# A Lindbladian From Feynman-Vernon

Jose A. Magpantay  Quezon City, Philippines<EMAIL_ADDRESS>
###### Abstract

The effective dynamics of a system interacting with a bath or environment is
presented in two ways, (1) the (LGKS) replacement of the von Neuman equation
for the density matrix and (2) the Feynman-Vernon path integral derivation, by
integrating out the bath degrees of freedom, to arrive at the propagator of
the system’s density matrix. In this paper I connect the two methods by
deriving a Lindbladian in a mechanical example, a point particle interacting
with a bath of harmonic oscillators, previously considered by Feynman and
Vernon (FV) and expounded on later by Caldeira and Leggett (CL). But the
(FV)/(CL) results only in non-Markov effect, memory terms from the bath
interaction. To derive a Lindbladian, I changed the interaction term they
considered to take into account the point particle interacting with the bath
harmonic oscillators to something more realistic. From the resulting path-
integral expression of the system’s propagator for the density matrix, the
Lindbladian and the non-Markov term are read for this simple problem. I also
point out the causes of these terms, the Markov Lindbladian from the very
local interaction of the point particle with the classical solutions of the
harmonic oscillator and the non-Markov term from the global interaction of the
point particle with the fluctuations of the classical solutions.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

1\. There are two ways to deal with the quantum statistics of open systems,
the first and derived earlier using path-integral techniques is the Feynman-
Vernon (FV) theory Feynman  and the second derived more than a decade later is
what became known as the Lindblad equation Lindblad , which was also
independently derived at about the same time by Gorini, Kossakowski and
Sudarshan (GKS) Gorini , from hereon will be referred to as the (LGKS)
approach. The (LGKS) approach derives the replacement of the von Neuman
equation for the density operator/matrix (see any statistical mechanics book,
for example Reichl ), under the general conditions of Hermiticity, Markov
process and complete positivity. The Feynman-Vernon starts from the system-
bath interaction and derive, using the path-integral method, the system’s
density matrix at any time t by integrating out the environment/bath degrees
of freedom. However, still at this time there has been no paper that relates
(LGKS) from (FV). The link is crucial for it will provide how the terms that
appear in the (LGKS), the Hamiltonian and the Lindblad operator, are given in
terms of the system-bath dynamics. But in a certain sense, the link does not
seem to be warranted for the simple reason that (FV), being a sum over
histories approach would naturally be dominated by memory effects, thus non-
Markov, while the (LGKS) is limited by the Markov process condition, along
with two other conditions.

Regardless of the above ideas, it is interesting to find out if the Lindblad
equation can be derived from the Feynman-Vernon path-integral approach to
quantum statistical mechanics. Maybe even in doing the sum over histories of
the bath, there may be non memory effects that can be isolated. But most
likely it depends on the interaction between the bath and the system. The
(FV)/(CL) point particle interacting with a bath of harmonic oscillators
clearly show the dominance of memory effects, the only path-integral result
for this problem is what I called the Lindblad Plus Magpantay , the Plus
represents the memory effect terms.

2\. I begin by summarizing known ideas about von Neuman type equations in
statistical mechanics. First, for a closed system, described by a Hamiltonian
$H(x,p)$, the von Neuman equation for the density operator $\rho$ is given by,
see for example Reichl

$i\hbar\dfrac{\partial\rho}{\partial t}=\left[H,\rho\right].$ (1)

The solution to equation (1)

$\rho(t)=\exp{(-\frac{i}{\hbar}Ht)}\rho(0)\exp{(+\frac{i}{\hbar}Ht)},$ (2)

from which we get the time evolution of the density matrix $\rho(x,y;t)$ as

$\begin{split}\rho(x,y;t)&=\int
dx^{\prime}dy^{\prime}K(x,x^{\prime};t)\rho(x^{\prime},y^{\prime};0)K^{*}(y^{\prime},y;t)\\\
&=\int
dx^{\prime}dy^{\prime}J(x,y;x^{\prime},y^{\prime};t)\rho(x^{\prime},y^{\prime};0).\end{split}$
(3)

The $K(x,x^{\prime};t)$ is the quantum propagator given by the path-integrals

$\begin{split}K(x,x^{\prime};t)&=\int_{endpts}(d\tilde{x})\exp{\frac{i}{\hbar}\int_{0}^{t}dt^{\prime}L(\tilde{x},\dot{\tilde{x}})}\\\
&=\int_{endpts}(d\tilde{x})(dp_{x})\exp{\frac{i}{\hbar}\int_{0}^{t}dt^{\prime}\left[p_{x}\dot{\tilde{x}}-H(\tilde{x},p_{x})\right]},\end{split}$
(4)

and $J(x,y;x^{\prime},y^{\prime};t)$ is the density matrix propagator and is
given by

$\begin{split}J(x,y;x^{\prime},y^{\prime};t)&=K(x,x^{\prime};t)K^{*}(y^{\prime},y;t)\\\
&=\int_{endpoints}(d\tilde{x})(d\tilde{y})\exp{\frac{i}{\hbar}\int_{0}^{t}dt^{\prime}\left[L(\tilde{x},\dot{\tilde{x}})-L(\tilde{y},\dot{\tilde{y}})\right]}\\\
&=\int_{endpoints}(d\tilde{x})(d\tilde{y})(dp_{x})(dp_{y})\exp{\frac{i}{\hbar}\int_{0}^{t}dt^{\prime}\left(\left[p_{x}\dot{\tilde{x}}-H(\tilde{x},p_{x})\right]-\left[p_{y}\dot{\tilde{y}}-H(\tilde{y},p_{y})\right]\right)}.\end{split}$
(5)

For an open system, i.e., a system interacting with a bath/environment, there
are two major approaches - the Feynman-Vernon theory and the Lindblad and
Gorini - Kossakowski - Sudarshan theory (LGKS). The LGKS provides the
analogous von Neumann type of equation for the system under conditions of (1)
Markov process, (2) Hermiticity and (3) complete positivity and it is given by

$\begin{split}i\hbar\dfrac{\partial\rho}{\partial
t}&=[H,\rho]+i\textbf{L}\rho\textbf{L}^{\dagger}-\frac{i}{2}\left(\textbf{L}^{\dagger}\textbf{L}\rho+\rho\textbf{L}^{\dagger}\textbf{L}\right)\\\
&=H_{eff}\rho-\rho
H^{\dagger}_{eff}+i\textit{L}\rho\textit{L}^{\dagger},\end{split}$ (6)

where L is the LIndbladian and $H_{eff}$ is given by

$H_{eff}=H-\frac{i}{2}\textbf{L}^{\dagger}\textbf{L}$ (7)

This form of the LGKS equation is due to Struntz Struntz , which provided the
path integral for the time differential evolution operator of the system’s
density operator, the analogue of equations (3) and (4) for a closed system.
This form is significant for it will guide the inclusion of memory effects,
the non-Markovian term, which must be present in general for a system
interacting with a bath/environment for an extended period of time. The path-
integral version of equation (6) is given in Struntz  as

$\rho(x,y;t)=\int
dx^{\prime}dy^{\prime}J_{L}(x,y;x^{\prime},y^{\prime};t)\rho(x^{\prime},y^{\prime};0),$
(8)

where the four-point plus time function $J_{L}$ is the propagator for the
density matrix of a Lindblad system and it is given by the system’s
coordinates and momenta path-integrals

$\begin{split}J_{L}(x,y;x^{\prime},y^{\prime};t)&=\int_{endpoints}(d\tilde{x})(d\tilde{y})(dp_{x})(dp_{y})\exp\frac{i}{\hbar}\int_{0}^{t}dt^{\prime}\big{\\{}\left[p_{x}\cdot\dot{\tilde{x}}-H(\tilde{x},p_{x})\right]\\\
&\quad-\left[p_{y}\cdot\dot{\tilde{y}}-H(\tilde{y},p_{y})\right]-i\left[(\textbf{L}^{\dagger}\textbf{L})(\tilde{x},p_{x})+(\textbf{L}^{\dagger}\textbf{L})(\tilde{y},p_{y})+\textbf{L}(\tilde{x},p_{x})\textbf{L}^{\dagger}(\tilde{y},p_{y})\right]\big{\\}}.\end{split}$
(9)

The $J_{L}$ is the propagator for the density matrix. To arrive at the
Lindbladian terms in $J_{L}$, use was made of equations (6) and (7). Note, the
coordinates and momenta end points are the same as the end point conditions in
equations (4) and (5). It is easy to see from equations (5) and (9) that both
$J$ for a closed system and $J_{L}$ for an open system with Markov conditions
are Markov.

Finally, consider an open system with memory term, for this I suggested an
integro-differential equation, with the extra term as the Plus

$i\hbar\dfrac{\partial\rho}{\partial t}=H_{eff}\rho-\rho
H^{\dagger}_{eff}+\textbf{L}\rho\textbf{L}^{\dagger}+\int_{0}^{t}d\tau\left[M(t,\tau)\rho(\tau)-\rho(\tau)M^{\dagger}(t,\tau)+N(t,\tau)\rho(\tau)N^{\delta}(t,\tau)\right].$
(10)

In a previous paper Magpantay  , I showed that it gives a a propagator for the
density matrix as given by

$\begin{split}J_{LP}(x,y;x^{\prime},y^{\prime};t)&=\int_{endpoints}(d\tilde{x})(d\tilde{y})(dp_{x})(dp_{y})\exp\frac{i}{\hbar}\int_{0}^{t}dt^{\prime}\Big{\\{}\left[p_{x}\cdot\dot{\tilde{x}}-H(\tilde{x},p_{x})\right]\\\
&\quad-\left[p_{y}\cdot\dot{\tilde{y}}-H(\tilde{y},p_{y})\right]-i\left[(\textbf{L}^{\dagger}\textbf{L})(\tilde{x},p_{x})+(\textbf{L}^{\dagger}\textbf{L})(\tilde{y},p_{y})+\textbf{L}(\tilde{x},p_{x})\textbf{L}^{\dagger}(\tilde{y},p_{y})\right]\\\
&\quad+\int_{0}^{t^{\prime}}dt^{\prime\prime}\Big{[}M(\tilde{x},p_{x};t^{\prime},t^{\prime\prime})-M^{\dagger}(\tilde{y},p_{y};t^{\prime},t^{\prime\prime})+N(\tilde{x},p_{x};t^{\prime},t^{\prime\prime})N^{\delta}(\tilde{y},p_{y};t^{\prime},t^{\prime\prime})\Big{\\}},\end{split}$
(11)

where the subscript LP in the above J stands for Lindblad Plus. The end points
condition in equation (11) is the same as that in equation (9) and equation
(5).

Equations (1) to (11) summarize the relevant ideas and methods relevant to
this problem. What they need is to derive the terms that appear in equations
(9) and (11) from the underlying system, bath interaction. The Feynman-Vernon
theory precisely provides the method for this. Unfortunately, the example they
used and expounded on by Caldeira-Leggett Caldeira  only gives non-Markov
terms. I will make use of their particular example and show that a Markov
term, a Lindbladian can be derived also. I do this by modifying the particle-
bath interaction they used.

