# Dilaton-induced open quantum dynamics

Christian Käding<EMAIL_ADDRESS>Technische Universität Wien,
Atominstitut, Stadionallee 2, 1020 Vienna, Austria  Mario Pitschmann
<EMAIL_ADDRESS>Technische Universität Wien, Atominstitut,
Stadionallee 2, 1020 Vienna, Austria  Caroline Voith
<EMAIL_ADDRESS>Technische Universität Wien, Atominstitut,
Stadionallee 2, 1020 Vienna, Austria

###### Abstract

In modern cosmology, scalar fields with screening mechanisms are often used as
explanations for phenomena like dark energy or dark matter. Amongst a zoo of
models, the environment dependent dilaton, screened by the Polyakov-Damour
mechanism, is one of the least constrained ones. Using recently developed path
integral tools for directly computing reduced density matrices, we study the
open quantum dynamics of a probe, modelled by another real scalar field,
induced by interactions with an environment comprising fluctuations of a
dilaton. As the leading effect, we extract a correction to the probe’s unitary
evolution, which can be observed as a frequency shift. Assuming the scalar
probe to roughly approximate a cold atom in matter wave interferometry, we
show that comparing the predicted frequency shifts in two experimentally
distinct setups has the potential to exclude large parts of the dilaton
parameter space.

dilaton, open quantum dynamics, non-equilibrium quantum field theory, atom
interferometry

##  I Introduction

In modern cosmology, scalar-tensor theories of gravity [1], in which scalar
fields are coupled to the gravitational metric tensor, are often proposed as
potential solutions for the problems of dark matter and dark energy, see Refs.
[2, 3] for an overview. In many cases, such scalar-tensor theories lead to a
universal coupling of the scalar to the trace $T^{\mu}_{\,\,\,\mu}$ of the
matter energy-momentum tensor. Consequently, these theories predict the
existence of gravity-like fifth forces. However, to date, we did not observe
any additional fundamental forces beyond the four known ones, which is
reflected in strong Solar System constraints [4, 5, 6].
A phenomenologically interesting way of explaining the apparent absence of
fifth forces in our Solar System is via a so-called screening mechanism. Such
mechanisms can arise in non-linear scalar field theories, and induce a
screening of the scalar’s fifth force, i.e. render it to be very weak, in
environments with large values of $T^{\mu}_{\,\,\,\mu}$, for example, for high
densities $\rho^{\text{ext}}=-T^{\mu}_{\,\,\,\mu}$ in dust-dominated regions
like our Solar System. By now, there are several scalar models which exhibit a
screening mechanism [7, 8]. Amongst the most prominent so-called screened
scalar fields are: the chameleon [9, 10]; the symmetron [11, 12, 13, 14, 15,
16, 17, 18], whose fifth force has recently been studied as an alternative to
particle dark matter [19, 20, 21, 22]; the environment dependent dilaton [13,
23, 24, 25, 26, 27, 28]; and the galileon [29, 30, 31]. Many of these models
have already been or are proposed to be tested in a plethora of experiments,
see, for example, Refs. [32, 7, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48]. In addition, in recent years, research programs were
initiated to study screened scalars as quantum fields [49, 50, 51, 52], and it
was even suggested to investigate screened scalar-tensor theories in analogue
gravity simulations [53].
Ref. [50] presented the idea for using open quantum dynamical effects [54]
like frequency shifts or decoherence [55], which might be induced by
interactions with an environment comprising light scalar fluctuations, for
further constraining chameleon screened scalar fields in atom interferometry
experiments. Besides presenting phenomenological research, Ref. [50] also
introduced a novel and powerful theoretical tool, namely a practicable and
first principle-based method for deriving quantum master equations for reduced
density matrices in scalar quantum field theory. This method relies on the
Schwinger-Keldysh formalism [56, 57] and the Feynman-Vernon influence
functional [58]. Based on the work in Ref. [50], Ref. [59] developed a way of
circumventing master equations by directly computing reduced density matrix
elements with influence functionals. In addition, Ref. [60] derived a similar
technique for computing total density matrices in interacting quantum field
theories.
In this article, we look at the environment dependent dilaton screened scalar
field as a yet barely constrained model. Following the approach made in Ref.
[50], we study the open quantum dynamics of a probe real scalar field $\phi$
when interacting with an environment comprising fluctuations $\chi$ of a
dilaton field $X$. More precisely, we use the scalar $\phi$ as a proxy for an
atom in an atom interferometry experiment (or any other object in a matter
interferometer, e.g., neutrons) and, applying the formalism from Ref. [59],
check how its reduced density matrix evolves from an initial time $0$ to a
final time $t$ due to the hypothetical interaction with a dilaton. We then
read off the most dominant open quantum dynamical effect induced in the probe
system $\phi$ in order to get an estimate of the dilaton parameter space that
could be constrained experimentally this way.
Certainly, using a scalar field $\phi$ as a proxy for a more complex object
like an atom has its shortcomings. However, since the dilaton only couples to
the trace of the atom’s energy-momentum tensor, a scalar field is a rough but,
in this case, suitable first approximation. In order to improve this
approximation, we consider the probe system being restricted to a single-
particle subspace and ignore all $\phi$-loop diagrams, which naturally appear
in a scalar quantum field theory, but are unrealistic for a complex and stable
object like a cold atom.
Providing a glance at the full potential that open quantum dynamical effects
can have for constraining physics beyond the standard models of particles and
cosmology, this article presents a first application of Ref. [59], and
motivates future studies with more realistic and sophisticated probe models.
The article is structured as follows: at first, we review the environment
dependent dilaton model in Sec. II. Subsequently, we take the required formula
from Ref. [59], discuss and evaluate it in Sec. III, and then use the results
for estimating constraints for the dilaton parameter space in Sec. IV.
Finally, in Sec. V, we draw our conclusions.

##  II Environment dependent dilatons

Dilatons originate from string theory [13, 24, 25], but have found
applications in cosmology [23, 27] and were shown, similar to symmetrons, to
be subject to the Polyakov-Damour mechanism [13], which coins them to be
environment dependent [26] and therefore a screened scalar field model.
The environment dependent dilaton has an effective potential [3]

$\displaystyle V_{\text{eff}}(\varphi;\rho^{\text{ext}})$ $\displaystyle=$
$\displaystyle\bar{V_{0}}\,e^{-\varphi/M_{Pl}}+\frac{(\varphi-\varphi_{*})^{2}}{2\mathcal{M}^{2}}\rho^{\text{ext}}\,\,\,,$
(1)

where $\bar{V_{0}}$ is a constant potential with a decreasing exponential
function $\exp(-\varphi/M_{Pl})$ as required by the strong coupling limit of
string theory. Here, $M_{Pl}$ denotes the Planck mass, $\mathcal{M}$ is a
coupling constant with dimension of a mass, for which we require
$\varphi\ll\mathcal{M}$, and $\varphi_{*}$ is a constant field value around
which the coupling of the dilaton to matter becomes feeble or even vanishes
entirely for $\varphi=\varphi_{*}$. This decoupling is the essence of the
Polyakov-Damour mechanism and leads to a suppression of the dilaton fifth
force.
The fact that a larger value of $\rho^{\text{ext}}$ actually corresponds to a
weaker coupling to matter can more easily be seen when substituting $\lambda
X:=\varphi-\varphi_{*}$ into Eq. (1). Introducing

$\displaystyle V(X)\,:=\,V_{0}\,e^{-\lambda X/M_{Pl}}$ $\displaystyle:=$
$\displaystyle\bar{V_{0}}\,e^{-(\lambda X+\varphi_{*})/M_{Pl}}$ (2)

with $\lambda$ being a dimensionless coupling constant, we find [28]

$\displaystyle V_{\text{eff}}(X;\rho^{\text{ext}})$ $\displaystyle=$
$\displaystyle V_{0}\,e^{-\lambda
X/M_{Pl}}+\frac{A_{2}\rho^{\text{ext}}}{2M_{Pl}^{2}}X^{2}\,\,\,,$ (3)

where a dimensionless constant $A_{2}:=\lambda^{2}M_{Pl}^{2}/\mathcal{M}^{2}$
was introduced that has to fulfill $A_{2}\gg 1$ in order to circumvent
existing Solar System-based constraints on scalar fifth forces [16]. From Eq.
(3) we find the dilaton vacuum expectation value (vev) as [28]

$\displaystyle\langle X\rangle$ $\displaystyle=$
$\displaystyle\frac{M_{Pl}}{\lambda}W\left(\frac{\lambda^{2}V_{0}}{A_{2}\rho^{\text{ext}}}\right)$
(4)

with the Lambert $W$-function

$\displaystyle W(x)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\frac{(-n)^{n-1}}{n!}x^{n}\,\,\,.$ (5)

