_{C}+m^{2}_{s}\cos^{2}\theta_{C}\end{array}\right]$
(8) $\displaystyle=$ $\displaystyle\frac{\Delta
m^{2}_{ds}}{4E}\begin{pmatrix}-\cos 2\theta_{C}&\sin 2\theta_{C}\\\ \sin
2\theta_{C}&\cos 2\theta_{C}\\\ \end{pmatrix}\big{]}$ (9)

with $\Delta m^{2}_{ds}=m^{2}_{s}-m^{2}_{d}$ is the mass square difference
between strange and down quark and is a positive definite quantity.

For relativistic down-type quarks (d,s) with $p\simeq E\gg m^{2}_{k}$ with
$k=d,s$ and $t\approx L$, a typical distance travelled by the quark mass
eigenstates, the transition probability for conversion of down-quark flavor to
strange-quark flavor is given by (Sarkar:2008xir, )

$P^{\rm
vac}\left(d\xrightarrow{}s\right)=\sin^{2}{(2\theta_{C})}\sin^{2}{\left(\frac{\Delta
m^{2}_{ds}L}{4E}\right)}$

The oscillation among quark flavor is possible in vaccum provided two of the
following conditions are satisfied:

  * •

The mixing angle ($\theta_{C}$) in vaccum is not be equal to $0$, $n\pi$ or
$\frac{n\pi}{2}$. The oscillation amplitude is determined by the Cabibbo
mixing angle $\theta_{C}$.

  * •

The mass square difference $\Delta m^{2}_{ds}=m^{2}_{s}-m^{2}_{d}\neq 0$. The
frequency of the quark flavor oscillation is controlled by this parameter and
is large for large value of $\Delta m^{2}_{ds}$.

Let us have an estimate of conversion probability in vaccum. With $\Delta
m^{2}_{ds}\simeq 10^{4}\,\mbox{MeV}^{2}$, $E\simeq 100$ MeV lead to $L_{\rm
osc}/2\simeq 4\pi$ fermi to have $P^{\rm
vac}\left(d\xrightarrow{}s\right)=\sin^{2}(2\theta_{C})\simeq 0.184$. This
implies that after travelling a distance of

$L_{\rm osc}/2=\frac{\big{(}2\pi\big{)}E}{\Delta m^{2}_{ds}}$

of $4\pi$ fermi, the probability of finding the quark flavor in the strange
quark flavor state $s$ is maximal while after traversing the full length
$L_{\rm osc}$, the system is back to its initial state. Thus, e.g., for down
quark energy of the order of 100 MeV, probability of the getting it converted
to strange quark can be as large as 18 percent after travelling a distance of
few tens of fermi even in vaccum which can be understood from the Fig.1.

Figure 1: Schematic illustration of quark oscillation in vacuum for conversion
probability of down quark to strange quark. We consider typical vaccum mixing
angle as the Cabibbo angle $\theta_{C}\simeq 0.22$, energy of down quark
flavor is around 100 MeV and distance travelled in terms of fermi. The key
point is that a down quark requires to travel a distance of few fermi to
convert its flavor to strange quark.

Let us consider the medium effects. A quark flavor while passing through the
medium of quark matter or hadronic matter, can interact through weak charge
current interaction with the medium quarks. This leads to change of mass
eigenstates of down and strange quarks resulting eventually in conversion of
down quark to strange quarks. The form of charge current effective Hamiltonian
in terms of up and down quarks is given by

$\displaystyle H_{{\rm eff}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\big{[}\bar{d}\gamma_{\mu}(1-\gamma_{5})u\big{]}\big{[}\bar{u}\gamma^{\mu}(1-\gamma_{5})d\big{]},$
(10)

with $G_{F}$ as the Fermi constant. Applying Fierz transformation
(Giunti:2007ry, ) and after averaging over up-quark background in the non-
relativistic limit, i.e.,
$\langle\overline{u}\gamma^{\mu}u\rangle\simeq\langle
u^{\dagger}u\rangle\equiv\rho_{u}$, we get

$\displaystyle\bar{H}_{{\rm eff}}$ $\displaystyle=$
$\displaystyle\sqrt{2}G_{F}\rho_{u}\bar{d}_{L}\gamma^{0}d_{L}=V^{u}_{cc}j^{0}_{d},$
(11)

where $d_{L}=\frac{1-\gamma_{5}}{2}d$, $j^{0}_{d}=\bar{d}_{L}\gamma^{0}d_{L}$
and $V^{u}_{cc}$ is the interaction matter potential given by (non-
relativistic limit leading to number density Pal:1991pm ; Kuo:1989qe )