3\. I will give first a summary of the (FV)/(CL) example. The system has a
single particle with coordinate x and the bath is described by coordinates
$X_{i}$ with $i=1,..,N$, N taken to be a large number. The bath is composed of
independent harmonic oscillators each with a particular frequency $\omega_{i}$
but all having the same mass m and the interaction of the particle, of mass M
(use of capital M does not imply more massive than bath oscillators) with the
bath, is given by a simple product term. The Lagrangians are

$\displaystyle
L_{b}=\frac{m}{2}\sum_{i=1}^{N}\left[\dot{X}_{i}^{2}-\omega_{i}^{2}X_{i}^{2}\right],$
(12a) $\displaystyle L_{int}=-x\sum_{i=1}^{N}c_{i}X_{i},$ (12b) $\displaystyle
L_{s}=\frac{M}{2}\dot{x}^{2}-V(x),$ (12c)

where $c_{i}$ are constants, The interaction term says that all bath
oscillators interact with the particle regardless of where the particle is.
This is clearly unrealistic.

THe Feynman-Vernon theory says the density matrix for the particle and bath is

$\rho(x,X;y,Y;t)=\int
dx^{\prime}dX^{\prime}dy^{\prime}dY^{\prime}K(x,X;x^{\prime},X^{\prime};t)\rho(x^{\prime},X^{\prime};y^{\prime},Y^{\prime};0)K^{*}(y^{\prime},Y^{\prime};y,Y;t),$
(13)

where the bath index i suppressed in the above equation for simplicity. Since
the system, bath interaction is turned on only at $t=0$, the initial density
matrix factorizes as

$\rho(x^{\prime},X^{\prime};y^{\prime},Y^{\prime};0)=\rho_{s}(x^{\prime},y^{\prime};0)\rho_{b}(X^{\prime},Y^{\prime};0)$
(14)

where the subscripts s and b that goes with $\rho$ refer to system and bath
respectively. Focusing on the particle’s density matrix, it is given by
tracing over the bath’s end points X and Y with X=Y, i.e.,

$\rho_{s}(x,y;t)=\int dX\rho(x,X;y,X;t).$ (15)

Using equation (13) and (14) gives

$\displaystyle\rho_{s}(x,y;t)=\int
dx^{\prime}dy^{\prime}J(x,y;x^{\prime},y^{\prime};t)_{FV}\rho_{s}(x^{\prime},y^{\prime};0),$
(16a) $\displaystyle J_{FV}(x,y;x^{\prime},y^{\prime};t)=\int
dXdX^{\prime}dY^{\prime}K(x,X;x^{\prime},X^{\prime};t)\rho_{b}(X^{\prime},Y^{\prime};0)K^{*}(y^{\prime},Y^{\prime};y,X;t),$
(16b)

where the subscript FV in above J stands for Feynman-Vernon and integrations
in the above J are over the bath end point conditions (note, the end point
condition at t is only X because of the tracing made). Before expressing this
$J_{FV}$ in terms of a path-integral, it is important to notice that it hints
it is not Markov because of the presence of the bath’s
$\rho_{b}(X_{1},Y_{1};0)$ between the two K’s. This should be expected because
the bath oscillators incessant interaction with the system’s degree of freedom
(given by x) will imprint long term memory effects on the system. It is
encouraging that the $J_{FV}$ in equation (16b) has the same character, non-
Markov, as the $J_{LP}$ in the Lindblad Plus as given by equation (11). But in
equation (11), it is explicitly constructed there is a bath effect that is
Markov. Maybe it can also be present here.

The quantum propagator K for the system plus bath is given by

$\begin{split}K(x,X;x^{\prime},X^{\prime};t)&=\left\langle
xX\Big{|}\exp{-\frac{i}{\hbar}H_{s+b}t}\Big{|}x^{\prime}X^{\prime}\right\rangle,\\\
&=\int_{endpoints}(d\tilde{x})(d\tilde{X})\exp{\frac{i}{\hbar}\int_{0}^{t}dt^{\prime}\left[L_{s}(\tilde{x})+L_{b}(\tilde{X})+L_{int}(\tilde{x},\tilde{X})\right]}.\end{split}$
(17)

The end points condition in above are
$\tilde{x}(t^{\prime}=0)=x^{\prime},\tilde{x}(t^{\prime}=t)=x,\tilde{X}(t^{\prime}=0)=X^{\prime},\tilde{X}(t^{\prime}=t)=X$,
and there are similar conditions for $\tilde{Y}$. Substituting equation (17)
and a similar equation for $K^{*}$ in equation (16b), the propagator for the
distribution function of the system becomes

$J_{FV}(x,y;x^{\prime},y^{\prime};t)=\int_{endpoints}(d\tilde{x})(d\tilde{y})\textbf{F}(\tilde{x},\tilde{y})\exp{\frac{i}{\hbar}\int
dt^{\prime}\left[L_{s}(\tilde{x}(t^{\prime}))-L_{s}(\tilde{y}(t^{\prime}))\right]},$
(18)

where the end points condition in above are
$\tilde{x}(t^{\prime}=0)=x^{\prime}$, $\tilde{x}(t^{\prime}=t)=x$ and similar
end point conditions for $\tilde{y}$. The functional F is known as the
influence functional and is given by the path-integral and ordinary integrals

$\begin{split}\textbf{F}(\tilde{x},\tilde{y})&=\int
dXdX^{\prime}dY^{\prime}\rho_{b}(X^{\prime},Y^{\prime};0)\int_{endpoints}(d\tilde{X})(d\tilde{Y})\exp\frac{i}{\hbar}\int
dt^{\prime}\Big{\\{}L_{b}(\tilde{X}(t^{\prime}))+L_{int}(\tilde{x}(t^{\prime}),\tilde{X}(t^{\prime}))\\\
&\quad-
L_{b}(\tilde{y}(t^{\prime}))-L_{int}(\tilde{y}(t^{\prime}),\tilde{Y}(t^{\prime}))\Big{\\}}\end{split}$
(19)

The end points in the above path-integral are defined by
$\tilde{X}(t^{\prime}=0)=X^{\prime},\tilde{X}(t^{\prime}=t)=X=\tilde{Y}(t^{\prime}=t),\tilde{Y}(t^{\prime}=0)=Y^{\prime}$,
and the ordinary integrals are over the initial and final end points. To carry
out the evaluation of equation (19), I use a background decomposition

$\tilde{X}_{i}=\bar{X}_{i}+\delta X_{i},$ (20)

where $\bar{X}_{i}$ (I now use the index i to make clear there are many
harmonic oscillators in the bath) is the harmonic oscillator state satisfying

$\dfrac{d^{2}\bar{X}_{i}}{dt^{\prime 2}}+\omega_{i}^{2}\bar{X}_{i}=0,$ (21)

and $\delta X_{i}$ a fluctuation about this classical solution. The solution
to equation (21) is

$\bar{X}_{i}(t^{\prime})=A_{i}\sin\omega_{i}t^{\prime}+B_{i}\cos\omega_{i}t^{\prime},$
(22)

where the constants $A_{i}$ and $B_{i}$ are determined from the end point
conditions $\bar{X}_{i}(0)=X^{\prime}_{i}$, $\bar{X}_{i}(t)=X_{i}$ and gives

$\displaystyle B_{i}=X^{\prime}_{i},$ (23a) $\displaystyle
A_{i}=X_{i}\dfrac{1}{\sin\omega_{i}t}-X^{\prime}_{i}\cot\omega_{i}t.$ (23b)

The interaction term becomes

$S_{int}=\int_{0}^{t}dt^{\prime}\tilde{x}(t^{\prime})[\bar{X}_{i}(t^{\prime})+\delta
X_{i}(t^{\prime})],$ (24)

and using equations (22) and (23) will show how the end points of the bath
oscillator appear with the particle’s coordinate.

Since the end point conditions for $\tilde{X}_{i}$ is accounted for by
$\bar{X}_{i}$, it means that the end point conditions for the fluctuations
$\delta X_{i}(t^{\prime}=0,ort)=0$. Substituting equation (20) in equations
(12; a, b) to compute the influence functional
$\textbf{F}(\tilde{x},\tilde{y})$ will involve the path integral of the
fluctuations $deltaX_{i}(t^{\prime})$ with a quadratic harmonic oscillator
term and a linear term $\tilde{x(t^{\prime})}\cdot\delta X_{i}(t^{\prime})$.
The Greens function $G(t^{\prime},t^{\prime\prime})$ of
$\dfrac{d^{2}}{dt^{\prime 2}}+\omega_{i}^{2}$ must be solved subject to the
boundary conditions of the fluctuations and it is given by

$G(t^{\prime},t^{\prime\prime})=\dfrac{1}{m\omega_{i}\sin\omega_{i}t}[\sin\omega_{i}(t-t^{\prime})\sin\omega_{i}t^{\prime\prime}],$
(25)

valid for $t^{\prime\prime}\leq t^{\prime}$ and t’ in the range $(0,t)$.

The bath end points integrations (ordinary integrals), follow after (1)
integration by parts of the kinetic and mass terms of $\bar{X}_{i}$ and (2)
making use of equations (22) and (23) in the first term of equation (24).

There are two more ingredients, (1) doing the same as above for the bath
degrees of freedom $\tilde{Y}_{i}$, with end point conditions
$\bar{Y}_{i}(t^{\prime}=0)=Y^{\prime}_{i}$, $\bar{Y}_{i}(t^{\prime}=t)=X_{i}$,
and (2) assuming that the bath degrees of freedom at $t^{\prime}=0$ is at
thermal equilibrium with distribution

$\rho_{b}(X^{\prime},Y^{\prime};0)=\prod_{i}\dfrac{m\omega_{i}}{2\pi\hbar\sinh\frac{\hbar\omega_{i}}{kT}}\exp{\left\\{-\dfrac{m\omega_{i}}{2\hbar\sinh\frac{\hbar\omega_{i}}{kT}}\left[(X_{i}^{\prime
2}+Y_{i}^{\prime
2})\cosh\frac{\hbar\omega_{i}}{kT}-2X^{\prime}_{i}Y^{\prime}_{i}\right]\right\\}}$
(26)

The ordinary integrals on the bath end point conditions can be done to give

$\begin{split}J_{FV}(x,y;x^{\prime},y^{\prime};t)&=\int_{endpoints}(d\tilde{x})(d\tilde{y})\exp\frac{i}{\hbar}\big{\\{}\int_{0}^{t}dt^{\prime}\left[L_{s}(\tilde{x},\dot{\tilde{x}})-L_{s}(\tilde{y},\dot{\tilde{y}})\right]\\\
&\quad+\int_{0}^{t}dt^{\prime}\int_{0}^{t^{\prime}}dt^{\prime\prime}[\tilde{x}(t^{\prime})-\tilde{y}(t^{\prime})]\left(i\sum_{i}\dfrac{c_{i}^{2}}{2m\omega_{i}}\left(\coth\frac{\hbar\omega_{i}}{2kT}\right)\cos\omega_{i}(t^{\prime}-t^{\prime\prime})\right)[\tilde{x}(t^{\prime\prime})-\tilde{y}(t^{\prime\prime})]\\\
&\quad+\int_{0}^{t}dt^{\prime}\int_{0}^{t^{\prime}}dt^{\prime\prime}[\tilde{x}(t^{\prime})-\tilde{y}(t^{\prime})]\left(\sum_{i}\frac{c_{i}^{2}}{2m\omega_{i}}\sin\omega_{i}(t^{\prime}-t^{\prime\prime})\right)[\tilde{x}(t^{\prime\prime})+\tilde{y}(t^{\prime\prime})]\big{\\}},\end{split}$
(27)

which is the same result given by Caldeira-Leggett. The imaginary number i
that goes with the first sum becomes a decay term, gives the decoherence
effect of the thermal initial state. Compairing with equation (11), the double
time-integrals show that the bath effect on the particle’s dynamics are purely
non-Markov. They are all memory terms, which can be explained by the fact that
the classical harmonic oscillator solutions $\bar{X}_{i}$ and the fluctuations
$\delta X_{i}$, all of them influence the particle coordinate $\tilde{x}$ all
the time, anywhere. Assuming no momentum dependence on the Markov and non-
Markov terms of equation (11), comparison with equation (27) gives a zero
Lindbladian and