Since $X=\langle X\rangle+\chi$, where $\chi$ is a small perturbation, Eq. (4)
lets us conclude that $X$ decreases when $\rho^{\text{ext}}$ increases.
Consequently, the leading $\langle X\rangle^{2}$-term in Eq. (3) becomes
smaller for larger values of $\rho^{\text{ext}}$, which indicates a decoupling
between dilaton and matter, i.e. a screening of the fifth force.
As a scalar-tensor theory, the dilaton is coupled to the gravitational metric
tensor via $\tilde{g}_{\mu\nu}=A^{2}(X)g_{\mu\nu},$ where $\tilde{g}$ and $g$
denote the metric in the Jordan and Einstein frame [1], respectively, and the
conformal factor is given by

$\displaystyle A(X)$ $\displaystyle=$ $\displaystyle
1+\frac{A_{2}}{2M_{Pl}^{2}}X^{2}\,\,\,.$ (6)

The total Einstein frame action describing gravity, the scalar $X$ and matter
is given by [9, 10]

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left[\frac{1}{2}M_{Pl}^{2}R-\frac{1}{2}g^{\mu\nu}\partial_{\mu}X\partial_{\nu}X-V(X)\right]+\int
d^{4}x\sqrt{-g}A^{4}(X)\tilde{\mathcal{L}}_{M}(\tilde{\phi},A^{2}(X)g_{\mu\nu})\,\,\,,\,\,\,\,\,\,$
(7)

where the last term describes the dynamics of the Jordan frame matter field
$\tilde{\phi}$, which in our case is a real scalar, and its interactions with
the dilaton. Here, the potential $V(X)$ is the self-interaction of $X$ given
by Eq. (2), and for the Jordan frame matter Lagrangian we use

$\displaystyle\tilde{\mathcal{L}}_{M}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\tilde{g}^{\mu\nu}\partial_{\mu}\tilde{\phi}\partial_{\nu}\tilde{\phi}-\frac{1}{2}\tilde{M}^{2}\tilde{\phi}^{2}\,\,\,.$
(8)

Since we will have to work with the dilaton perturbatively if we later want to
discuss it in the formalism presented in Ref. [59], we are required to assume
$\lambda X/M_{Pl}\ll 1$, such that we can expand

$\displaystyle V(X)$ $\displaystyle\approx$ $\displaystyle
V_{0}\left[1-\frac{\lambda}{M_{Pl}}X+\frac{1}{2}\frac{\lambda^{2}}{M_{Pl}^{2}}X^{2}+\mathcal{O}\bigg{(}\frac{\lambda^{3}}{M_{Pl}^{3}}X^{3}\bigg{)}\right]\,\,\,.$
(9)

Following the general procedure outlined in Ref. [50] for Eqs. (6) to (9),
while only keeping operators of dimension $4$ or lower, we can derive Einstein
frame free actions for the matter field $\phi$ and the dilaton fluctuation
$\chi$ as well as an interaction action between both types of fields:

$\displaystyle S_{\phi}[\phi]$ $\displaystyle:=$
$\displaystyle\int_{x}\left[-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}M^{2}\phi^{2}\right]\,\,\,,$
(10) $\displaystyle S_{\chi}[\chi]$ $\displaystyle:=$
$\displaystyle\int_{x}\left[-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi-\frac{1}{2}m^{2}\chi^{2}\right]\,\,\,,$
(11) $\displaystyle S_{\text{int}}[\phi,\chi]$ $\displaystyle:=$
$\displaystyle\int_{x\in\Omega_{t}}\left[-\frac{1}{2}\alpha_{1}M\chi\phi^{2}-\frac{1}{4}\alpha_{2}\chi^{2}\phi^{2}\right]\,\,\,,$
(12)

where, using $g_{\mu\nu}\equiv\eta_{\mu\nu}$ from now on,

$\displaystyle\int_{x}$ $\displaystyle:=$ $\displaystyle\int d^{4}x\,\,\,,$
(13)

the masses are

$\displaystyle M^{2}$ $\displaystyle:=$
$\displaystyle\left(1+\frac{A_{2}}{M_{Pl}^{2}}\langle
X\rangle^{2}\right)\tilde{M}^{2}\,\,\,,\,\,\,\,\,\,\,\,\,m^{2}\,\approx\,\frac{1}{M_{Pl}^{2}}\left(V_{0}\lambda^{2}+A_{2}\rho^{\text{ext}}\right)\,\,\,,$
(14)

the coupling constants take on the forms

$\displaystyle\alpha_{1}$ $\displaystyle:=$ $\displaystyle
2M\frac{A_{2}}{M_{Pl}^{2}}\langle
X\rangle\left(1-\frac{A_{2}}{M_{Pl}^{2}}\langle
X\rangle^{2}\right)\,\,\,,\,\,\,\,\,\,\,\,\,\alpha_{2}\,:=\,\frac{M}{\langle
X\rangle}\alpha_{1}\,\,\,,$ (15)

and we restrict integrations over spacetime coordinates to
$\Omega_{t}:=[0,t]\times\mathbb{R}$ since we are only interested in the finite
interval between initial time $0$ and final time $t$. Note that at the order
considered in the expansion made in Eq. (9) the dilaton does not self-interact
and, therefore, we do not need to consider a self-interaction action for
$\chi$.

##  III Open quantum dynamics

Most realistic quantum systems should be treated as open, i.e. as interacting
with one or several environments. Such interactions can induce open quantum
dynamical effects like frequency (or phase) shifts or decoherence in a system.
While the theory of open quantum systems naturally finds many applications in
non-relativistic quantum physics, see e.g. Ref. [61, 62, 63, 64, 65, 66, 67],
it also receives increased attention in quantum field theory [68, 69, 70, 71,
72, 73, 50, 74, 75, 76], and adjacent areas like Early Universe cosmology [77,
78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89], black holes [90, 91, 92, 93,
94, 95, 96], or heavy-ion physics [97, 98, 99, 100, 101, 102, 103, 104, 105,
106].
Quantum systems, especially open ones, are often described by density
operators $\hat{\rho}(t)$, which are advantageous over a wave function
description since they can not only describe pure but also mixed states. This
is necessary in order to fully capture phenomena like decoherence. In case of
an open quantum system, we usually work with reduced density operators, which
are obtained by tracing out the environmental degrees of freedom.
Projecting a (reduced) density operator into a basis, for example, a single-
particle momentum basis as we will use in this article, gives elements of a
density matrix:

$\displaystyle\rho(\mathbf{p};\mathbf{p}^{\prime};t)$ $\displaystyle=$
$\displaystyle\bra{\mathbf{p}}\hat{\rho}(t)\ket{\mathbf{p}^{\prime}}\,\,\,.$
(16)

The time evolution of density matrices can be described by quantum master
equations. However, such equations are often analytically intricate or even
impossible to solve. In order to avoid such complications, using the
technology presented in Ref. [50], Ref. [59] developed a formalism that
enables us to directly compute reduced density matrices in terms of the
Feynman-Vernon influence functional [58], which itself is based on the
Schwinger-Keldysh formalism [56, 57].
In this article, we consider an open quantum system $\phi$, which is a proxy
for an atom (or another, compared to the dilaton relatively heavy matter
particle like a neutron) with zero temperature in order to justify a
restriction to the single-particle subspace. As was pointed out in Refs. [50]
and [59], in this case, contractions of the system field can only give rise to
Feynman and Dyson propagators:

$\displaystyle\contraction{}{\phi}{{}^{+}_{x}}{\phi}\phi^{+}_{x}\phi^{+}_{y}$
$\displaystyle=$ $\displaystyle
D^{++}_{xy}\,=\,D^{\mathrm{F}}_{xy}\,=\,-\mathrm{i}\int_{k}\frac{e^{\mathrm{i}k\cdot(x-y)}}{k^{2}+M^{2}-\mathrm{i}\epsilon}\,\,\,,$
(17)
$\displaystyle\contraction{}{\phi}{{}^{-}_{x}}{\phi}\phi^{-}_{x}\phi^{-}_{y}$
$\displaystyle=$ $\displaystyle
D^{--}_{xy}\,=\,D^{\mathrm{D}}_{xy}\,=\,+\mathrm{i}\int_{k}\frac{e^{\mathrm{i}k\cdot(x-y)}}{k^{2}+M^{2}+\mathrm{i}\epsilon}\,\,\,,$
(18)

where

$\displaystyle\int_{k}$ $\displaystyle:=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\,\,\,,$ (19)

and $+$ and $-$ denote the positive and negative branches of the Schwinger-
Keldysh closed time path, respectively.
The system is interacting with an environment comprised of dilaton
fluctuations $\chi$. As was also done in Ref. [50] for the chameleon, we
assume the dilaton to have a, in general, non-zero temperature $T$, for
example, due to thermalization with the walls of a vacuum chamber in an atom
interferometry experiment. This means, open system and environment are out of
thermal equilibrium in our discussion. Since environmental degrees of freedom
can be contracted for any combination of $+$ and $-$ [50, 59], the dilaton can
give Feynman and Dyson, but also Wightman propagators:

$\displaystyle\contraction{}{\chi}{{}^{+}_{x}}{\chi}\chi^{+}_{x}\chi^{+}_{y}$
$\displaystyle=$ $\displaystyle\Delta^{++}_{xy}\,=\,\Delta^{\rm
F}_{xy}\,=\,-\mathrm{i}\int_{k}e^{ik\cdot(x-y)}\left[\frac{1}{k^{2}+m^{2}-\mathrm{i}\epsilon}+2\pi\mathrm{i}f(|k^{0}|)\delta(k^{2}+m^{2})\right]\,\,\,,$
(20)
$\displaystyle\contraction{}{\chi}{{}^{-}_{x}}{\chi}\chi^{-}_{x}\chi^{-}_{y}$
$\displaystyle=$ $\displaystyle\Delta^{--}_{xy}\,=\,\Delta^{\rm
D}_{xy}\,=\,+\mathrm{i}\int_{k}e^{\mathrm{i}k\cdot(x-y)}\left[\frac{1}{k^{2}+m^{2}+\mathrm{i}\epsilon}-2\pi\mathrm{i}f(|k^{0}|)\delta(k^{2}+m^{2})\right]\,\,\,,$
(21)
$\displaystyle\contraction{}{\chi}{{}^{+}_{x}}{\chi}\chi^{+}_{x}\chi^{-}_{y}$
$\displaystyle=$
$\displaystyle\langle\chi_{y}\chi_{x}\rangle\,=\,\Delta^{+-}_{xy}\,=\,\Delta^{<}_{xy}\,=\,\int_{k}e^{\mathrm{i}k\cdot(x-y)}2\pi\text{sgn}(k^{0})f(k^{0})\delta(k^{2}+m^{2})\,\,\,,$
(22)
$\displaystyle\contraction{}{\chi}{{}^{-}_{x}}{\chi}\chi^{-}_{x}\chi^{+}_{y}$
$\displaystyle=$
$\displaystyle\langle\chi_{x}\chi_{y}\rangle\,=\,\Delta^{-+}_{xy}\,=\,\Delta^{>}_{xy}~{}=~{}\Delta^{<}_{yx}\,=\,(\Delta^{<})^{*}_{xy}\,\,\,,$
(23)

where

$\displaystyle f(k^{0})$ $\displaystyle:=$ $\displaystyle\frac{1}{e^{\beta
k^{0}}-1}\,=\,-[1+f(-k^{0})]$ (24)

is the Bose-Einstein distribution function with $\beta=1/T$ being the inverse
temperature.
Following Ref. [59], the reduced density matrix element for $\phi$ at time $t$
in the single-particle momentum subspace, under the assumption that the
system’s particle number does not change in the interval $[0,t]$, can be
obtained by evaluating

$\displaystyle\rho(\mathbf{p};\mathbf{p}^{\prime};t)$ $\displaystyle=$
$\displaystyle\lim_{\begin{subarray}{c}x^{0(\prime)}\,\to\,t^{+}\\\
y^{0(\prime)}\,\to\,0^{-}\end{subarray}}\int_{\mathbf{k}\mathbf{k}^{\prime}}\frac{\rho(\mathbf{k};\mathbf{k}^{\prime};0)}{(2E_{\mathbf{k}}^{\phi})(2E_{\mathbf{k}^{\prime}}^{\phi})}$
(25)
$\displaystyle\times\int_{\mathbf{x}\mathbf{x}^{\prime}\mathbf{y}\mathbf{y}^{\prime}}e^{-\mathrm{i}(\mathbf{p}\cdot\mathbf{x}-\mathbf{p}^{\prime}\cdot\mathbf{x}^{\prime})+\mathrm{i}(\mathbf{k}\cdot\mathbf{y}-\mathbf{k}^{\prime}\cdot\mathbf{y}^{\prime})}\partial_{x^{0},E^{\phi}_{\mathbf{p}}}\partial_{x^{0\prime},E^{\phi}_{\mathbf{p}^{\prime}}}^{*}\partial_{y^{0},E^{\phi}_{\mathbf{k}}}^{*}\partial_{y^{0\prime},E^{\phi}_{\mathbf{k}^{\prime}}}$
$\displaystyle\times\int\mathcal{D}\phi^{\pm}e^{\mathrm{i}\left\\{S_{\phi}[\phi^{+}]-S_{\phi}[\phi^{-}]\right\\}}\phi^{+}_{x}\phi^{-}_{x^{\prime}}\mathcal{F}[\phi^{\pm};t]\phi^{+}_{y}\phi^{-}_{y^{\prime}}\,\,\,,$

where the Feynman-Vernon influence functional

$\displaystyle\mathcal{F}[\phi^{\pm};t]$ $\displaystyle=$
$\displaystyle\left\langle
e^{\mathrm{i}\big{\\{}S_{\text{int}}[\phi^{+},\chi^{+};t]-S_{\text{int}}[\phi^{-},\chi^{-};t]\big{\\}}}\right\rangle_{\chi}$
(26)

is given in terms of the expectation value

$\displaystyle\langle A[\chi^{a}]\rangle_{\chi}$ $\displaystyle:=$
$\displaystyle\int
d\chi^{\pm}_{t}d\chi^{\pm}_{0}\delta(\chi_{t}^{+}-\chi_{t}^{-})\rho_{\chi}[\chi^{\pm}_{0};0]\int^{\chi^{\pm}_{t}}_{\chi^{\pm}_{0}}\mathcal{D}\chi^{\pm}A[\chi^{a}]e^{\mathrm{i}\left\\{S_{\chi}[\chi^{+}]-S_{\chi}[\chi^{-}]\right\\}}$
(27)

with $\rho_{\chi}[\chi^{\pm}_{0};0]$ denoting the initial environmental
density matrix under the assumption that system and environment were initially
uncorrelated. The index of a field, e.g. $t$ on $\chi_{t}$, labels the time
slice on which the field eigenstate was taken.
We expand the Feynman-Vernon influence functional up to second order in the
coupling constants $\alpha_{1}$ and $\alpha_{2}$, and find

$\displaystyle\mathcal{F}[\phi^{\pm};t]$ $\displaystyle=$ $\displaystyle
1-\mathrm{i}\frac{\alpha_{2}}{4}\sum_{a=\pm}a\int_{x}(\phi^{a}_{x})^{2}\Delta^{\rm
F}_{xx}-\frac{1}{2}\sum_{a,b=\pm}ab\int_{xy}\biggl{[}\frac{\alpha_{1}^{2}M^{2}}{4}(\phi^{a}_{x})^{2}(\phi^{b}_{y})^{2}\Delta^{ab}_{xy}$
(28)
$\displaystyle+\frac{\alpha_{2}^{2}}{16}(\phi^{a}_{x})^{2}(\phi^{b}_{y})^{2}\left(\Delta^{\rm
F}_{xx}\Delta^{\rm
F}_{yy}+2(\Delta^{ab}_{xy})^{2}\right)\bigg{]}+\mathcal{O}\left(\alpha^{3}_{1,2}\right)\,\,\,.$

Substituting Eq. (28) into Eq. (25), using Wick’s theorem [107], and dropping
every term containing $\phi$-loops, we obtain