$\displaystyle V^{u}_{cc}=\sqrt{2}G_{F}\rho_{u}.$ (12)

In the presence of the medium, the evolution equation for the down and strange
quarks becomes similar to eq.(5) as,

$\displaystyle i\frac{\partial}{\partial t}\begin{pmatrix}d^{\prime}\\\
s^{\prime}\\\ \end{pmatrix}=H_{\rm matt}\begin{pmatrix}d^{\prime}\\\
s^{\prime}\\\ \end{pmatrix},$ (13)

where, the effective Hamiltonian in medium is given by

$\displaystyle H_{\rm mat}$ $\displaystyle=$ $\displaystyle H_{\rm
vac}+\begin{pmatrix}V^{u}_{cc}&0\\\ 0&0\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}-\frac{\Delta m^{2}_{ds}}{4E}\cos
2\theta_{C}+2\sqrt{2}G_{F}\rho_{u}&\frac{\Delta m^{2}_{ds}}{4E}\sin
2\theta_{C}\\\ \frac{\Delta m^{2}_{ds}}{4E}\sin 2\theta_{C}&\frac{\Delta
m^{2}_{ds}}{4E}\cos 2\theta_{C}\\\ \end{pmatrix}.$

Figure 2: Resonant amplification of quark mixing as a result of medium effects
with the variation of number density of up-quark $\rho_{u}$ taken in terms of
$\mbox{fm}^{-3}$.

Let us introduce a notation characterising the matter effects as,

$\displaystyle A=2\sqrt{2}G_{F}\rho_{u}E.$ (14)

Using eq.(MSW effect with quark matter: Neutron Star as a case study), the
modified Hamiltonian is read as,

$\displaystyle H_{\rm matt}=\frac{1}{4E}\begin{pmatrix}A-\Delta m^{2}_{ds}\cos
2\theta_{C}&\Delta m^{2}_{ds}\sin 2\theta_{C}\\\ \Delta m^{2}_{ds}\sin
2\theta_{C}&-A+\Delta m^{2}_{ds}\cos 2\theta_{C}\\\ \end{pmatrix}.$ (15)

The resulting energy eigenvalues of $H_{\rm matt}$ are as follows

$\displaystyle E^{M}_{d,s}=\frac{1}{4E}\left[A\mp\sqrt{(-A+\Delta
m^{2}_{ds}\cos 2\theta_{C})^{2}+\bigg{(}\Delta m^{2}_{ds}\sin
2\theta_{C}\bigg{)}^{2}}\right]$ (16)

With the medium effects, the Cabibbo mixing angle $\theta_{C}$ will be
modified as follows,

$\displaystyle\tan 2\theta^{M}_{C}=\frac{\Delta m^{2}_{ds}\sin
2\theta_{C}}{-A+\Delta m^{2}_{ds}\cos 2\theta_{C}},$ (17)

Figure 3: Energy eigenvalues for quarks including medium effects and its
variation with respect to change of number density of up-quark $\rho_{u}$.

After all these simplifications, the expression for the probability for
$P_{ds}$ becomes

$\displaystyle P^{\rm
mat}\left(d\xrightarrow{}s\right)=\sin^{2}2\theta^{M}_{C}\sin^{2}{\left(\frac{\Delta^{M}_{ds}L}{4E}\right)}$
(18)

Here using the fact that $E_{s}-E_{d}\approx(m^{2}_{s}-m^{2}_{d})/2E$, we
obtain the modified mass squared difference in the presence of matter as

$\displaystyle\Delta^{M}_{ds}=\sqrt{(-A+\Delta m^{2}_{ds}\cos
2\theta_{C})^{2}+(\Delta m^{2}_{ds}\sin 2\theta_{C})^{2}}.$ (19)

It is noted that vaccum oscillation in d-s quarks is not sensitive to sign of
$\Delta m^{2}_{ds}$ and octant of $\theta_{C}$. However matter effects is
sensitive to both of them. The resonance condition in presence of medium
effects is derived to be