$\displaystyle
M(\tilde{x};t^{\prime},t^{\prime\prime})=\tilde{x}(t^{\prime})\sum_{i}\frac{c_{i}^{2}}{2m\omega_{i}}\left[i\coth\frac{\hbar\omega_{i}}{2kT}\cos\omega_{i}(t^{\prime}-t^{\prime\prime})+\sin\omega_{i}(t^{\prime}-t^{\prime\prime})\right]\tilde{x}(t^{\prime\prime}),$
(28a) $\displaystyle
N(\tilde{x};t^{\prime},t^{\prime\prime})N^{\delta}(\tilde{y};t^{\prime},t^{\prime\prime})=\begin{pmatrix}A\tilde{x}(t^{\prime})&B\tilde{x}(t^{\prime\prime})\end{pmatrix}\cdot\begin{pmatrix}A\tilde{y}(t^{\prime\prime})\\\
B\tilde{y}(t^{\prime})\end{pmatrix},$ (28b)

with the values of A and B given by

$\displaystyle
A^{2}=\sum_{i}\frac{c_{i}^{2}}{2m\omega_{i}}\left[-i\coth\frac{\hbar\omega_{i}}{2kT}\cos\omega_{i}(t^{\prime}-t^{\prime\prime})+\sin\omega_{i}(t^{\prime}-t^{\prime\prime})\right],$
(29a) $\displaystyle
B^{2}=\sum_{i}\frac{c_{i}^{2}}{2m\omega_{i}}\left[-i\coth\frac{\hbar\omega_{i}}{2kT}\cos\omega_{i}(t^{\prime}-t^{\prime\prime})-\sin\omega_{i}(t^{\prime}-t^{\prime\prime})\right]$
(29b)

This is the reason that I stated in Magpantay  that the (FV) and (CL) example
yielded ony the Lindblad Plus.

4\. In this part I show how to get a Lindbladian from the (FV) and (CL)
example. As I already pointed out in part 3, the interaction term equation (12
b) is unrealistic, the particle interacts with all oscillators all the time
regardless of where the particle is.

I assume that bath surrounds the particle with coordinate x, with the number
of oscillators N being large and the particle needs equal time, call it
$\tau$, to bounce off from one bath oscillator to another, the collision time
is infinitesimal and call it $\epsilon$. It is clear that $\epsilon$ is much
less than $\tau$, the spacing between oscillators is much bigger than the size
of the oscillators. I will label the particle’s collision time sequentially as
$(\tau_{1},\tau_{2},....,\tau_{i},....\tau_{N}=t)$, where $\tau_{i}=i\tau$
seconds is the time the ith oscillator collides with the particle, the first
collision happens after $\tau$ seconds from the start. I will also assume that
the oscillators all have the same frequency $\omega$, this will simplify the
calculations and will be clear later, when I discuss the general case of many
different frequencies. From equation (24), the interaction term becomes

$S_{int}=\sum_{i}\epsilon\tilde{x}(\tau_{i})\left[(\dfrac{-1}{\sin\omega
t}X_{i}+\cot\omega
tX^{\prime}_{i})\sin\omega\tau_{i}-X^{\prime}_{i}\cos\omega\tau_{i}\right]+\int_{0}^{t}dt^{\prime}\tilde{x}(t^{\prime})\delta
X_{i}(t^{\prime}),$ (30)

the index i, which goes with time $\tau_{i}$ is now the index for the end
point conditions of the bath coordinates. This equation also says all the end
point conditions are still determined at $t^{\prime}=0$ and at $t^{\prime}=t$,
which is also used in part 4. But the ith oscillator only interacts with the
particle during the collision at $\tau_{i}$ for an infinitesimal time
$\epsilon$, and then this oscillator continues its oscillation without
interacting any more with the particle, which by time $\tau_{i+1}$ is now
interacting with the $i+1$ oscillator. However, note the integral in equation
(30), the weak fluctuation is assumed to interact throughout with the
particle.

Now in calculating the bath + particle propagator
$K^{*}(y^{\prime},Y^{\prime};y,X;t)$, the time interval (0,t) will also be
divided by N $\tau$ collision intervals. Since all oscillators have the same
mass m and frequencies $\omega$, we can lump the ith label with the collision
time $\tau_{i}$. Now follow the same calculation as before, the quadratic term
in $\delta X_{i}(t^{\prime})$ and linear term with equation (30) will yield
the same as equation (25). This will be the only double time integral in
$J_{FV}$, memory term. The end point conditions
$X_{i},X^{\prime}_{i},Y^{\prime}_{i}$ that appear in
$\frac{m}{2}[\bar{X}_{i}(t)\dot{\bar{X}_{i}}(t)-\bar{X}_{i}(0)\dot{\bar{X}_{i}}(0)]$
and those that appear in an equation similar to the first term of (30) but now
in terms of the end point conditions for $Y$ will now be time ordered also by
$\tau_{i}$. When the end point ordinary integrals are done, it will involve
end point conditions for the same time $\tau_{i}$, thus the result involve
only a single sum over time $\tau_{i}$, which can be written as a single time
integral. This is a Lindbladian.

Now, the details. The quadratic kinetic term in $\bar{X}_{i}$, which upon
integration by parts gives
$\frac{m}{2}[\bar{X}_{i}(t)\dot{\bar{X}_{i}}(t)-\bar{X}_{i}(0)\dot{\bar{X}_{i}}(0)]$
is equal to

$QuadK=\exp\frac{i}{\hbar}\sum_{i=0}^{N}\Big{\\{}\frac{m\omega}{2}(\cot\omega
t)(X_{i}^{2}+X^{\prime}_{i}{2})-\dfrac{m\omega}{\sin\omega
t}X_{i}X^{\prime}_{i},$ (31)

and there will be a similar equation for the $Y$ terms, except has a
$\frac{-i}{\hbar}$. The linear in $\bar{X}_{i}$ terms are given by

$Lin=\exp\frac{i}{\hbar}\sum_{i=0}^{N}\epsilon
c_{i}\tilde{x}(\tau_{i})\left[(\dfrac{-1}{\sin\omega t}X_{i}+\cot\omega
tX^{\prime}_{i})\sin\omega\tau_{i}-X^{\prime}_{i}\cos\omega\tau_{i}\right],$
(32)

and there is a similar equation for the $Y,y$ term. The fluctuation path-
integral makes use of equation (25) and gives

$Fluct=\exp\frac{i}{\hbar}\sum_{i=0}^{N}c_{i}^{2}\frac{1}{m\omega}\dfrac{1}{\sin\omega
t}\int_{0}^{t}dt^{\prime}\int_{0}^{t^{\prime}}dt^{\prime\prime}\tilde{x}(t^{\prime})\sin\omega(t-t^{\prime})\sin\omega
t^{\prime\prime}\tilde{x}(t^{\prime\prime}),$ (33)

and a similar expression involving $\tilde{y}$. The index i that appear in
(31) and (32) are connected with the time index $\tau_{i}$, thus each end
points integrals $X_{i},X^{\prime}_{i},Y^{\prime}_{i}$ are done for each time
and the final result is a single sum over $\tau_{i}$. The end point integrals
are simple. The terms suggest the $X_{i}$ integral is done first, which will
give a relationship between $X^{\prime}_{i}andY^{\prime}_{i}$. To get the
final answer, do the final integral in terms of $X^{\prime}_{i}$ then go back
to the intermediate step after the $X_{i}$ integral, do the final integral
with $Y^{\prime}_{i}$ and since the results should be equal, add the two
results and divide by two to get the final answer for the generating
functional J for the modified (FV)/(CL) example

$\begin{split}J_{ME}(x,y;x^{\prime},y^{\prime};t)&=\int_{endpoints}(d\tilde{x})(d\tilde{y})\exp\frac{i}{\hbar}\int_{0}^{t}\big{\\{}dt^{\prime}\left[L_{s}(\tilde{x},\dot{\tilde{x}})-L_{s}(\tilde{y},\dot{\tilde{y}})\right]\\\
&\quad+\begin{pmatrix}\\\
\tilde{x}(t^{\prime})&\tilde{x}(t^{\prime})\end{pmatrix}\cdot\left[\sum_{i}\dfrac{c_{i}^{2}\epsilon^{2}}{4m\omega}\left(-2\sin
2\omega t^{\prime}+\cot\omega t\sin^{2}\omega
t^{\prime}\right)\delta(t^{\prime}-\tau_{i})\right]\begin{pmatrix}\tilde{x}(t^{\prime})\\\
\tilde{x}(t^{\prime})\end{pmatrix}\\\ &\quad+\begin{pmatrix}\\\
\tilde{y}(t^{\prime})&\tilde{y}(t^{\prime})\end{pmatrix}\cdot\left[\sum_{i}\dfrac{c_{i}^{2}\epsilon^{2}}{4m\omega}\left(-2\sin
2\omega t^{\prime}+\cot\omega t\sin^{2}\omega
t^{\prime}\right)\delta(t^{\prime}-\tau_{i})\right]\begin{pmatrix}\tilde{y}(t^{\prime})\\\
\tilde{y}(t^{\prime})\end{pmatrix}\\\ &\quad+\begin{pmatrix}\\\
\tilde{x}(t^{\prime})&\tilde{y}(t^{\prime})\end{pmatrix}\cdot\left[\sum_{i}\dfrac{c_{i}^{2}\epsilon^{2}}{4m\omega}\left(-2\sin
2\omega t^{\prime}+\cot\omega t\sin^{2}\omega
t^{\prime}\right)\delta(t^{\prime}-\tau_{i})\right]\begin{pmatrix}\tilde{y}(t^{\prime})\\\
\tilde{x}(t^{\prime})\end{pmatrix}\big{\\}}\\\
&\quad\exp\frac{-1}{\hbar}\int_{0}^{t}dt^{\prime}\big{\\{}\begin{pmatrix}\\\
\tilde{x}(t^{\prime})&\tilde{x}(t^{\prime})\end{pmatrix}\cdot\left[\sum_{i}\dfrac{c_{i}^{2}\epsilon^{2}}{4m\omega}\cos
2\omega
t^{\prime}\coth\frac{\hbar\omega}{2kT}\delta(t^{\prime}-\tau_{i})\right]\begin{pmatrix}\tilde{x}(t^{\prime})\\\
\tilde{x}(t^{\prime})\end{pmatrix}\\\ &\quad+\begin{pmatrix}\\\
\tilde{y}(t^{\prime})&\tilde{y}(t^{\prime})\end{pmatrix}\cdot\left[\sum_{i}\dfrac{c_{i}^{2}\epsilon^{2}}{4m\omega}\cos
2\omega
t^{\prime}\coth\frac{\hbar\omega}{2kT}\delta(t^{\prime}-\tau_{i})\right]\begin{pmatrix}\tilde{y}(t^{\prime})\\\
\tilde{y}(t^{\prime})\end{pmatrix}\\\ &\quad+\begin{pmatrix}\\\
\tilde{x}(t^{\prime})&\tilde{y}(t^{\prime})\end{pmatrix}\cdot\left[\sum_{i}\dfrac{c_{i}^{2}\epsilon^{2}}{4m\omega}\cos
2\omega
t^{\prime}\coth\frac{\hbar\omega}{2kT}\delta(t^{\prime}-\tau_{i})\right]\begin{pmatrix}\tilde{y}(t^{\prime})\\\
-\tilde{x}(t^{\prime})\end{pmatrix}\big{\\}}\\\
&\quad+exp\frac{i}{\hbar}\sum_{i=0}^{N}\dfrac{c_{i}^{2}}{m\omega\sin\omega
t}\int_{0}^{t}dt^{\prime}\int_{0}^{t^{\prime}}dt^{\prime\prime}\begin{pmatrix}\\\
\tilde{x}(t^{\prime})&\tilde{y}(t^{\prime})\end{pmatrix}\cdot\sin\omega(t-t^{\prime})\sin\omega
t^{\prime\prime}\begin{pmatrix}\tilde{x}(t^{\prime})\\\
-\tilde{y}(t^{\prime\prime})\end{pmatrix}.\end{split}$ (34)

Equation (34) clearly shows the single time integral characteristic of a
LIndbladian, which is read off by comparing with equation (11) under the no
momentum dependence. This Lindbladian has a time-decoherence effect due to the
initial thermal distribution of the bath, same as the (FV)/(CL) example but
there coming from the purely non-Markov effect. The non-Markov terms of
equation (35) is quite simple because it comes about from the fluctuation
terms of the bath harmonic oscillator interacting incessantly with the
particle coordinate $\tilde{x}(t^{\prime})$. In the (FV)/(CL) example, even
the classical harmonic oscillator interacts all the time and all at the same
time with the particle.