$\displaystyle\rho(\mathbf{p};\mathbf{p}^{\prime};t)$ $\displaystyle=$
$\displaystyle\lim_{\begin{subarray}{c}x^{0(\prime)}\to t^{+}\\\
y^{0(\prime)}\to
0^{-}\end{subarray}}\int_{\mathbf{k}\mathbf{k}^{\prime}}\frac{\rho(\mathbf{k};\mathbf{k}^{\prime};0)}{(2E_{\mathbf{k}}^{\phi})(2E_{\mathbf{k}^{\prime}}^{\phi})}$
(29)
$\displaystyle\times\int_{\mathbf{x}\mathbf{x}^{\prime}\mathbf{y}\mathbf{y}^{\prime}}e^{-\mathrm{i}(\mathbf{p}\cdot\mathbf{x}-\mathbf{p}^{\prime}\cdot\mathbf{x}^{\prime})+\mathrm{i}(\mathbf{k}\cdot\mathbf{y}-\mathbf{k}^{\prime}\cdot\mathbf{y}^{\prime})}\partial_{x^{0},E_{\mathbf{p}}^{\phi}}\partial^{*}_{x^{0\prime},E_{\mathbf{p}^{\prime}}^{\phi}}\partial^{*}_{y^{0},E_{\mathbf{k}}^{\phi}}\partial_{y^{0\prime},E_{\mathbf{k}^{\prime}}^{\phi}}$
$\displaystyle\times\Bigg{\\{}D^{\mathrm{F}}_{xy}D^{\mathrm{D}}_{x^{\prime}y^{\prime}}-\mathrm{i}\frac{\alpha_{2}}{2}\int_{z}\left[D^{\mathrm{F}}_{xz}D^{\mathrm{F}}_{zy}D^{\mathrm{D}}_{x^{\prime}y^{\prime}}-(x,y\longleftrightarrow
x^{\prime},y^{\prime})^{\ast}\right]\Delta^{\rm F}_{zz}$
$\displaystyle-\frac{\alpha_{1}^{2}M^{2}}{8}\int_{zz^{\prime}}\bigg{[}\Big{(}D^{\mathrm{D}}_{x^{\prime}y^{\prime}}\big{(}8D^{\mathrm{F}}_{xz}D^{\mathrm{F}}_{zz^{\prime}}D^{\mathrm{F}}_{z^{\prime}y}+2D^{\mathrm{F}}_{xy}D^{\mathrm{F}}_{zz^{\prime}}D^{\mathrm{F}}_{zz^{\prime}}\big{)}\Delta^{\rm
F}_{zz^{\prime}}+(x,y\longleftrightarrow x^{\prime},y^{\prime})^{\ast}\Big{)}$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\big{(}8D^{\mathrm{F}}_{xz}D^{\mathrm{F}}_{zy}D^{\mathrm{D}}_{x^{\prime}z^{\prime}}D^{\mathrm{D}}_{z^{\prime}y^{\prime}}+4D^{\mathrm{F}}_{xy}D^{\mathrm{F}}_{zz}D^{\mathrm{D}}_{x^{\prime}z^{\prime}}D^{\mathrm{D}}_{z^{\prime}y^{\prime}}\big{)}\Delta^{+-}_{zz^{\prime}}\bigg{]}$
$\displaystyle-\frac{\alpha_{2}^{2}}{32}\int_{zz^{\prime}}\bigg{[}\Big{(}D^{\mathrm{D}}_{x^{\prime}y^{\prime}}\big{(}8D^{\mathrm{F}}_{xz}D^{\mathrm{F}}_{zz^{\prime}}D^{\mathrm{F}}_{z^{\prime}y}+2D^{\mathrm{F}}_{xy}D^{\mathrm{F}}_{zz^{\prime}}D^{\mathrm{F}}_{zz^{\prime}}\big{)}\big{(}\Delta^{\rm
F}_{zz}\Delta^{\rm F}_{z^{\prime}z^{\prime}}+2(\Delta^{\rm
F}_{zz^{\prime}})^{2}\big{)}$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+(x,y\longleftrightarrow
x^{\prime},y^{\prime})^{\ast}\Big{)}$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\big{(}8D^{\mathrm{F}}_{xz}D^{\mathrm{F}}_{zy}D^{\mathrm{D}}_{x^{\prime}z^{\prime}}D^{\mathrm{D}}_{z^{\prime}y^{\prime}}+4D^{\mathrm{F}}_{xy}D^{\mathrm{F}}_{zz}D^{\mathrm{D}}_{x^{\prime}z^{\prime}}D^{\mathrm{D}}_{z^{\prime}y^{\prime}}\big{)}\big{(}\Delta^{\rm
F}_{zz}\Delta^{\rm
F}_{z^{\prime}z^{\prime}}+2(\Delta^{+-}_{zz^{\prime}})^{2}\big{)}\bigg{]}\Bigg{\\}}\,\,\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$

Next, we evaluate the integrals in Eq. (29). The full result of this
computation can be found in Appendix A. However, for the present discussion we
focus only on the following terms:

$\displaystyle\rho(\mathbf{p};\mathbf{p}^{\prime};t)$ $\displaystyle=$
$\displaystyle\rho(\mathbf{p};\mathbf{p^{\prime}};0)e^{-\mathrm{i}(E_{\mathbf{p}}^{\phi}-E_{\mathbf{p^{\prime}}}^{\phi})t}$
(30)
$\displaystyle\times\Bigg{[}1-\frac{\mathrm{i}\alpha_{2}}{4}\left(\frac{1}{E_{\mathbf{p}}^{\phi}}-\frac{1}{E_{\mathbf{p^{\prime}}}^{\phi}}\right)t\Delta^{\mathrm{F}}_{zz}-\frac{\alpha_{2}^{2}}{32}\left(\frac{1}{E_{\mathbf{p}}^{\phi}}-\frac{1}{E_{\mathbf{p^{\prime}}}^{\phi}}\right)^{2}t^{2}\Delta^{\mathrm{F}}_{zz}\Delta^{\mathrm{F}}_{z^{\prime}z^{\prime}}\Bigg{]}+\mathcal{O}\left(\alpha^{2}_{1,2}\right)\,\,\,.\,\,\,\,\,\,$

A diagrammatic depiction of the last two terms in the square brackets can be
found in Fig. 1.

(a)

(b)

(c)

(d)

(e)

Figure 1: Diagrammatic representation of of the last two terms in the square
brackets in Eq. (30); solid/dashed lines represent $\phi$-/$\chi$-propagators.
Crossed boxes depict insertions of the initial density matrix.

Following Ref. [50], we deal with the tadpoles in Eq. (30) by adding a counter
term

$\displaystyle\delta S_{\text{int}}[\phi,\chi]$ $\displaystyle:=$
$\displaystyle\ \frac{\alpha_{2}}{4}\int_{x}\Delta^{F(T=0)}_{xx}\phi^{2}$ (31)

to the interaction action in Eq. (12), where $(T=0)$ indicates the
temperature-independent part of the $\chi$-propagator. The thermal part

$\displaystyle\Delta^{\mathrm{F}(T\neq 0)}_{xx}$ $\displaystyle=$
$\displaystyle\int_{k}2\pi
f(|k^{0}|)\delta(k^{2}+m^{2})\,=\,\frac{T^{2}}{2\pi^{2}}\int_{m/T}^{\infty}d\xi\frac{\sqrt{\xi^{2}-(\frac{m}{T})^{2}}}{e^{\xi}-1}$
(32)

is by default finite in the ultraviolet. Using this, and identifying the terms
in the square brackets of Eq. (30) as the expansion of an exponential
function, we find

$\displaystyle\rho(\mathbf{p};\mathbf{p}^{\prime};t)$ $\displaystyle=$
$\displaystyle\rho(\mathbf{p};\mathbf{p^{\prime}};0)\exp\left\\{-\mathrm{i}\Bigg{[}E_{\mathbf{p}}^{\phi}-E_{\mathbf{p^{\prime}}}^{\phi}+\frac{\alpha_{2}}{4}\left(\frac{1}{E_{\mathbf{p}}^{\phi}}-\frac{1}{E_{\mathbf{p^{\prime}}}^{\phi}}\right)\Delta^{\mathrm{F}(T\neq
0)}_{zz}\Bigg{]}t\right\\}+\mathcal{O}\left(\alpha^{2}_{1,2}\right)\,\,\,.$
(33)

Now we recall that we performed all computations with the rescaled mass $M$ as
given in Eq. (14). However, cp. with Ref. [50], experiments are actually
sensitive to the absolute mass $\tilde{M}$. Rewriting Eq. (33), we obtain

$\displaystyle\rho(\mathbf{p};\mathbf{p}^{\prime};t)$ $\displaystyle=$
$\displaystyle\rho(\mathbf{p};\mathbf{p^{\prime}};0)\exp\left\\{-\mathrm{i}\Bigg{[}\tilde{E}_{\mathbf{p}}^{\phi}-\tilde{E}_{\mathbf{p}^{\prime}}^{\phi}\right.$
(34)
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left.+\tilde{M}^{2}\frac{A_{2}}{2M_{Pl}^{2}}\left(\langle
X\rangle^{2}+\Delta^{\mathrm{F}(T\neq
0)}_{zz}\right)\left(\frac{1}{\tilde{E}_{\mathbf{p}}^{\phi}}-\frac{1}{\tilde{E}_{\mathbf{p}^{\prime}}^{\phi}}\right)\Bigg{]}t\right\\}+\mathcal{O}\left(\alpha^{2}_{1,2}\right)\,\,\,,\,\,\,\,\,\,\,\,\,\,\,\,$

where $\tilde{E}_{\mathbf{p}}^{\phi}=\sqrt{\mathbf{p}^{2}+\tilde{M}^{2}}$, we
replaced $\alpha_{2}$ by Eq. (15), and neglected the term
$\sim\mathcal{O}(A_{2}^{2}\tilde{M}^{2}\langle X\rangle^{2}/M_{Pl}^{4})$. What
we found in Eq. (34) is a first order correction to the unitary evolution
term, i.e. a phase shift $\Delta u\cdot t$ with a frequency shift $\Delta u$.
Other effects like de-/recoherence and momentum diffusion first appear at
second order, see Appendix A, and are therefore expected to be subdominant.