$\displaystyle\Delta m^{2}_{ds}\cos 2\theta_{C}$ $\displaystyle=A\equiv
2\sqrt{2}G_{F}\rho_{u}E$ $\displaystyle=2.65\times
10^{-4}\bigg{(}\frac{\rho_{u}}{\mbox{fm}^{-3}}\bigg{)}\,\bigg{(}\frac{E}{\mbox{MeV}}\bigg{)}\,\mbox{MeV}^{2}\,.$

where $\rho_{u}$ is the number density of up-quark background with which both
down quark is propagating leading to significant medium effects. The resonance
condition for amplification of mixing angle $\sin^{2}2\theta_{M}$ due to
medium effects is displayed in Fig.2. The variation of mass eigenstates with
eigenvalues $E^{M}_{d,s}$ demonstrating the conversion of down quark to
strange quark is presented in Fig.3. The conversion probability is presented
in Fig.4 for illustration of medium effects in quark matter oscillation. In
the limit $\sin\theta_{C}\to 0$, the off-diagonal terms can be neglected in
comparison to diagonal terms in the effective Hamiltonian in presence of
matter. This implies that the resulting energy eigenvalues $E^{M}_{d}$,
$E^{M}_{s}$ and the corresponding eigenstates $(d^{\prime},s^{\prime})$ are
same as their mass eigenstates $(d,s)$. However, due to large medium effects
i.e, $\rho_{u}>>0$, the eigenvalue of down quark $E^{M}_{d}$ can become larger
than $E^{M}_{s}$ causing conversion of down quark flavor to strange quark
flavor. This cross-over occurs at critical density of u-quarks as,

$\rho^{c}_{u}=\frac{\Delta m^{2}_{ds}\cos 2\theta_{C}}{2\sqrt{2}G_{F}E},$ (20)

Figure 4: Conversion probability of down-quarks converted to strange quark in
matter (solid line) and vaccum (dashed line) with variation of energy of quark
flavor. The input parameters considered for medium effects are $\Delta
m^{2}_{ds}\simeq 8.6\times 10^{3}\,\mbox{MeV}^{2}$ is the mass-square
difference between down and strange quark, $\rho_{u}\simeq
10^{5}\,\mbox{fm}^{-3}$ and distance as $0.1$ fermi. For vaccum, we used the
same mass-squared difference parameter but with different distance of 12 fermi
for fixing the first oscillation maximum peak at 100 MeV energy of quark
flavor.

We next examine here the effect of quark flavor oscillation in neutron star.
The key parameters relevant for $d-s$ quark oscillation using matter effects
inside neutron star are

$\Delta m^{2}_{ds},\theta_{C},E_{d},\rho_{u}\,.$

Out of these four parameters, the mass-square difference between strange and
down quark ($\Delta m^{2}_{ds}$) and the Cabibbo mixing angle $\theta_{C}$ are
precisely known while the other two parameters $E_{d}$ and $\rho_{u}$ is to
estimated for the neutron star medium.

Figure 5: Resonance amplification of mixing angle $\sin^{2}(2\theta^{M}_{C})$
in presence of high density matter in terms $Y$ quantifying up-quark number
density in terms of nuclear matter density. Here, we take $X=0.1,0.3,1.0$
which is defined as how much down quark can have energy in terms of its fermi
momentum.

The number density of up quarks inside neutron star is written as
$\rho_{u}=Y\,\rho_{0}$ where $\rho_{0}=0.16\times\mbox{fm}^{-3}$ is the
saturation density of nuclear matter and $Y$ is the parameter defining up-
quark density in terms of nuclear matter number density. Let us note that up-
quark number density $\rho_{u}\approx\rho$ is the number density at the core
of neutron star as there is one up-quark per neutron. We can take
$\rho=5\,\rho_{0}=\rho_{u}$ so that $Y$ is of the order of $5$.