5\. Now suppose the oscillators have different frequencies
$(\omega_{1},....,\omega_{k},...)$, and that the particle only interacts with
an oscillator if it is around or near the vicinity of that oscillator of
frequency $\omega_{k}$ in time $\epsilon$ then moves on to the next oscillator
in time $\tau$. The calculation will still show a Lindbladian because the
calculation done in part 4 still can be done, to a certain extent. Using
equations (20), (22) (23) in the first term equation (24), the integral can
made into a sum over time $(\tau_{1},\tau_{2},...,\tau_{i},...)$ where the
index i is the same as the index in the frequencies $\omega_{i}$ when
following the evolution from bath oscillators
$\tilde{X_{i}}(t^{\prime}=0)=X^{\prime}_{i}$ to
$\tilde{X_{i}}(t^{\prime}=t)=X_{i}$. But when we follow the the evolution from
bath oscillators $\tilde{Y_{k}}(t^{\prime}=0)=Y^{\prime}_{k}$ to
$\tilde{Y_{k}}(t^{\prime}=t)=Y_{k}$, the time assignments
$(\tau_{1},\tau_{2},....\tau_{i},...)$ will not have the same indices that
goes with the frequencies and end points. Still the sequential time
assignments can be made but for a given time $\tau_{i}$ the end point
conditions that appear and the frequencies will have two different indices.
For that particular time $\tau_{i}$, we may have
$X_{i},X^{\prime}_{i},Y^{\prime}_{i}$ and
$X_{k},X^{\prime}_{k},Y^{\prime}_{k}$. But still the end point integrals can
be done two at a time but more difficult and the results will not be as nice
and simple as in part 4. In other words, the very local interaction of the
particle with the oscillators will still yield a Lindbladian, The non-Markov
term will still be the same as in part 4.

6\. There are many Lindblad systems discussed in the literature, for examples,
see Chruscinski . These examples can be called phenomenological. It would be
nice if a derivation from a system interacting with an environment can be
shown for them. Maybe, the interaction can give another aspect, i.e., they may
have a non-Markov contribution.

In conclusion, it is nice to find that the two methods, the path-integral or
sum over histories by Feynman-Vernon and the (LGKS), in dealing with quantum
statistical mechanics can be related and are consistent.

###### Acknowledgements.

I have always wanted to find out how to derive (LGKS) from (FV). In learning
this, Gravity, a loyal company, kept me focused and distracted, both needed
for me to keep on working. Thank you Gravity.

## References

  * (1) Feynman, R. P. and Vernon, F., The Theory of a General Quantum System Interacting with a Linear Dissipative System, Annals of Physics, 24, 118, 1963.
  * (2) Lindblad, G., On The Generators of Quantum Dynamical Semigroups, Commun. Math. Physics, 48, 119, 1976.
  * (3) Gorini, V., Kossakowski, A. and Sudarshan, E. C. G., Completely Positive Dynamical Semigroups of N-Level Systems, J. of Math. Phys., 17, 821, 1976.
  * (4) Reichl, L. E., A Modern Course in Statistical Physics, University of Texas Press, Austin, 1980.
  * (5) Struntz, W. T., Path-Integral semiclassical and stochastic propagators for Markovian open quantum systems, J. Phys. A, 30, 4053, 1997.
  * (6) Magpantay, Jose A., Lindblad Plus From Feynman-Vernon, 2020, arxiv: 2003.00194.
  * (7) Caldeira, A. and Leggett, A. J., Path Integral Approach To Quantum Brownian Motion, Physica, 121A, 587, 1983.
  * (8) Chruscinski, D. and Pascazio, S., A Brief History of the GKLS Equation, arxiv: 1710.05993v2, 2017.


# Degenerate interpretations of O3 spectral features in exoplanet atmosphere
observations due to stellar UV uncertainties: a 3D case study with TRAPPIST-1e

G. J. Cooke School of Physics and Astronomy, University of Leeds, Leeds, LS2
9JT, UK
Institute of Astronomy, University of Cambridge, UK.
D. R. Marsh School of Physics and Astronomy, University of Leeds, Leeds, LS2
9JT, UK
C. Walsh School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT,
UK
A. Youngblood NASA Goddard Space Flight Center, Solar System Exploration
Division, 8800 Greenbelt Road, Greenbelt, MD 20771, USA

###### Abstract

TRAPPIST-1e is a potentially habitable terrestrial exoplanet orbiting an
ultra-cool M Dwarf star and is a key target for observations with the James
Webb Space Telescope (JWST). One-dimensional photochemical modelling of
terrestrial planetary atmospheres has shown the importance of the incoming
stellar UV flux in modulating the concentration of chemical species, such as
O3 and H2O. In addition, three-dimensional (3D) modelling has demonstrated
anisotropy in chemical abundances due to transport in tidally locked exoplanet
simulations. We use the Whole Atmosphere Community Climate Model Version 6
(WACCM6), a 3D Earth System Model, to investigate how uncertainties in the
incident UV flux, combined with transport, affect observational predictions
for TRAPPIST-1e (assuming an initial Earth-like atmospheric composition). We
use two semi-empirical stellar spectra for TRAPPIST-1 from the literature. The
UV flux ratio between them can be as large as a factor of 5000 in some
wavelength bins. Consequently, the photochemically-produced total O3 columns
differ by a factor of 26. Spectral features of O3 in both transmission and
emission spectra vary between these simulations (e.g. differences of 19 km in
transmission spectra effective altitude for O3 at 0.6 µm). This leads to
potential ambiguities when interpreting observations, including overlap with
scenarios that assume alternative O2 concentrations. Hence, to achieve robust
interpretations of terrestrial exoplanetary spectra, characterisation of the
UV spectra of their host stars is critical. In the absence of such stellar
measurements, atmospheric context can still be gained from other spectral
features (e.g. H2O), or by comparing direct imaging and transmission spectra
in conjunction.

Exoplanets (498) — Exoplanet atmospheres (487) — Transmission spectroscopy
(2133) — Exoplanet atmospheric composition (2021)

††thanks: E-mail<EMAIL_ADDRESS>
, ,

##  1 Introduction

On account of their frequency and relative ease of characterisation, planetary
systems orbiting M dwarf stars are prime targets in the search for potentially
habitable exoplanets. TRAPPIST-1 is an M8.5V star, orbited by seven
terrestrial exoplanets, and each of them could be tidally locked to their host
star (Gillon et al., 2017). TRAPPIST-1e is of particular interest because with
current knowledge it is more likely than the close-in planets to have retained
its atmosphere due to lower predicted escape rates (Dong et al., 2018), and is
probably more able to sustain surface liquid water when compared to the outer
planets which receive less stellar irradiation (Gillon et al., 2017; Wolf,
2017). To begin to characterise the TRAPPIST-1 exoplanets and determine the
composition of their atmospheres, at the time of writing, several
observational programs with JWST are scheduled, including the observation of
four transits of TRAPPIST-1e in 2023 (see program
1331)111https://www.stsci.edu/jwst/science-execution/program-
information.html?id=1331, accessed Wed April 12 2023.

In general, the detection of molecular oxygen (O2) on an exoplanet is of
profound interest because of its importance for life on Earth (Segura et al.,
2003; Meadows et al., 2018). Ozone (O3) is produced in exoplanetary
atmospheres by UV radiation which is able to dissociate O2, and it has been
calculated that in some situations the detection of O3 is easier to achieve
than a detection of O2; for example, for low O2 concentrations like those
potentially present during Earth’s Proterozoic eon, but where O3
concentrations are still detectable (Kozakis et al., 2022). Thus, in such
cases, it has been proposed that the detection of O3 may be used as a proxy to
confirm the presence of O2 (Leger et al., 1993; Segura et al., 2003; Meadows
et al., 2018; Quanz et al., 2021).

One-dimensional (1D) photochemical modelling has demonstrated that planetary
atmospheric composition (including O3 and H2O) is influenced by the strength
and shape of the incoming ultraviolet (UV) radiation from the host star (see
Grenfell et al., 2014; Rugheimer et al., 2013; Kozakis et al., 2022; Meadows
et al., 2018, and references therein). For example, Teal et al. (2022) used
MUSCLES Treasury survey M-dwarf spectra combined with UV spectra
reconstructions as stellar spectra input to Atmos (a coupled 1D photochemistry
and climate model), and similarly demonstrated that UV irradiation can
modulate hydrocarbon haze concentrations. Because the atmospheric composition
with respect to altitude affects molecular detectability in remote sensing,
the link between O3 abundance and O2 abundance will be difficult to ascertain
because it depends on several parameters, including the catalytic cycles that
remove O3 (e.g. HOx, NOx, ClOx and BrOx chemical families), and atmospheric
pressure. A well-characterised spectrum of the host star is required for
confident modelling of planetary climate (Eager-Nash et al., 2020),
atmospheric chemistry (Kozakis et al., 2022), and atmospheric escape (Dong et
al., 2018). However, the host star’s spectral energy distribution may not be
known to high precision when analysing and interpreting exoplanet
observations.

The UV flux from TRAPPIST-1 remains uncertain because of the intrinsic
faintness of the star ($V=18.798$ mag; Costa et al., 2006). Peacock et al.
(2019), henceforth known as P19, modelled the spectrum of TRAPPIST-1. To do
this, they used the PHOENIX stellar atmospheric code (Baron & Hauschildt,
2007; Hauschildt, 1993; Hauschildt & Baron, 2006), and added a treatment of
the chromosphere to produce synthetic stellar spectra of cool dwarf stars,
including TRAPPIST-1, whose ultraviolet light have negligible flux
contribution from the photosphere. More recently, Wilson et al. (2021),
hereafter known as W21, used new observations of TRAPPIST-1 to create a
semiempirical spectrum for use in atmospheric modelling simulations. Whilst
neither spectrum wholly represents the true stellar irradiation environment of
the TRAPPIST-1 planets, the W21 spectrum is in significantly better agreement
with available observations of TRAPPIST-1.