##  IV Predicted constraints

Atom interferometry experiments have successfully been used to constrain a
variety of screened scalar field models [42, 39, 38, 41, 36, 35, 44]. As was
also done in Ref. [50], we now estimate whether the frequency shift found in
Eq. (34) could potentially be observed in an atom interferometer. Since such
experiments use low-energy atoms, we make a non-relativistic approximation
$\tilde{M}^{2}\gg\mathbf{p}^{2}$, such that

$\displaystyle\frac{1}{\tilde{E}_{\mathbf{p}}^{\phi}}$ $\displaystyle\approx$
$\displaystyle\frac{1}{\tilde{M}}\left(1-\frac{\mathbf{p}^{2}}{2\tilde{M}^{2}}\right)\,\,\,,\,\,\,\,\,\,\,\,\,\frac{1}{\tilde{E}_{\mathbf{p}}^{\phi}}-\frac{1}{\tilde{E}_{\mathbf{p}^{\prime}}^{\phi}}\,\approx\,\frac{v^{2}}{2\tilde{M}}$
(35)

with $v:=\frac{||\mathbf{p}|-|\mathbf{p}^{\prime}||}{\tilde{M}}$ being the
speed difference between the two atomic states. Substituting this into Eq.
(34), gives for the frequency shift:

$\displaystyle\Delta u$ $\displaystyle:=$
$\displaystyle\tilde{M}\frac{A_{2}}{4M_{Pl}^{2}}\left[\langle
X\rangle^{2}+\Delta^{\mathrm{F}(T\neq 0)}_{zz}\right]v^{2}\,\,\,.$ (36)

For the estimation we choose commonly used values for the parameters of the
experiment: a vacuum chamber with radius $L=10$ cm [38, 35], and a Rubidium-87
atom with a mass of $\tilde{M}=87\,m_{u}$ [44], where $m_{u}$ is the atomic
mass unit, as the probe object. For the speed difference between the atomic
states we use $v=50$ mm $\text{s}^{-1}$ [108]. In addition, from Refs. [108,
109] we infer that in modern atom interferometry experiments frequency shifts
as small as $\Delta u_{\text{min}}\approx 10^{-8}$ Hz can be measured. For the
dilaton’s vev we follow the ansatz made in Ref. [50] and assume that the
experiment takes place in a sufficiently small part in the center of the
vacuum chamber, such that we can assume $\langle X\rangle$ to be approximately
spatially constant. In addition, we only consider parts of the parameter space
for which the Compton wavelength $\lambda_{\mathrm{C}}=1/m$ of the dilaton
fulfils $\lambda_{\mathrm{C}}\leq L$, such that the field is able to take on
its proper vev and not an intermediate value. In order to also consider cases
for which $\lambda_{\mathrm{C}}>L$, we would need a sufficiently good model
for the evolution of the field profile from within the vacuum chamber walls to
the center of the chamber, as was developed for the chameleon, for example, in
Ref. [39]. However, the sophisticated numerical studies required for this are
beyond the scope of the present article.
A frequency shift is only a useful observable if we can compare it to another
measurement without any frequency shift or at least with a frequency shifted
by a different magnitude. Since the expression in Eq. (36) depends on some
parameters that can easily be varied, two measurements with different
predicted frequency shifts can be compared. If the resulting predicted
difference in frequency shifts cannot be observed, this puts constraints on
the dilaton parameter space. The parameters that we choose to vary are the
temperature $T$ of the vacuum chamber walls, which affects the thermal part of
the $\chi$-propagators, and the pressure $P$, which, if we assume that the
residual gas inside the vacuum chamber has thermalized with the walls,
determines the value of $\rho^{\text{ext}}$ due to
$\rho^{\text{ext}}=Pm_{\text{mol}}/(TR_{\text{gas}})$ with $m_{\text{mol}}$
being the molar mass of the residual gas and $R_{\text{gas}}$ the universal
gas constant, and consequently leads to different values of $\langle X\rangle$
and $m$. For the temperature, we consider four different values:
$T_{1}=0.5\times 10^{-3}$ K, $T_{2}=100$ K, $T_{3}=300$ K, and $T_{4}=500$ K.
Ref. [44] reports a residual $\text{H}_{2}$
($m_{\text{mol}}(\text{H}_{2})=2.016$ g/mol [110]) gas pressure of
$P_{2}=9.6\times 10^{-10}$ mbar for their experiment. However, as most extreme
values, vacua with $P_{1}\approx 10^{-17}$ mbar [111] can be reached, and atom
interferometry experiments have been performed in warm vapors of rubidium
($m_{\text{mol}}(\text{Rb})=85.468$ g/mol [112]) atoms with pressures as high
as $P_{3}\approx 10^{-2}$ mbar [113].
We select two pairs of pressure and temperature, and compare the resulting
difference in frequency shifts, which can only be observed if

$\displaystyle|\Delta u(P_{a},T_{b})-\Delta u(P_{c},T_{d})|$ $\displaystyle=:$
$\displaystyle u(P_{a},T_{b};P_{c},T_{d})\,\geq\,\Delta u_{\text{min}}$ (37)

is fulfilled. Furthermore, our previous assumptions require us to demand
$\lambda\langle X\rangle/M_{Pl}\ll 1$ and $\sqrt{A_{2}}\langle
X\rangle/M_{Pl}\ll 1$. Using these restrictions, and choosing values for
$V_{0}$, we plot the parts of the $(\lambda,A_{2})$-space for which Eq. (37)
is fulfilled. The resulting plots for a few selected examples can be found in
Figs. 2 \- 4. In case of an experimental null result, the shaded areas in
these figures exclude the respective parts of the environment dependent
dilaton parameter space.

(a)

(b)

Figure 2: Exclusion plots for the environment dependent dilaton
$(\lambda,A_{2})$-parameter space resulting from differences in frequency
shifts $u(P_{1},T_{2};P_{1},T_{1})$ (blue) and $u(P_{1},T_{4};P_{1},T_{1})$
(orange); (a): $V_{0}=1\,\text{eV}^{4}$, (b): $V_{0}=1\,(\text{MeV})^{4}$

In Fig. 2 we present cases for which the lowest possible pressure $P_{1}$ is
used, but the temperature is varied. For Fig. 2(a) we select
$V_{0}=1\,\text{eV}^{4}$, and consider $u(P_{1},T_{2};P_{1},T_{1})$ (blue
plot) and $u(P_{1},T_{4};P_{1},T_{1})$ (orange plot), while for Fig. 2(b) we
use $V_{0}=1\,(\text{MeV})^{4}$, and keep everything else the same. Looking at
these figures, we observe that in case of a null result, large parts of the
dilaton parameter space could be excluded. Comparing frequency shifts with
larger temperature differences, i.e. $(T_{4},T_{1})$, extends the exclusion
area. However, considering the lower temperature difference $(T_{2},T_{1})$
adds comparatively small areas to the exclusion plots, which cannot be
constrained with only $(T_{4},T_{1})$. For the $V_{0}=1\,\text{eV}^{4}$ case,
the horizontal exclusion area, which is more than $10$ orders of magnitude in
$A_{2}$ wide, stretches beyond $\lambda=10^{20}$, while for
$V_{0}=1\,(\text{MeV})^{4}$ it does not even reach $\lambda\approx 1$.

(a)

(b)

Figure 3: Exclusion plots for the environment dependent dilaton
$(\lambda,A_{2})$-parameter space resulting from differences in frequency
shifts $u(P_{2},T_{2};P_{2},T_{1})$ (blue) and $u(P_{2},T_{4};P_{2},T_{1})$
(orange); (a): $V_{0}=1\,(\text{keV})^{4}$, (b): $V_{0}=1\,(\text{MeV})^{4}$

In Fig. 3 we consider $P_{2}$ and again vary the temperature, such that blue
plots represent $u(P_{2},T_{2};P_{2},T_{1})$ and orange ones
$u(P_{2},T_{4};P_{2},T_{1})$. Fig. 3(a) depicts the case
$V_{0}=1\,(\text{keV})^{4}$ and (b) $V_{0}=1\,(\text{MeV})^{4}$.
Interestingly, the main difference between blue and orange shaded areas is
that each orange one has a narrow (in $A_{2}$) horizontal section that cannot
be found for the blue ones. Similarly to Fig. 2, going to higher values of
$V_{0}$ lowers the values of $\lambda$ that can be excluded by this section.
However, here, the horizontal stripe is much shorter in Fig. 3(a) than in (b).

Figure 4: Exclusion plots for the environment dependent dilaton
$(\lambda,A_{2})$-parameter space resulting from difference in frequency
shifts $u(P_{2},T_{4};P_{2},T_{3})$; the blue plot represents
$V_{0}=1\,\text{eV}^{4}$ and the orange plot $V_{0}=1\,(\text{MeV})^{4}$.

Finally, in Fig. 4 we again consider the pressure $P_{2}$, but compare
frequency shifts for the highest temperatures $T_{3}$ and $T_{4}$, i.e. we
look at $u(P_{2},T_{4};P_{2},T_{3})$. The blue plot depicts the case
$V_{0}=1\,\text{eV}^{4}$ and the orange one $V_{0}=1\,(\text{MeV})^{4}$. Both
exclusion areas appear to be similar but shifted by more than $10$ orders of
magnitude on the $\lambda$-axis.
We do not find any exclusion areas for the highest pressure $P_{3}$ in
combination with the other parameter values we chose. This can be explained by
the rather large $\rho^{\text{ext}}$ that results from the high pressure and
temperatures in the considered warm rubidium vapors. A too dense environment
suppresses the dilaton field, affecting the non-thermal part in Eq. (36), and
increases its mass, which decreases the thermal part, such that the frequency
shift predicted becomes too small to be observable.