The energy of down-quark ($E_{d}$) can be taken as a fraction of its fermi
momentum $k^{F}_{d}$ as,

$E_{d}=X\,k^{F}_{d}\approx
X\,\big{(}3\,\pi^{2}\,\rho_{d}\big{)}^{1/3}=X\,\big{(}\frac{3}{2}\,\pi^{2}\,\rho\big{)}^{1/3}\,.$

The modified expression for parameter $A$ is written in terms of $X$ and $Y$
instead of $\rho_{u}$ and $E_{d}$ as

$\displaystyle A$ $\displaystyle=$
$\displaystyle\big{(}2\sqrt{2}\big{)}\,G_{F}\,\big{(}\frac{3}{2}\,\pi^{2}\,\big{)}^{1/3}\,X\,Y^{4/3}\rho^{4/3}_{0}\,$
(21)

The new contribution of matter potential can be inserted in medium effect
mixing angle $\sin^{2}(2\theta^{M}_{C})$ and is displayed in Fig.5. For a
representative values, $X\simeq 0.5$, $Y\simeq 5$ and $L\simeq 10$ km as the
typical radius of the neutron star, the estimated value of conversion
probability to have strange quark flavor is
$P^{M}\left(d\xrightarrow{}s\right)\simeq\frac{1}{2}\sin^{2}{2\theta^{M}_{C}}\simeq
0.02$. Here we have use the fact that $L$ is much much greater than typical
oscillation length, the frequency part of the probability i.e,
$\sin^{2}{\left(\frac{\Delta^{M}_{ds}L}{4E}\right)}\simeq 1/2$. This means
that about 2 percent of down quarks are converted to strange quark inside the
neutron star i.e., $\rho_{d}\approx 0.2\rho_{s}$ while travelling a distance
from core to the surface of the star. Such a result however depends upon
adiabatic approximations i.e, of uniform density throughout the star. Non-
adiabatic approximation may change this results.

Let us next compare strange quark number density arising from beta equilibrium
condition for the neutron star matter. In terms of Fermi momentum and number
density, the charge neutrality gives

$n_{p^{+}}=n_{e^{-}}\,\quad\mbox{or,}k_{F,p^{+}}=k_{F,e^{-}}\,.$

In the other hand, chemical equilibrium yields $\mu_{n}=\mu_{p^{+}}+\mu_{e}$.
Since the chemical potential is related to the number densities as
$\rho_{i}=\gamma k^{3}_{F,i}/(6\pi^{2})$ with
$k_{F,i}=\sqrt{\mu^{2}_{i}-m^{2}_{i}}$ with $i=d,s$ and $\gamma$ is the
degeneracy factor related to color and spin. Using these basic relations, the
number densities of down- and strange quark are related to each other inside
neutron star as,

$\displaystyle\frac{\rho_{d}}{\rho_{s}}=\frac{k^{3}_{F,d}}{k^{3}_{F,s}}=\frac{\big{(}\mu^{2}_{d}-m^{2}_{d}\big{)}^{3/2}}{\big{(}\mu^{2}_{s}-m^{2}_{s}\big{)}^{3/2}}\approx
1+\frac{3}{2}\frac{\Delta m^{2}_{ds}}{\mu^{2}_{q}}$ (22)

Using the values of current quark masses, $m_{d}\simeq 5$ MeV, $m_{s}\simeq
95$ MeV and $\mu^{2}_{q}\simeq\mbox{500\, MeV}^{2}$, we get

$\frac{\rho_{d}}{\rho_{s}}=1+0\,.05$

Thus, the beta equilibrium condition leads to existence of five percent of
strange quark number density compare to down-quark number density.

To summarise we have presented here quark oscillations in vaccum for two
generations of quarks similar to neutrino oscillation. It turns out that for a
down quark of energy 100 MeV, the oscillation length is a few Fermi’s.
However, in vaccum such oscillation is not observable as the strong
interaction become dominant. On the hand, it is possible to have such flavor
oscillation in dense matter. Indeed, following MSW mechanism, it turns out
that the resonance oscillation to take place for a very large density of up
quark background of the order of $\rho_{u}\simeq 10^{5}\,\mbox{fm}^{-3}$.

We finally estimated the conversion probability of down quark to strange quark
within typical set of parameters for neutron star and found that about 2
percent of down quark can be converted to strange quark inside the core of the
neutron star. This is of the same order as one can expect from the condition
of beta equilibrium inside the neutron star. Such an extra possible source of
strange quark from flavor oscillation can have interesting consequences
regarding equation of state at high densities relevant for neutron star
phenomenology. Such an alternate source of strange quarks could possibly lead
to higher densities of strange baryons inside neutron star. It can have
consequences regarding ”Hyperon puzzle” in the context of observation of high
mass ($\sim 2\,M_{\odot}$) neutron star Vidana:2018bdi .

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