In addition to the incoming stellar spectrum, the 3D transport and chemistry
of the exoplanet’s atmosphere is important for understanding the distribution
and abundance of chemical species. Chen et al. (2019) investigated exoplanets
orbiting at the inner edge of the habitable zone using a 3D chemistry climate
model (WACCM4), showing how different assumed UV spectra can influence
atmospheric mixing ratios of species such as H2O, O3, and H. Additionally, 3D
simulation studies have demonstrated the influence of UV radiation on 3D
transport (Chen et al., 2019), stratospheric temperature (Godolt et al., 2015;
Chen et al., 2019), and the distribution and abundance of chemical species
(Chen et al., 2018, 2019, 2021; Braam et al., 2022; Ridgway et al., 2022).
Tidally locked terrestrial exoplanets modelled in 3D exhibit atmospheric jets
that transport heat and chemical constituents to the night side (Showman &
Polvani, 2011; Eager-Nash et al., 2020; Yates et al., 2020). Proedrou & Hocke
(2016), Chen et al. (2018) and Yates et al. (2020) found that O3, which is
photochemically generated on the dayside, can be transported to the night
side, where its lifetime increases due to the lack of UV irradiation and a
reduction in catalytic cycle destruction. This body of previous work motivates
the need to use 3D models when investigating the climate and chemistry of
specific exoplanets, in particular, with respect to their molecular
observability linked to the oxygenation state of the atmosphere.

In this study we simulate the atmosphere of TRAPPIST-1e using the WACCM6 Earth
System Model and the stellar spectra from P19
(222doi:10.17909/t9-j6bz-5g89model 1A, version 1) and W21
(333https://zenodo.org/record/4556130#.Y_9hOOb7RmM version 7) which differ in
UV flux in some wavelength bins by up to a factor of 5000. Our aim is to
quantify the effects of such uncertainties in the strength of UV from the host
star on the climate and composition of the atmosphere. This is the first time
a 3D global climate model has been used to simultaneously assess the influence
of uncertain UV spectra and transport on the climate and chemistry of
TRAPPIST-1e. Additionally, we simulate possible future observations
(transmission and emission spectra) of TRAPPIST-1e using the outputs from the
WACCM6 simulations. We discuss the implications of uncertainties in the
stellar UV spectrum in the interpretation of future observations of
terrestrial exoplanets.

##  2 Methods

###  2.1 UV spectra input

This work employs two different assumed stellar spectra as input to the
simulations. P19 (Peacock et al., 2019) generated three different models of
TRAPPIST-1’s spectrum (Model 1A, 2A, and 2B). Model 1A was created such that
the emission was benchmarked to the Ly-$\alpha$ reconstruction that was
presented in Bourrier et al. (2017), who used the Hubble Space Telescope (HST)
Space Telescope Imaging Spectrograph instrument to observe the TRAPPIST-1
Ly-$\alpha$ line. Using an alternative approach to construct Model 2A and
Model 2B, P19 aligned the stellar emission to be within with the range of
distance-corrected Galaxy Evolution Explorer NUV photometry of stars with a
spectral type akin to TRAPPIST-1, whilst remaining compatible with FUV upper
limits. The EUV estimates were extracted from empirical scaling relationships
based on X-ray and Ly-$\alpha$ emission.

W21 (Wilson et al., 2021) used new HST (1100 – 5500 Å) and XMM-Newton (10 – 50
Å) observations of TRAPPIST-1 from the Mega-MUSCLES (Measurements of the
Ultraviolet Spectral Characteristics of Low-Mass Exoplanetary Systems)
Treasury Survey (Froning et al., 2019; Wilson et al., 2021) to construct a 5 Å
– 100 µm spectrum of the star. Four models were used to fill in gaps in
wavelength coverage, including a PHOENIX model for wavelengths $>5500$ Å
($>0.55$ µm). Because of TRAPPIST-1’s relatively low luminosity, W21
substituted the noisy 1100 – 4200 Å HST spectrum with a semi-empirical,
noiseless spectrum that reproduced the measured flux of detected UV emission
lines and agreed with the upper limits on the stellar continuum established by
the HST spectra.

W21 found that whilst the P19 Model 1A shows good agreement with the C II and
Ca II stellar lines, several other lines are inconsistent with measured
fluxes, exhibiting a discrepancy $\sim 10$ times more than the W21 upper
limits permit. In the WACCM6 simulations, we decided to use both the P19 and
W21 spectra in order to illustrate important differences that may occur in
instances where the exoplanet’s photochemical environment is highly uncertain.

###  2.2 WACCM6

TRAPPIST-1 is an ultra cool M dwarf at a distance of 12.4 pc. Its stellar
properties are summarised in Table LABEL:Properties_table. We assume that
TRAPPIST-1e receives $900~{}\textrm{W\,m}^{-2}$ of irradiation (0.66
$S_{\oplus}$, where $S_{\oplus}$ is the total insolation received by the
Earth). This is consistent with the value used in the TRAPPIST-1 Habitable
Atmosphere Intercomparison project (THAI; Fauchez et al., 2020a; Turbet et
al., 2022), although note that the latest data available in the NASA Exoplanet
Archive (Akeson et al., 2013) lists a value of $0.646\pm 0.025\ S_{\oplus}$
(Agol et al., 2021).

We use the Earth System Model WACCM6 to simulate the climate of TRAPPIST-1e
with the properties indicated in Table LABEL:Properties_table, using given
values from from Delrez et al. (2018) and Grimm et al. (2018). Note that the
simulations were started in 2020, before Agol et al. (2021) published their
work. WACCM6 is a specific configuration of the Community Earth System Model
version 2 (CESM2). The model release we use in this work is
CESM2.1.3444http://www.cesm.ucar.edu/models/cesm2/. We use a pre-industrial
atmosphere (approximating the atmosphere of 1850 in terms of pollutants and
greenhouse gas mixing ratios) with the modern ocean and land configuration, a
horizontal resolution of $1.875^{\circ}$ by $2.5^{\circ}$ (latitude by
longitude), and 70 vertical atmospheric levels. The ocean and atmosphere are
fully interactive, meaning that they respond to physical perturbations such as
temperature, or in the case of the atmosphere, chemical perturbations. Because
TRAPPIST-1e is suspected to be tidally locked (Gillon et al., 2017), we lock
the substellar point by fixing the solar zenith angle in each grid cell, and
we set the exoplanet’s obliquity and orbital eccentricity to zero (note that
the eccentricity may be non-zero, albeit $<0.01$; Luger et al., 2017). The
substellar point is placed the Pacific Ocean at $180^{\circ}$ longitude and
$0^{\circ}$ latitude. The chemical mechanism, which is described in Emmons et
al. (2020), has 98 chemical species and 298 chemical reactions (including
photochemical reactions). Absorption by CO2 and H2O in the Schumann-Runge
bands is included (Ji et al., 2023). The full details of the model set-up,
alongside simulation scripts, are available via
GitHub555https://github.com/exo-
cesm/CESM2.1.3/tree/main/Tidally_locked_exoplanets.

Table 1: Orbital and planetary parameters used in this study, for the exoplanet TRAPPIST-1e and its star (TRAPPIST-1). The parameters are from Delrez et al. (2018), Grimm et al. (2018), and Agol et al. (2021). The mass and radius are chosen to be consistent with the THAI project (Fauchez et al., 2020a; Sergeev et al., 2022b; Turbet et al., 2022). $\odot$ and $\oplus$ subscripts denote values relative to the Sun and Earth, respectively. Parameter | Value
---|---
Stellar luminosity [$L_{\odot}$] | 0.000553
Stellar effective temperature [K] | 2566
Stellar radius [$R_{\odot}$] | 0.119
Stellar mass [$M_{\odot}$] | 0.089
Stellar metallicity | 0.04
Planetary orbital period [days] | 6.099
Planetary radius [$R_{\oplus}$] | 0.91
Planetary mass [$M_{\oplus}$] | 0.772
Planetary surface gravity [m s-2] | 9.1454

We scale the P19 and W21 spectra to the irradiance received by TRAPPIST-1e
($900~{}\textrm{W\,m}^{-2}$), rebinning them to match the wavelength grid
required for WACCM6 simulations. The resulting spectra are shown in Fig. 1.
For the wavelength regions over which O2 and O3 photolyse, the integrated flux
is listed in Table 2.

Figure 1: The top of atmosphere irradiance per unit wavelength as a function
of wavelength for the TRAPPIST-1 semi-empirical stellar spectra from Peacock
et al. (2019, orange) and Wilson et al. (2021, light blue), each of which are
scaled to the total integrated irradiance received by TRAPPIST-1e
($900~{}\mathrm{W\,m}^{-2}$). These spectra were rebinned in order to match
the wavelength grid used in WACCM6 simulations. The grey shaded region
indicates UV wavelengths (100 – 400 nm).

We present ten simulations with WACCM6, five with the P19 spectrum, and five
with the W21 spectrum – see Table LABEL:Run_table for a summary. The initial
conditions for the five simulations with each spectrum consist of one with the
standard initial pre-industrial Earth composition (PI), then three lower O2
simulations with a composition with 10, 100, and 1000 times less O2 (10% PAL,
1% PAL, and 0.1% PAL), and one in which the planet is not tidally locked
(noTL). These simulations allow us to assess the influence of different
strengths of incoming UV spectra, the influence of tidal locking, and the
effects of reducing O2 in order to quantify any degeneracies between O2, O3,
and incident UV light. Each simulation is run for over 250 model Earth years,
and the last ten years of the simulation are used for time-averaged results
(Figs 2 – 6 and Figs 9 – 11). The WACCM6 simulations were run on 120 cores at
a model cost of 1,332 pe-hours per simulated year. A 250 year simulation
therefore costs 333,000 pe-hours.