##  V Conclusions

Screened scalar fields are frequently used models in cosmology. Their
screening mechanisms enable them to circumvent tight Solar-System constraints
on their gravity-like fifth forces. However, it is possible to probe their
parameter spaces in laboratory-based experiments like atom interferometry.
While other screened scalar field models like the chameleon or the symmetron
are already experimentally well-studied, the parameter space of the
environment dependent dilaton, motivated by string theory and screened by the
Polyakov-Damour mechanism, is still largely unconstrained.
Recently, Ref. [59] presented a new path integral-based method for directly
computing reduced density matrices of open quantum systems. In the present
article, we applied this formalism to an open system modelled by the real,
massive scalar field $\phi$ coupled to an environment comprising fluctuations
$\chi$ of the dilaton field $X$. Following the study in Ref. [50], it was our
intention to use $\phi$ as proxy for an atom in matter wave interferometry.
For an atom, this was a rough, but, in this particular physical situation,
suitable, first approximation since the dilaton only couples to the mass
density. We improved the approximation by restricting the discussion to the
single-particle momentum subspace, and neglecting all diagrams containing
$\phi$-loops since those would not appear at the considered perturbative order
for complex composite objects like atoms. Using Ref. [59], we computed the
single-particle density matrix element in a momentum basis at final time $t$
under the assumption of an initial single atom at time $0$. The resulting
expression was presented in Appendix A. However, for the actual discussion, we
focused only on the leading effect, which turned out to be a correction to the
unitary evolution of $\phi$. In an interferometry experiment, such a
correction would be visible as a frequency shift if the measurement was
compared to another one with a different or without any shifted frequency.
For realistic parameters of an atom interferometry experiment, we used this
frequency shift in order to predict exclusion plots for the dilaton parameter
space. While these predictions should be taken with care due to us
approximating an atom by a scalar field, our investigation offers a glimpse at
the full potential that open quantum dynamical effects have for studying and
constraining physics beyond the standard models of particles and cosmology.
Since the predicted exclusion plots look very promising, the present article,
besides being the very first practical application of the formalism developed
in Ref. [59], serves as a strong motivation for future studies with more
realistic probe systems. Those will provide us with new, powerful tools for
our search after screened scalar fields and even other candidate models for
dark energy or dark matter.

###### Acknowledgements.

The authors are grateful to H. Abele and P. Haslinger for useful discussions.
This article was supported by the Austrian Science Fund (FWF): P 34240-N, and
is based upon work from COST Action COSMIC WISPers CA21106, supported by COST
(European Cooperation in Science and Technology).