Table 2: For the two different spectra used in the simulations, the integrated flux (in units of W m-2) is given for each spectrum for the total flux and six different wavelength bands: Schumman-Runge continuum (S-RC), Schumman-Runge bands (S-RB), Herzberg continuum (HC), Hartley band (HaB), Huggins band (HuB), and the Chappuis band (CB). For reference, the Earth receives 1360 $\textrm{Wm}^{-2}$ of irradiation. Photons in the Schumman-Runge continuum, Schumann-Runge bands, and Herzberg continuum are able to photolyse O2. Photons in the Hartley band, Higgins band, and Chappuis band, are able to photolyse O3. Spectrum | Total | S-RC | S-RB | HC | HaB | HuB | CB
---|---|---|---|---|---|---|---
| 10.5–99975 nm | 130–175 nm | 176-192 nm | 200-240 nm | 200-310 nm | 310–340 nm | 400-650 nm
Sun | 1361 | 0.0092 | 0.0373 | 1.2997 | 19.8916 | 23.4556 | 453.7313
TP-1 P19 | 900 | 0.0382 | 0.0239 | 0.0841 | 0.3953 | 0.0829 | 2.1355
TP-1 W21 | 900 | 0.0025 | 0.0001 | 0.0002 | 0.0050 | 0.0133 | 2.3045
Flux ratio $\frac{\textrm{P19}}{\textrm{W21}}$ | 1.00 | 15.54 | 451.81 | 528.85 | 79.59 | 6.21 | 0.93
Table 3: Ten simulations have been performed, each with an obliquity of $0^{\circ}$ and a circular orbit, using two different spectra for TRAPPIST-1 from P19 and W21 (see text for UV spectra input details). Eight simulations were set up in a tidally locked configuration with a 6.1 day rotation rate. The “PI” simulations have an initial pre-industrial Earth composition and are the same apart from using the P19 and W21 spectra. The “noTL” simulations have an initial pre-industrial Earth composition, are not tidally locked, and have a diurnal cycle (rotational period of 1 day). For the lower O2 scenarios, the volume mixing ratio of atmospheric O2 is reduced by a factor of 10, 100, and 1000 (the “10% PAL”,“1% PAL”, and “0.1% PAL” simulations, respectively). PAL means present atmospheric level, where the present atmospheric level of oxygen is a volume mixing ratio of 0.21. Simulation | Spectrum |  O2 mixing ratio [PAL] | Orbital parameters
---|---|---|---
P19 PI | P19 | 1 | Tidally locked, 6.1 day rotational period
P19 10% PAL | P19 | 0.1 | Tidally locked, 6.1 day rotational period
P19 1% PAL | P19 | 0.01 | Tidally locked, 6.1 day rotational period
P19 0.1% PAL | P19 | 0.001 | Tidally locked, 6.1 day rotational period
P19 noTL | P19 | 1 | Not tidally locked, 1 day rotational period
W21 PI | W21 | 1 | Tidally locked, 6.1 day rotational period
W21 10% PAL | W21 | 0.1 | Tidally locked, 6.1 day rotational period
W21 1% PAL | W21 | 0.01 | Tidally locked, 6.1 day rotational period
W21 0.1% PAL | W21 | 0.001 | Tidally locked, 6.1 day rotational period
W21 noTL | W21 | 1 | Not tidally locked, 1 day rotational period

###  2.3 Planetary Spectrum Generator

We use the Planetary Spectrum Generator (PSG; Villanueva et al., 2018)
GlobES666 https://psg.gsfc.nasa.gov/apps/globes.php 3D mapping tool to compute
transmission and emission spectra from the WACCM6 atmospheric simulations,
using instantaneous data (a ‘snapshot’ at a model time step rather than an
average over model time steps) for each produced spectrum. The WACCM6
instantaneous data are rebinned to a resolution of $10^{\circ}$ in longitude
only and we keep the same latitudinal grid resolution ($1.875^{\circ}$). This
is done to reduce the data size so it is compatible with GlobES in PSG, and is
the same approach as used in Cooke et al. (2023). The same data at the same
time step is used for both transmission spectra and emission spectra in this
paper. PSG ingests the data and integrates across the whole observable disk to
produce a reflection or emission spectrum. It uses the grid cells at the
terminator to produce a transmission spectrum. Because model grid cells either
side of the terminator would contribute to the opacity of the atmosphere, this
is not the most realistic way to represent the atmospheric geometry during a
planetary transit (Caldas et al., 2019). However, it is adequate for the
relatively low temperature contrasts between the day and night sides for the
TRAPPIST-1e atmospheres simulated here.

The molecules we use for the computation of the transmission and emission
spectra are N2, O2, CO2, H2O, O3, CH4, and N2O which is a list that includes
possible biosignatures and indicators of habitability (Kaltenegger, 2017). The
default HITRAN opacity data (Gordon et al., 2017) is used for each molecule,
as well as all available collision-induced absorption coefficients (e.g.
O2-O2, N2-N2, and O2-N2). The radiative transfer model used in PSG is the
Planetary and Universal Model of Atmospheric Scattering (PUMAS) model. The
correlated-k method is used by PUMAS for the spectral resolving powers used in
this paper (Caldas et al., 2019). If high resolution is required, it would
instead use the line-by-line method. Scattering effects are included, as are
ice clouds and water clouds. The effective radius of the cloud particles is
assumed to be 5 µm for water and 100 µm for ice clouds. Scripts and data to
generate the PSG files are provided with the data associated with this
article.

##  3 Results

###  3.1 Atmospheric chemistry

Figure 2: The P19 (left column) and W21 (right column) irradiation scenarios
are compared to each other for the PI (top row) and 0.1% PAL (bottom row)
cases. The zonal mean (averaged over longitude) of the production of O from O2
photolysis is displayed, with latitude in ∘ on the horizontal axis and
pressure in hPa on the vertical axis. O2 photolysis takes place at wavelengths
less than 242 nm. The white regions show where the production has dropped
below $10^{8}$ molecules m-3 s-1.

Atmospheric O3 is created and destroyed via the Chapman cycle (Chapman, 1930),
which is initiated by ultraviolet light (of wavelength $\lambda$) dissociating
an O2 molecule:

(1) (2) (3) (4)

The latter two photolysis reactions do not contribute to O3 destruction
because the O produced rapidly recombines with O2 to produce O3 via reaction
3.1. However, the reaction between O and O3 (reaction 3.1) does lead to a loss
of O3, and it can be sped up through catalytic agents, which are denoted below
as X. An example of a catalytic cycle is shown here:

$\begin{split}\ce{X+O3&->XO+O2},\\\ \ce{XO+O&->X+O2},\\\ \textrm{Overall:}\
\ce{O3+O&->2O2}.\end{split}$ (5)

Catalytic agents may be NO, H, OH, Cl, or Br (Brasseur & Solomon, 2005). These
species can be produced through other photolysis reactions (e.g., H2O
photolysis producing H and OH).

Figure 3: The P19 (left column) and W21 (right column) irradiation scenarios
are compared to each other for the PI (top row) and 0.1% PAL (bottom row)
cases. The zonal mean (average over longitude) of the O3 number density is
shown, with latitude in ∘ on the horizontal axis and pressure in hPa on the
vertical axis. The white regions show where the number density has dropped
below $10^{16}$ molecules m-3.

Fig. 2 shows the zonal mean of O production from O2 photolysis in the P19 PI,
P19 0.1% PAL, W21 PI, and W21 0.1% PAL simulations. In the two PI simulations,
O2 can be photolysed lower down in the P19 case because there is more incoming
UV radiation than in the W21 case. When O2 is reduced to 0.1% PAL, it can be
seen that the peak of O2 photolysis moves to higher pressures (downward in
altitude). Where O is produced impacts the number density of O3. Fig. 3 shows
the zonal mean of the O3 number density for the same set of simulations as in
Fig. 2. Reducing O2 in the P19 cases causes a drop in O3 number density at
pressures above 1 hPa, whilst the opposite occurs in the W21 cases. This is
because of the pressure dependency on the reaction that produces O3 (reaction
3.1).

Figure 4: The global mean temperature (a) is plotted against pressure for the
P19 PI (orange), P19 noTL (dark red), P19 10% PAL (brown), P19 1% PAL
(yellow), and P19 0.1% PAL (red) simulations, and the W21 PI (light blue), W21
noTL (black), W21 10% PAL (blue), W21 1% PAL (lilac), and W21 0.1% PAL (grey)
simulations. The globally averaged mixing ratios for O (b), O2 (c), O3 (d),
H2O (e), CH4 (f), N2O (g), and CO2 (h), are also shown.

In Fig. 4 we show the global mean vertical profiles for temperature, and the
mixing ratio of O, O2, O3, H2O, CH4, N2O, and CO2. On global average, the
temperature profile shows deviations of up to 9 K in the troposphere and up to
19 K below the thermosphere between the W21 PI and P19 PI simulations. The W21
PI middle atmosphere (between the troposphere and thermosphere) is colder than
the P19 PI middle atmosphere due to reduced O3 heating because of the lower O3
concentration. The P19 noTL simulation has a lower temperature in the
troposphere by up to 23 K and in the middle atmosphere by up to 6 K, resulting
in lower concentrations of H2O compared to the P19 PI simulation. Generally,
between 200 and 10 hPa, the relative H2O number density in each simulation
correlates with the relative temperature profile in each simulation (i.e. a
lower temperature results in less H2O).

The O3 mixing ratio profile in Fig. 4 shows large deviations between the P19
PI (orange) and W21 PI (light blue) simulations, with a difference of a factor
of 116 in the mixing ratio at the surface. The O3 number density peaks at
$2.0\times 10^{19}$ molecules m-3 at 50 hPa in the P19 PI case and at
$6.3\times 10^{17}$ molecules m-3 at 50 hPa in the W21 PI case. Despite the
amount of O3 present in the P19 PI simulated atmosphere, the temperature
inversion in the middle atmosphere is $<8$ K because the UV intensity is not
sufficient to provide enough O3 UV heating to create a similar temperature
inversion to Earth’s stratosphere ($\approx 60$ K change between the
tropopause and stratopause). The P19 noTL simulation (dark red) shows that
tidal locking increases the amount of O3 in the middle atmosphere, whilst
reducing the O3 concentration in the troposphere. The non-tidally locked cases
are colder in the lower atmosphere, and warmer in the middle atmosphere. This
reduces the rate of catalytic destruction reactions and allows O3 formation to
occur faster.

Figure 5: The O3 column in Dobson Units [DU] across the simulation’s
latitudinal and longitudinal grid is plotted for all the simulations used in
this work (the P19 scenarios are in the left column and the W21 scenarios are
in the right column). 1 DU is equal to $2.6867\times 10^{20}$ molecules m-2.
In all simulations, O3 column maxima occur at high latitudes. The substellar
point for the tidally locked cases is at $180^{\circ}$ longitude and
$0^{\circ}$ latitude on Earth’s coordinate grid (in the centre of each
Robinson projection). Note that the colour bars are extended for some
simulations in order to show the O3 column structure in each scenario. Each
panel has a different colour bar range. Figure 6: The oxygen (O2)
concentration is shown against the ozone (O3) column in each simulation
(orange for P19 scenarios and blue for W21 scenarios). The ‘error bar’ lines
show the range between the minimum and maximum O3 columns in each simulation.
The circular dots show the mean O3 columns. The O3 columns simulated by WACCM6
for Earth, from Cooke et al. (2022), are shown in black for comparison.

Fig. 5 displays the longitudinal and latitudinal variation of the O3 column
for the simulations averaged over the last ten years of each scenario. O3 is
inhomogeneously distributed horizontally, which highlights the importance of
using 3D models. The P19 PI simulation has a global mean O3 column of 1310 DU
(approximately 4.4 times Earth’s global mean value of $\approx 300$ DU, where
1 DU = $2.687\times 10^{20}$ molecules m-2), whilst the W21 PI simulation
predicts a global mean O3 column of 50 DU. Both of the O3 column maxima, at
7152 DU and 134 DU for the P19 PI and W21 PI simulations, respectively, occur
near the southern pole.

Fig. 6 shows the total range of O3 columns produced across the atmospheric
latitude-longitude grid against the O2 volume mixing ratio. The O3 column
depends on the incident UV radiation, O2 concentration, atmospheric pressure,
temperature, and transport, as well as the rates of destruction. In the W21 PI
case, the UV radiation is absorbed high in the atmosphere where the pressure
is low, so little O3 is produced and the Ox (O + O3) is primarily in O. As O2
is decreased, the O3 column increases. In contrast, in the P19 simulations,
which have larger incoming fluxes of UV, as O2 increases above 1% PAL, the O3
column increases, similar to the relationship on Earth. Both of these data
sets are rather different when compared to previous results that have
simulated the atmospheres of planets around late M dwarf stars (e.g.,
Rugheimer & Kaltenegger, 2018; Kozakis et al., 2022, although note that the
total instellation in those simulations were set to Earth’s modern value).
Whilst there are likely many interacting parameters which will cause
dissimilar O3 column predictions, the very large discrepancies with 1D models
could be due to 3D transport. This seems plausible because Braam et al. (2023)
reported a stratospheric circulation in simulations of a terrestrial Proxima
Centauri b scenario, in which the winds move O3 from the day side to the night
side, with similar circulation effects occurring in the WACCM6 simulations
here (the large-scale dynamics of the atmosphere will be explored in future
work).