##  Appendix A Density matrix elements

Evaluating Eq. (29) gives us

$\displaystyle\rho(\mathbf{p};\mathbf{p^{\prime}};t)\,=\,\rho(\mathbf{p};\mathbf{p^{\prime}};0)e^{-\mathrm{i}(E_{\mathbf{p}}^{\phi}-E_{\mathbf{p^{\prime}}}^{\phi})t}\Biggl{\\{}1-(2\pi)^{3}\delta^{(3)}(\mathbf{0})\Biggl{[}\frac{\alpha^{2}_{2}}{8}\int_{\mathbf{q}}\frac{\sin^{2}(E_{\mathbf{q}}^{\phi}t)}{4(E_{\mathbf{q}}^{\phi})^{4}}\Delta^{\mathrm{F}}_{zz}\Delta^{\mathrm{F}}_{z^{\prime}z^{\prime}}$
$\displaystyle+\frac{\alpha_{1}^{2}M^{2}}{2}\int_{\mathbf{k}\mathbf{q}}\frac{1}{2E_{\mathbf{k}}^{\phi}E_{\mathbf{q}}^{\phi}E_{\mathbf{q}-\mathbf{k}}^{\chi}}\Biggl{(}\frac{\sin^{2}\left(\frac{(E_{\mathbf{k}}^{\phi}+E_{\mathbf{q}}^{\phi}+E_{\mathbf{q}-\mathbf{k}}^{\chi})t}{2}\right)}{(E_{\mathbf{k}}^{\phi}+E_{\mathbf{q}}^{\phi}+E_{\mathbf{q}-\mathbf{k}}^{\chi})^{2}}+\sum_{a=\pm}\frac{\sin^{2}\left(\frac{(E_{\mathbf{k}}^{\phi}+E_{\mathbf{q}}^{\phi}+aE_{\mathbf{q}-\mathbf{k}}^{\chi})t}{2}\right)}{(E_{\mathbf{k}}^{\phi}+E_{\mathbf{q}}^{\phi}+aE_{\mathbf{q}-\mathbf{k}}^{\chi})^{2}}f(E_{\mathbf{q}-\mathbf{k}}^{\chi})\Biggr{)}$
$\displaystyle+\frac{\alpha_{2}^{2}}{4}\int_{\mathbf{k}\mathbf{q}\mathbf{l}}\frac{1}{2E_{\mathbf{k}}^{\phi}E_{\mathbf{k}+\mathbf{q}+\mathbf{l}}^{\phi}E_{\mathbf{q}}^{\chi}E_{\mathbf{l}}^{\chi}}\Biggl{(}\sum_{a,b=\pm}\frac{\sin^{2}\left(\frac{(E_{\mathbf{k}}^{\phi}+E_{\mathbf{k}+\mathbf{q}+\mathbf{l}}^{\phi}+aE_{\mathbf{q}}^{\chi}+bE_{\mathbf{l}}^{\chi})t}{2}\right)}{2(E_{\mathbf{k}}^{\phi}+E_{\mathbf{k}+\mathbf{q}+\mathbf{l}}^{\phi}+aE_{\mathbf{q}}^{\chi}+bE_{\mathbf{l}}^{\chi})^{2}}f(E_{\mathbf{q}}^{\chi})f(E_{\mathbf{l}}^{\chi})$
$\displaystyle\qquad+\frac{\sin^{2}\left(\frac{(E_{\mathbf{k}}^{\phi}+E_{\mathbf{k}+\mathbf{q}+\mathbf{l}}^{\phi}+E_{\mathbf{q}}^{\chi}+E_{\mathbf{l}}^{\chi})t}{2}\right)}{2(E_{\mathbf{k}}^{\phi}+E_{\mathbf{k}+\mathbf{q}+\mathbf{l}}^{\phi}+E_{\mathbf{q}}^{\chi}+E_{\mathbf{l}}^{\chi})^{2}}+\sum_{a=\pm}\frac{\sin^{2}\left(\frac{(E_{\mathbf{k}}^{\phi}+E_{\mathbf{k}+\mathbf{q}+\mathbf{l}}^{\phi}+E_{\mathbf{q}}^{\chi}+aE_{\mathbf{l}}^{\chi})t}{2}\right)}{(E_{\mathbf{k}}^{\phi}+E_{\mathbf{k}+\mathbf{q}+\mathbf{l}}^{\phi}+E_{\mathbf{q}}^{\chi}+aE_{\mathbf{l}}^{\chi})^{2}}f(E_{\mathbf{l}}^{\chi})\Biggr{)}\Biggr{]}$
$\displaystyle-\frac{\mathrm{i}\alpha_{2}}{4}\left(\frac{1}{E_{\mathbf{p}}^{\phi}}-\frac{1}{E_{\mathbf{p^{\prime}}}^{\phi}}\right)t\Delta^{F}_{zz}-\frac{\alpha_{2}^{2}}{16}\Biggl{[}\frac{1}{2}\left(\frac{1}{(E_{\mathbf{p}}^{\phi})^{2}}-\frac{2}{E_{\mathbf{p}}^{\phi}E_{\mathbf{p^{\prime}}}^{\phi}}+\frac{1}{(E_{\mathbf{p^{\prime}}}^{\phi})^{2}}\right)t^{2}$
$\displaystyle\qquad+\left(\left(\frac{-\mathrm{i}}{2(E_{\mathbf{p}}^{\phi})^{3}}t+\frac{1-e^{-2\mathrm{i}E_{\mathbf{p}}^{\phi}t}}{4(E_{\mathbf{p}}^{\phi})^{4}}\right)+(\mathbf{p}\longleftrightarrow\mathbf{p^{\prime}})^{*}\right)\Biggr{]}\Delta^{\mathrm{F}}_{zz}\Delta^{\mathrm{F}}_{z^{\prime}z^{\prime}}$
$\displaystyle+\mathrm{i}\alpha_{1}^{2}M^{2}\int_{\mathbf{q}}\Biggl{[}\frac{1}{8E_{\mathbf{p}}^{\phi}E_{\mathbf{q}}^{\phi}E_{\mathbf{p}-\mathbf{q}}^{\chi}}\Biggl{(}\sum_{a=\pm}\frac{\frac{\mathrm{i}}{aE_{\mathbf{p}}^{\phi}+E_{\mathbf{q}}^{\phi}+E_{\mathbf{p}-\mathbf{q}}^{\chi}}\left(1-e^{-\mathrm{i}(aE_{\mathbf{p}}^{\phi}+E_{\mathbf{q}}^{\phi}+E_{\mathbf{p}-\mathbf{q}}^{\chi})t}\right)+t}{aE_{\mathbf{p}}^{\phi}+E_{\mathbf{q}}^{\phi}+E_{\mathbf{p}-\mathbf{q}}^{\chi}}$
$\displaystyle\qquad+\sum_{a,b=\pm}\frac{\frac{\mathrm{i}}{aE_{\mathbf{p}}^{\phi}+E_{\mathbf{q}}^{\phi}+bE_{\mathbf{p}-\mathbf{q}}^{\chi}}\left(1-e^{-\mathrm{i}(aE_{\mathbf{p}}^{\phi}+E_{\mathbf{q}}^{\phi}+bE_{\mathbf{p}-\mathbf{q}}^{\chi})t}\right)+t}{aE_{\mathbf{p}}^{\phi}+E_{\mathbf{q}}^{\phi}+bE_{\mathbf{p}-\mathbf{q}}^{\chi}}f(E_{\mathbf{p}-\mathbf{q}}^{\chi})\Biggr{)}+(\mathbf{p}\longleftrightarrow\mathbf{p^{\prime}})^{*}\Biggr{]}$
$\displaystyle+\mathrm{i}\frac{\alpha_{2}^{2}}{2}\int_{\mathbf{k}\mathbf{q}}\Biggl{[}\frac{1}{8E_{\mathbf{p}}^{\phi}E_{\mathbf{p}-\mathbf{k}-\mathbf{q}}^{\phi}E_{\mathbf{k}}^{\chi}E_{\mathbf{q}}^{\chi}}\Biggl{(}\sum_{a=\pm}\frac{\frac{\mathrm{i}}{aE_{\mathbf{p}}^{\phi}+E_{\mathbf{p}-\mathbf{k}-\mathbf{q}}^{\phi}+E_{\mathbf{k}}^{\chi}+E_{\mathbf{q}}^{\chi}}\left(1-e^{-\mathrm{i}(aE_{\mathbf{p}}^{\phi}+E_{\mathbf{p}-\mathbf{k}-\mathbf{q}}^{\phi}+E_{\mathbf{k}}^{\chi}+E_{\mathbf{q}}^{\chi})t}\right)+t}{2(aE_{\mathbf{p}}^{\phi}+E_{\mathbf{p}-\mathbf{k}-\mathbf{q}}^{\phi}+E_{\mathbf{k}}^{\chi}+E_{\mathbf{q}}^{\chi})}$
$\displaystyle\qquad+\sum_{a,b,c=\pm}\frac{\frac{\mathrm{i}}{aE_{\mathbf{p}}^{\phi}+E_{\mathbf{p}-\mathbf{k}-\mathbf{q}}^{\phi}+bE_{\mathbf{k}}^{\chi}+cE_{\mathbf{q}}^{\chi}}\left(1-e^{-\mathrm{i}(aE_{\mathbf{p}}^{\phi}+E_{\mathbf{p}-\mathbf{k}-\mathbf{q}}^{\phi}+bE_{\mathbf{k}}^{\chi}+cE_{\mathbf{q}}^{\chi})t}\right)+t}{2(aE_{\mathbf{p}}^{\phi}+E_{\mathbf{p}-\mathbf{k}-\mathbf{q}}^{\phi}+bE_{\mathbf{k}}^{\chi}+cE_{\mathbf{q}}^{\chi})}f(E_{\mathbf{k}}^{\chi})f(E_{\mathbf{q}}^{\chi})$
$\displaystyle\qquad+\sum_{a,b=\pm}\frac{\frac{\mathrm{i}}{aE_{\mathbf{p}}^{\phi}+E_{\mathbf{p}-\mathbf{k}-\mathbf{q}}^{\phi}+E_{\mathbf{k}}^{\chi}+bE_{\mathbf{q}}^{\chi}}\left(1-e^{-\mathrm{i}(aE_{\mathbf{p}}^{\phi}+E_{\mathbf{p}-\mathbf{k}-\mathbf{q}}^{\phi}+E_{\mathbf{k}}^{\chi}+bE_{\mathbf{q}}^{\chi})t}\right)+t}{aE_{\mathbf{p}}^{\phi}+E_{\mathbf{p}-\mathbf{k}-\mathbf{q}}^{\phi}+E_{\mathbf{k}}^{\chi}+bE_{\mathbf{q}}^{\chi}}f(E_{\mathbf{q}}^{\chi})\Biggr{)}-(\mathbf{p}\longleftrightarrow\mathbf{p^{\prime}})^{*}\Biggr{]}\Biggr{\\}}$
$\displaystyle+\alpha_{1}^{2}M^{2}\int_{\mathbf{q}}\rho(\mathbf{p}-\mathbf{q};\mathbf{p^{\prime}}-\mathbf{q};0)e^{-\mathrm{i}(E_{\mathbf{p}}^{\phi}-E_{\mathbf{p^{\prime}}}^{\phi})t}\sum_{a=\pm}\left(e^{\mathrm{i}(E_{\mathbf{p}}^{\phi}-E_{\mathbf{p}-\mathbf{q}}^{\phi}-aE_{\mathbf{q}}^{\chi})t}-1\right)$
$\displaystyle\qquad\times\left(e^{-\mathrm{i}(E_{\mathbf{p^{\prime}}}^{\phi}-E_{\mathbf{p^{\prime}}-\mathbf{q}}^{\phi}-aE_{\mathbf{q}}^{\chi})t}-1\right)\frac{af(aE_{\mathbf{q}}^{\chi})}{8E_{\mathbf{p}-\mathbf{q}}^{\phi}E_{\mathbf{p^{\prime}}-\mathbf{q}}^{\phi}E_{\mathbf{q}}^{\chi}(E_{\mathbf{p}}^{\phi}-E_{\mathbf{p}-\mathbf{q}}^{\phi}-aE_{\mathbf{q}}^{\chi})(E_{\mathbf{p^{\prime}}}^{\phi}-E_{\mathbf{p^{\prime}}-\mathbf{q}}^{\phi}-aE_{\mathbf{q}}^{\chi})}$
$\displaystyle+\frac{\alpha_{2}^{2}}{2}\int_{\mathbf{k}\mathbf{q}}\rho(\mathbf{p}-\mathbf{k}-\mathbf{q};\mathbf{p^{\prime}}-\mathbf{k}-\mathbf{q};0)e^{-\mathrm{i}(E_{\mathbf{p}}^{\phi}-E_{\mathbf{p^{\prime}}}^{\phi})t}$
$\displaystyle\qquad\times\sum_{a,b=\pm}\left(e^{\mathrm{i}(E_{\mathbf{p}}^{\phi}-E_{\mathbf{p}-\mathbf{k}-\mathbf{q}}^{\phi}-aE_{\mathbf{k}}^{\chi}-bE_{\mathbf{q}}^{\chi})t}-1\right)\left(e^{-\mathrm{i}(E_{\mathbf{p^{\prime}}}^{\phi}-E_{\mathbf{p^{\prime}}-\mathbf{k}-\mathbf{q}}^{\phi}-aE_{\mathbf{k}}^{\chi}-bE_{\mathbf{q}}^{\chi})t}-1\right)$
$\displaystyle\qquad\times\frac{abf(aE_{\mathbf{k}}^{\chi})f(bE_{\mathbf{q}}^{\chi})}{16E_{\mathbf{p}-\mathbf{k}-\mathbf{q}}^{\phi}E_{\mathbf{p^{\prime}}-\mathbf{k}-\mathbf{q}}^{\phi}E_{\mathbf{k}}^{\chi}E_{\mathbf{q}}^{\chi}(E_{\mathbf{p}}^{\phi}-E_{\mathbf{p}-\mathbf{k}-\mathbf{q}}^{\phi}-aE_{\mathbf{k}}^{\chi}-bE_{\mathbf{q}}^{\chi})(E_{\mathbf{p^{\prime}}}^{\phi}-E_{\mathbf{p^{\prime}}-\mathbf{k}-\mathbf{q}}^{\phi}-aE_{\mathbf{k}}^{\chi}-bE_{\mathbf{q}}^{\chi})}\,\,\,.$

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Assume (8) holds for its children, $u_{1}$ and $u_{2}$. Consider the case
where

$d_{\text{KT}}\left(f(u_{1})\oplus
f(u_{2}),\tau_{v}\right)<d_{\text{KT}}\left(f(u_{2})\oplus
f(u_{1}),\tau_{v}\right),$ (9)

so that Line 16 sets $f(v)$ as $f(u_{1})\oplus f(u_{2})$. We then have

$\begin{split}d_{\text{KT}}\left(f(v),\tau_{v}\right)&=d_{\text{KT}}\left(f(u_{1})\oplus
f(u_{2}),\tau_{v}\right)\\\
&\overset{\lx@cref{creftype~refnum}{lem:ktdist_1}}{=}d_{\text{KT}}\left(\tau_{v},\tau_{u_{1}}\oplus\tau_{u_{2}}\right)+d_{\text{KT}}\left(f(u_{1}),\tau_{u_{1}}\right)+d_{\text{KT}}\left(f(u_{2}),\tau_{u_{2}}\right).\end{split}$
(10)

The first term in (10) reaches the minimum since (9) holds. Moreover, the last
two terms also reach the minimum since (8) holds for $u_{1}$ and $u_{2}$.
These conditions imply $f(v)$ satisfies (8). Similar analysis can be applied
for the case where $d_{\text{KT}}\left(f(u_{1})\oplus
f(u_{2}),\tau_{v}\right)>d_{\text{KT}}\left(f(u_{2})\oplus
f(u_{1}),\tau_{v}\right)$. This along with the based cases finish the proof.