###  3.2 Transmission spectra

In Fig. 7 we show idealised transmission spectra between 0.1 – 11 µm generated
using the WACCM6 simulations with PSG (excluding the non-tidally locked
cases). The model date chosen for the transit is arbitrary. Because there are
time-dependent fluctuations for several variables in the WACCM6 simulations
(e.g. clouds, chemical mixing ratios), we could have investigated time
variability in the transmission spectra. However, Fauchez et al. (2022) showed
that such variability is within the measurement uncertainties of JWST. The
spectra are binned to approximate a resolving power of $R=250$ to show detail
in spectral features, where $R=\lambda/\Delta\lambda$, $\lambda$ is the
wavelength, and $\Delta\lambda$ is width of the wavelength bin. A 5 ppm error
bar corresponding to the lowest achievable noise with JWST instruments, which
may be between 5 – 20 ppm as calculated by Matsuo et al. (2019), Schlawin et
al. (2020), Schlawin et al. (2021), and Rustamkulov et al. (2022), is
indicated.

The differences in the effective altitude of O3 spectral features between the
P19 PI (orange) and W21 PI (light blue) transmission spectra are $-4$ km,
$+19$ km, $+15$ km, and $+17$ km for the 0.25 µm, 0.6 µm, 4.71 µm, and 9.6 µm
O3 features, respectively. Despite the W21 PI simulation having an O3 column
$\approx 26$ times lower than the P19 PI simulation, the W21 PI UV feature
(centred at 0.25 µm) due to the Hartley band (0.2 – 0.31 µm) actually has the
largest effective altitude between 0.2 – 0.3 µm of all the simulations. This
is because the Hartley band saturates quickly and the W21 PI atmosphere has
more total O3 than the P19 PI atmosphere above $\approx 0.5$ hPa. Between 0.3
– 0.35 µm, the temperature dependence of the Hartley and Huggins bands reduces
the effective height of the W21 PI transmission spectra due to the colder
middle atmosphere in the W21 PI simulation. At 0.6 µm, a significant detection
of O3 with JWST in the W21 PI simulation scenario would be improbable given
that the noise floor is larger than the height of the feature. Therefore,
assuming that the W21 spectrum is closest to the true spectrum of TRAPPIST-1,
or the case that the true stellar UV emission is weaker, a null detection of
the 0.6 µm O3 feature should not rule out the presence of O2 abundances at
levels as high as the present-day Earth. For H2O, the spectral features are
stronger in the P19 PI transmission spectra compared to W21 PI by up to 8 km
which is a result of a larger number density of H2O in the middle atmosphere.
Despite the difference in temperature and O3 number density profiles between
the noTL simulation and PI simulations, the transmission spectra are
remarkably similar (within $\pm$4 km effective altitude, not shown for
clarity). The P19 0.1% PAL transmission spectrum (red) produces a
quantitatively similar transmission spectrum feature at 9.6 µm to the W21 PI
simulation (light blue), even though there is a 1000 times difference in O2
mixing ratio between the two cases. The same can be said of the following
pairs at 9.6 µm: W21 1% PAL and P19 PI; W21 0.1% PAL and P19 1% PAL; W21 10%
PAL and P19 10% PAL. There is a noticeable difference between the spectra at
4.71 µm and 9 µm, but this would require reaching the most optimistic 5 ppm
noise floor in order to demonstrate that the two O2 scenarios are
distinguishable when there exist uncertain stellar UV flux estimates. Also
note that the effective height of the O2-X collision-induced absorption
feature at 6.4 µm (see Fauchez et al., 2020b, for more details) is 7 km
shallower when O2 is at 0.1% PAL in the P19 and W21 scenarios, compared to the
PI cases.

Figure 7: The transmission spectrum atmospheric effective altitude is plotted
against wavelength between 0.2 µm and 11 µm for the P19 PI (orange), P19 10%
PAL (brown), P19 1% PAL (yellow), and P19 0.1% PAL (red) simulations, and the
W21 PI (light blue), W21 10% PAL (blue), W21 1% PAL (lilac), and W21 0.1% PAL
(grey) simulations. The non-tidally locked cases are excluded for clarity, but
show little differences compared to the equivalent tidally locked case. The
transit depth, in terms of contrast with respect to the star, is indicated on
the right vertical axis in parts per million (ppm). The spectra are binned to
a spectral resolving power of $R=250$. Spectral features are indicated in
grey. The wavelength range of the proposed Habitable Worlds Observatory (HWO;
blue shaded region), and that of the JWST NIRSpec instrument (yellow shaded
region), are shown. They overlap in the green shaded region. The wavelength
range of the JWST MIRI instrument is indicated in the magenta shaded region.
The wavelength spacing between 0.2 – 1 µm is changed between 1 – 11 µm in
order to clearly show the UV and visible regions. The black bar represents the
uncertainty that would be present on a measurement that has reached the noise
floor of the instrument, where the noise floor is indicated as 5 ppm. Note
this error bar is an estimate of the performance of the telescope and does not
indicate a measurement. Figure 8: The left panel shows the PSG simulations of
planetary spectral radiance from emission spectra focused on the 4.71 µm O3
feature (which overlaps with a CO2 feature, both shown by the grey shaded
regions) for the P19 PI (orange), P19 10% PAL (brown), P19 1% PAL (yellow),
and P19 0.1% PAL (red) simulations, and the W21 PI (light blue), W21 noTL
(black), W21 10% PAL (blue), W21 1% PAL (lilac), and W21 0.1% PAL (grey)
simulations. The right panel shows the same for the 9.6 µm O3 feature, shown
by the grey shaded region. Telescopes and their instruments which may be able
to probe atmospheres in the infrared include: the Extremely Large Telescope
(ELT; Brandl et al., 2021) METIS instrument, the Large Interferometer for
Exoplanets (LIFE; Konrad et al., 2022), and the JWST (Morley et al., 2017)
NIRSpec and MIRI instruments.

###  3.3 Emission spectra

In Fig. 8 we show the emission spectra from each simulation for the
atmospheric absorption O3 features at 4.71 µm and 9.6 µm at $90^{\circ}$
orbital phase (the maximum planet-star separation as viewed in an edge-on
system with a circular orbit). The 4.71 µm feature overlaps with a CO2
feature, but O3 is the dominant absorber at 4.71 µm. The P19 PI simulation
(orange) predicts higher O3 columns than the W21 PI case (light blue); hence,
the depth of the features relative to the continuum in the W21 PI case are
weaker than the P19 PI scenario by a factor of 4.2 and 4.5 at 4.71 µm and 9.6
µm, respectively. With respect to the P19 0.1% PAL emission spectrum (red),
the P19 PI emission spectrum (orange) has a greater depth by a factor of 1.3
and 1.1 at 4.71 µm and 9.6 µm, respectively, even though the P19 PI simulation
has a mean O3 column which is 3.2 times higher than the P19 0.1% PAL
simulation (1310 DU versus 405 DU). The 3D effects are important here because
the largest O3 columns are found at the poles, but PSG is integrating over the
whole observable disk. At 4.71 µm, the noTL absorption features are deeper
than the tidally locked PI cases by a factor of 2.0 and 1.2 in the P19 and W21
scenarios, respectively. At 9.6 µm, these values are 1.2 and 1.0. In terms of
the relative depths of O3 features shown in Fig. 8, increasing depth is
generally shown with increasing mean O3 columns (see Fig. 5).

The tidally locked simulations exhibit strong convection and high clouds
around the substellar point, whereas the P19 noTL simulation has mainly low
clouds with comparatively little high cloud coverage. Therefore, this results
in the P19 noTL simulation (dark red) having the deepest O3 emission spectral
features.

It is important to note that other orbital phases may show quantitatively
different results because variability in spectral features may occur due to
climate variations throughout the orbit (e.g. Cooke et al., 2023).
Additionally, longer term variations may be expected due to the possible
presence of a ‘longitudinally asymmetric stratospheric oscillation’ (Cohen et
al., 2022).

##  4 Discussion

We have demonstrated that large differences in assumed stellar UV spectra can
lead to different predictions for the strength of O3 spectral features. These
spectral features may overlap at an assumed minimum observational uncertainty
of 5 ppm, despite the fact that the O2 concentration differs by factors of up
to 1000. In this section, we compare our results to previous work, consider
known uncertainties, and discuss work that should be done in preparation for
future exoplanet observations.

1D photochemical modelling of M dwarf terrestrial exoplanet atmospheres has
demonstrated that CH4 and N2O could have greater abundances in the middle
atmosphere compared to the modern Earth’s atmosphere (e.g. Segura et al.,
2005; Wunderlich et al., 2019), which we also find in our WACCM6 simulations
of TRAPPIST-1e (see Fig. 4). Teal et al. (2022) showed changes of over two
orders of magnitude in the middle atmosphere O3 mixing ratios when modelling a
modern Earth-like exoplanet that receives $1\ S_{\oplus}$ of irradiation
around GJ 176 (an M2.5V star), but with various UV irradiation scenarios. They
derived transmission spectra predictions from their atmospheric simulations
and found the maximum transit depth differences to be $<2$ ppm, which is below
the noise floor (5 ppm or greater) for JWST. This is in contrast to the W21 PI
and P19 PI transmission spectra results shown here, where the strengths of
estimated O3 features here are distinct at the 5 ppm level. Whilst different
atmospheric modelling methods are used in Teal et al. (2022) compared with
this work, the difference in observational significance is primarily due to
the size of the stars modelled: the radius of GJ 176 is $0.45\ R_{\odot}$, and
the radius of TRAPPIST-1 is $0.1192\ R_{\odot}$. Teal et al. (2022) also
demonstrated that hazy Archean Earth atmospheres were more sensitive to
changes in the incoming UV spectra compared to the modern Earth’s atmosphere,
which warrants future investigations for how uncertainties in the UV spectrum
of the host star affect hazy atmospheres in 3D models.

In terms of 3D modelling, the THAI series (Fauchez et al., 2020a; Sergeev et
al., 2022b; Turbet et al., 2022) has investigated the climate of TRAPPIST-1e
using four different 3D GCMs, assuming either an N2 or CO2 dominated
atmosphere and not including interactive chemistry, where the composition
evolves depending on chemical and photochemical reactions. The surface
temperatures in the WACCM6 simulations are similar although slightly lower
(219 – 231 K global mean compared to 230 – 240 K in the THAI simulations. See
appendix A and Fig. 9 for more details), which may be due to differences in
assumptions regarding the surface (including the distribution of the
continents and the fact that an interactive ocean is used here, in contrast to
a slab ocean with no meridional heat transport) or the composition of the
atmosphere. With a previous version of WACCM (CESM1), Chen et al. (2019)
investigated a planet with a 43.87 day orbital period around a star with an
effective temperature of 4000 K and an insolation of $1.9\ S_{\oplus}$, as
opposed to the $0.66\ S_{\oplus}$ used here, and assessed the impact of
uncertain host-star UV flux on the atmosphere. Chen et al. (2019) showed that
two different spectra (representing a quiescent and an active M dwarf star)
impacted the middle atmospheric concentrations of O3, OH, N2O, CH4, and H2O.
They calculated transmission spectra for the two simulated atmospheres,
finding that the only observable difference was for the O3 feature at 9.6 µm
(although the UV O3 feature is not shown in their figure 11). On the other
hand, the transmission spectra simulations shown here in Fig. 7 display
noticeable spectral differences for O3 at 0.25, 0.6, 4.7, 9.0 and 9.6 µm, as
well as for H2O between 5 – 6 µm. The differences in predicted observations
between our work and that of Chen et al. (2019) likely arise due to the
differences in exoplanetary system setup, the different stellar spectra, and
the calculated lower O3 columns from Chen et al. (2019), compared to the
simulated atmospheres here. 43.87 days is in the ‘slow rotator’ regime (for
the definition of tidally locked rotation regimes see Haqq-Misra et al.,
2018), and 6.1 days for TRAPPIST-1e can correspond to either the ‘Rhines
rotator’ or ‘fast rotator’ regime (Sergeev et al., 2022a). Thus, our results,
alongside those from Chen et al. (2019), demonstrate that 3D modelling results
are sensitive to the choice of the assumed stellar UV spectra for potentially
habitable tidally locked exoplanets across early and late M dwarf stars and
different rotation periods. Future work should also investigate the influence
of orbital perturbations away from a synchronous 1:1 spin-orbit resonance
(e.g. Chen et al., 2023) on composition.

The anisotropy of chemical molecules will affect the emission spectra and
photometry of terrestrial exoplanets, by modulating when the exoplanet is
brightest and dimmest at particular wavelengths (Selsis et al., 2011; Chen et
al., 2018). The phase curve photometry amplitude, and where the maximum
brightness occurs during the orbit, depends on the particular rotation regime
that the exoplanet exists within (Haqq-Misra et al., 2018). Thermal emission
features are influenced by molecular abundance, the atmospheric temperature,
and the temperature difference between the emitting and absorbing region
(e.g., for Earth, the infrared emission emanates from the troposphere, whilst
the absorbing region is the O3 layer in the stratosphere). This temperature
difference is larger for M dwarf exoplanets which exhibit a less pronounced
stratospheric temperature inversion due to lower incident UV emission for the
same total instellation.

Note that detecting O3 will be difficult with JWST within the nominal 5 year
mission lifetime (although JWST is expected to continue science operations for
at least 10 years), even for a modern Earth scenario (Lin et al., 2021), and
Fauchez et al. (2019) found that gases other than CO2 may require hundreds or
thousands of transits to be detectable. Simulations of high-resolution
observations with the extremely large class of telescopes indicate that O2 at
0.76 µm may be detectable in the case of TRAPPIST-1e within $\sim 100$
transits (Snellen et al., 2013; Rodler & López-Morales, 2014; Serindag &
Snellen, 2019).

The derived Mega-MUSCLES spectrum of TRAPPIST-1 (W21; Wilson et al., 2021) is
constrained by more observations than the P19 spectrum, but both spectra have
significant flux uncertainties. Whilst neither spectrum used in this study is
likely to wholly represent the true stellar irradiation environment of
TRAPPIST-1e, there are at least observational constraints on the ‘ground
truth’ of its parent star’s spectrum. For many planetary systems, there will
only be estimates from stellar models, and this will cause significant
problems for predicting the photochemical environment of potentially habitable
exoplanets. Furthermore, in each wavelength bin, we have assumed that the flux
does not vary with time. Due to M dwarf stellar activity, such an assumption
is unlikely to be accurate (Loyd et al., 2018). The O3 abundance will be
perturbed by the inclusion of incident stellar flares (Segura et al., 2010;
Tilley et al., 2019; Chen et al., 2021; Ridgway et al., 2022) which we have
not investigated here. Based on previous results, it seems that stellar flares
will exacerbate the interpretation of observed spectra, so future work on
incoming UV uncertainties could evaluate the additional impact of stellar
flares. The present modelling uncertainties in the O2-O3 non-linear
relationship arising from differences in predictions between 1D and 3D models
(Cooke et al., 2022; Kozakis et al., 2022; Yassin Jaziri et al., 2022; Ji et
al., 2023) will compound this issue. In addition to previous work, our
simulations, which focus on the specific target of TRAPPIST-1e, further
motivates the need for a dedicated next generation observatory with UV
capabilities to characterise exoplanet host stars.

UV flux measurements from a telescope such as the $\sim 6$ m UV/VIS/NIR
telescope (currently referred to as the Habitable Worlds Observatory) that was
recommended by the Decadal Survey (National Academies of Sciences,
Engineering, and Medicine, 2021) will aid the interpretation of observed
exoplanet spectra and help to infer the concentration of O2 and trace gases in
the atmosphere without direct measurements (Kozakis et al., 2022). However,
this telescope is not expected to be operational until the late 2030s at the
earliest. Determining the EUV fluxes from a host star (which will require a
dedicated observatory; Youngblood et al., 2019) will also provide important
information about atmospheric escape, habitability, and help to examine the
atmospheric history of the exoplanets in the system.

Before next generation telescopes are online, there are other clues available
to characterise oxygenated terrestrial atmospheres if the interpretation of
the spectral features (e.g. O3) leaves degeneracies in the parameter space
between O2 concentration, O3 concentration, UV irradiation, and O3 depleting
catalytic cycles. For example, the major differences between the P19 PI and
the P19 0.1% PAL transmission spectra are between the H2O, O2, and O3
features. Moreover, the estimated inter-simulation trends with wavelength in
transmission spectra are not mirrored in emission spectra predictions. Namely,
the depth relative to the continuum in emission spectra for O3 at 4.71 µm and
9.6 µm contrasts with the relative strength of associated transmission spectra
features between the simulations. This means that if both transmission spectra
and emission spectra are acquired with adequate precision, multi-wavelength
observations combined with atmospheric retrieval methods (Quanz et al., 2021)
will be useful when delineating between possible atmospheric composition
scenarios. Nevertheless, Batalha et al. (2018) showed that confident estimates
on atmospheric composition from emission spectra observed with JWST MIRI LRS
will prove difficult to achieve for temperate exoplanets, using TRAPPIST-1 f
as an example.

Finally, future modeling work should explore the impact of chemical boundary
conditions on simulated transmission and emission spectra. For example, the
choice of boundary conditions, such as the upward flux and abundances of
species in the HOx, NOx, ClOx, and BrOx families, will not only affect the O3
distribution but also the atmospheric temperature and incoming radiation.
Provided the atmospheric profile of O3 derived from atmospheric retrievals is
well-constrained, by assuming minimal and maximal O3 catalytic cycle
destruction, limits could be placed on O2.

##  5 Conclusions

For the first time using a 3D chemistry-climate model (WACCM6) to simulate
TRAPPIST-1e (assuming an initial Earth-like composition) and including two
different incoming UV spectra, we demonstrated that using a single observed O3
feature outside of the UV range to extrapolate to undetected molecules, such
as O2, will lead to degeneracies over multiple orders of magnitude in the
parameter space for atmospheric composition. The UV spectrum in both of the
incoming stellar spectra varies by up to a factor of $\approx 500$ for
important photolysis bands, and up to $\approx 5000$ for individual wavelength
bins. Whilst the atmospheric columns of many species (including O2 and CO2)
are virtually unaffected by the difference between the two spectra, for an O2
mixing ratio of 0.21, the O3 columns differ by a factor of 26 due to different
O3 production rates that are sensitive to the incoming spectrum. Consequently,
the associated O3 transmission spectral features differ in effective altitude
by up to $19$ km, whilst the O3 features in emission spectra differ by a
factor of up to 4.5 in relative depth. One implication is that a non-detection
of O3 at visible wavelengths may not indicate the absence of an oxygenated
atmosphere. Furthermore, tidal locking of the model results in substantially
different emission spectra features which are shallower relative to the
emission continuum.

Without the direct detection of O2, additional context for determining the
oxygenation state of the atmosphere can be gained from either 1) future
missions that are able to better characterise the UV spectra of faint stars,
or 2) sensitive direct imaging observations combined with transmission spectra
observations targeting individual features.

## Acknowledgments

G.J.C. acknowledges the studentship funded by the Science and Technology
Facilities Council of the United Kingdom (STFC). C.W. acknowledges financial
support from the University of Leeds and from the Science and Technology
Facilities Council (grant numbers ST/T000287/1 and MR/T040726/1). This work
was undertaken on ARC4, part of the High Performance Computing facilities at
the University of Leeds, UK.

We would like to acknowledge high-performance computing support from Cheyenne
(doi:10.5065/D6RX99HX) provided by NCAR’s Computational and Information
Systems Laboratory, sponsored by the National Science Foundation. The CESM
project is supported primarily by the National Science Foundation (NSF). This
material is based upon work supported by the National Center for Atmospheric
Research (NCAR), which is a major facility sponsored by the NSF under
Cooperative Agreement 1852977.

This research has made use of the NASA Exoplanet Archive, which is operated by
the California Institute of Technology, under contract with the National
Aeronautics and Space Administration under the Exoplanet Exploration Program.

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##  Appendix A Surface temperatures

The time-averaged surface temperatures for all the simulations (see Table
LABEL:Run_table for simulation details) are shown in Fig. 9. On time average,
the global mean surface temperatures are between 219 – 231 K, with the minima
ranging between 171 – 180 K, and the maxima ranging between 252 – 277 K. The
P19 and W21 noTL simulations are everywhere colder than 273 K, whilst all
other simulations do have some areas of the surface which are warmer than 273
K (the freezing point of water under 1 bar of atmospheric pressure).

Figure 9: The surface temperatures in kelvin are shown for the P19 PI, P19
0.1% PAL, P19 noTL, and W21 PI simulations. In the tidally locked cases, a
relatively warm region exists around the substellar point, which receives
constant irradiation. The substellar point for the tidally locked cases is at
$180^{\circ}$ longitude and $0^{\circ}$ latitude (in the centre of each
Robinson projection). In the P19 and W21 noTL cases, the substellar point
moves because there is a diurnal cycle. These simulations have the coldest
surface temperatures (171 and 170 K, respectively), but the smallest range in
surface temperatures.

##  Appendix B Water vapour and high clouds

Fig. 10 shows the water vapour (H2O) column for the PI and 0.1% PAL
simulations for both the P19 and W21 scenarios. The H2O column is
predominantly situated around and above the substellar point. This is because
the incoming solar radiation heats the surface, and water vapour evaporates.
The ascending air parcels carry the water vapour upwards, where the parcels
cool and form clouds.

Figure 10: The H2O column, given in terms of molecules m-2, is shown for the
PI and 0.1% PAL simulations in the P19 (left) and W21 (right) scenarios. White
shows a relatively low H2O column, whilst blue shows relatively large amounts
of H2O column. For reference, Earth’s H2O column reaches a peak of about 50 kg
m-2.

Fig. 11 shows the high cloud fraction for the PI and 0.1% PAL simulations for
both the P19 and W21 scenarios. High clouds are defined as clouds at pressures
less than 400 hPa. Clouds, through shortwave (incoming solar radiation) and
longwave (outgoing terrestrial radiation) forcing, provide a net warming of 8
– 12 W m-2 in the tidally locked scenarios, and 5 – 7 W m-2 in the non-tidally
locked scenarios.

Figure 11: The high cloud fraction, where high clouds are specified as clouds
below pressures of 400 hPa, is shown for the PI and 0.1% PAL simulations in
the P19 (left) and W21 (right) scenarios. Blue shows low cloud fraction,
whilst white shows large amounts of cloud coverage. Clouds provide a net
warming in the tidally locked scenarios.