∎

##  Appendix F Lemmas in computational cost analysis

This section provides lemmas for Theorem 6.1, which shows that the asymptotic
cost of the density matrix algorithm is upper-bounded by that of the
canonicalization-based algorithm. We demonstrate in Lemma F.3 that when the
input tensor network has a tree structure, both the density matrix algorithm
and the canonicalization-based algorithm exhibit the same asymptotic cost for
truncating the bond sizes in the tree tensor network.

The Lemma F.1 and Lemma F.2 below are used to prove Lemma F.3.

######  Lemma F.1.

Consider a tensor network with a tree structure $T=(V_{T},E_{T})$. Assuming
that changing a tree tensor network into the canonical form will not change
any bond size of the network. For two adjacent vertices $z,v$, forming
$\texttt{canonical\\_form}_{T}(v,z)$ has the same asymptotic cost as forming
$\texttt{density\\_matrix}_{T}(v,z)$.

###### Proof.

For each edge set $E^{\prime}\subseteq E_{T}$, we let
$s(E^{\prime})=\exp(w(E^{\prime}))$ denote the bond size of $E^{\prime}$. We
also let $\mathbf{M}_{v}$ denote the tensor at each vertex $v\in V_{T}$.

For the pair of adjacent vertices $v,z$, assume that
$\texttt{canonical\\_form}_{T}(u,v)$ already exist for all $u\in
N(v)\setminus\\{z\\}$. Let $\mathbf{R}_{u}$ denote the non-orthogonal core of
$\texttt{canonical\\_form}_{T}(u,v)$. To construct the form
$\texttt{canonical\\_form}_{T}(v,z)$, we first contract $\mathbf{M}_{v}$ with
$\mathbf{R}_{u}$ for each $u\in N(v)\setminus\\{z\\}$, which yields a cost of
$\Theta\left(\sum_{u\in N(v)\setminus\\{z\\}}s(E_{T}(v))s(E_{T}(u,v))\right)$,
and then use a QR decomposition to orthogonalize the tensor at $v$, which
yields a cost of $\Theta\left(s(E_{T}(v))s(E_{T}(v,z))\right)$. These steps
make the overall cost

$\Theta\left(\sum_{u\in N(v)}s(E_{T}(v))s(E_{T}(u,v))\right).$ (11)

We now consider the computation of $\texttt{density\\_matrix}_{T}(v,z)$ under
the assumption that for all $u\in N(v)\setminus\\{z\\}$,
$\mathbf{L}_{u}=\texttt{density\\_matrix}_{T}(u,v)$ already exist. Below we
consider the three different cases,

  * •

when $N(v)\setminus\\{z\\}=\emptyset$, the computation involves the
contraction $\mathbf{M}_{v}\mathbf{M}_{v}^{T}$,

  * •

when $N(v)\setminus\\{z\\}=\\{u\\}$, the computation involves the contraction
$(\mathbf{M}_{v}\mathbf{L}_{u})\mathbf{M}_{v}^{T}$,

  * •

when $N(v)\setminus\\{z\\}=\\{u_{1},u_{2}\\}$, the computation involves the
contraction
$\mathbf{M}_{v}(\mathbf{L}_{u_{1}}\otimes\mathbf{L}_{u_{2}})\mathbf{M}_{v}^{T}$,
which can be efficiently computed by performing the contractions
$\mathbf{M}_{v}$ with $\mathbf{L}_{u_{1}}$ and $\mathbf{M}_{v}$ with
$\mathbf{L}_{u_{2}}$ first, and then contracting the outputs.

For all the cases above, the overall cost is $\Theta\left(\sum_{u\in
N(v)}s(E_{T}(v))s(E_{T}(u,v))\right)$, which equals the cost of the canonical
form. Since both $\texttt{canonical\\_form}_{T}(v,z)$ and
$\texttt{density\\_matrix}_{T}(v,z)$ have the same recursive relation,
computing $\texttt{canonical\\_form}_{T}(u,v)$ has the same cost as that of
the $\texttt{density\\_matrix}_{T}(u,v)$ for $u\in N(v)\setminus\\{z\\}$. This
finishes the proof.

∎

######  Lemma F.2.

Consider a tensor network with a tree structure $T=(V_{T},E_{T})$, where each
$z\in V_{T}$ represents a tensor $\mathbf{M}_{z}$. Let $v\in V_{T}$ be a leaf
vertex that represents $\mathbf{M}_{v}\in\mathbb{R}^{a_{v}\times b_{v}}$,
where $a_{v}$ denotes the size of the uncontracted modes and $b_{v}$ denotes
the size of the contracted modes incident on $v$, and let
$u=\texttt{parent}(T,v)$. Given that $\texttt{density\\_matrix}_{T}(u,v)$ has
been computed, computing the orthogonal matrix $\mathbf{U}_{v}$ (Line 11 or 18
of Algorithm 4) has a cost of $\Theta\left(a_{v}b_{v}^{2}\right)$.

###### Proof.

For the case where $a_{v}=O\left(b_{v}\right)$, the algorithm first computes
$\mathbf{L}_{v}=\texttt{density\\_matrix}_{T}(v)$ with a cost of
$\Theta\left(a_{v}b_{v}^{2}+a_{v}^{2}b_{v}\right)$, and then computes
$\mathbf{U}_{v}$ via a low-rank factorization on
$\mathbf{L}_{v}\in\mathbb{R}^{a_{v}\times a_{v}}$ with the maximum rank being
$r=O\left(a_{v}\right)$, which costs $\Theta\left(a_{v}^{2}r\right)$. The
overall cost is
$\Theta\left(a_{v}b_{v}^{2}+a_{v}^{2}b_{v}+a_{v}^{2}r\right)=\Theta\left(a_{v}b_{v}^{2}\right).$

For the case where $a_{v}=\Omega\left(b_{v}\right)$, the algorithm first
performs a QR decomposition of $\mathbf{M}_{v}$ into
$\mathbf{U}_{v}\in\mathbb{R}^{a_{v}\times
b_{v}},\mathbf{R}_{v}\in\mathbb{R}^{b_{v}\times b_{v}}$ with a cost of
$\Theta\left(a_{v}b_{v}^{2}\right)$, then computes the leading singular
vectors of $\mathbf{R}_{v}\mathbf{L}_{u}$ that is denoted
$\hat{\mathbf{U}}_{v}\in\mathbb{R}^{b_{v}\times r}$, which costs
$\Theta\left(b_{v}^{3}\right)$. Finally, $\mathbf{U}_{v}$ is updated as the
product $\mathbf{U}_{v}\hat{\mathbf{U}}_{v}$ with a cost of
$\Theta(a_{v}b_{v}r)$. Overall the cost is
$\Theta\left(a_{v}b_{v}^{2}+b_{v}^{3}+a_{v}b_{v}r\right)=\Theta\left(a_{v}b_{v}^{2}\right)$.
This finishes the proof. ∎

Figure 25:  Illustration of the difference between the canonicalization-based
algorithm and the density matrix algorithm. The upper path denotes truncating
the edge $(u,v)$ using canonicalization, and the lower path uses the density
matrix algorithm. In the lower path, the orthogonal matrix is calculated as
the leading singular vectors/eigenvectors of the density matrix
$\texttt{density\\_matrix}_{T}(z)$.

######  Lemma F.3.

Consider a given tree tensor network $T=(V_{T},E_{T})$. Let
$\sigma:V_{T}\to\\{1,\ldots,|V_{T}|\\}$ be a post-order DFS traversal of $T$
that shows the the tensor update ordering. Assuming that changing a tree
tensor network into its canonical form will not change any bond size of the
network, the asymptotic cost of the density matrix algorithm (Algorithm 4) for
truncating the modes in $T$ is the same as that of the canonicalization-based
algorithm (Algorithm 1) if both algorithms use the same update ordering
$\sigma$, and the same maximum bond size $\chi$.

###### Proof.

Consider the step to update the tensor at a given vertex $v\in V_{T}$. Let
$\mathbf{M}_{v}\in\mathbb{R}^{a_{v}\times b_{v}}$, where $a_{v}$ denote the
size of the uncontracted modes and $b_{v}$ denote the size of the contracted
modes of $\mathbf{M}_{v}$. Also let $r=\min\left(a_{v},b_{v},\chi\right)$ and
$u=\texttt{parent}(T,v)$. We break down the cost of Algorithm 4 and Algorithm
1 into 3 parts, and show that for each of the three parts, the costs of the
two algorithms are asymptotically equal.

In Algorithm 1, the steps include 1) forming
$\texttt{canonical\\_form}_{T}(u,v)$, 2) multiplying $\mathbf{M}_{v}$ with
$\mathbf{R}_{u}\in\mathbb{R}^{b_{v}\times b_{v}}$, the non-orthogonal core of
the canonical form, and 3) performing a rank-$\chi$ approximation to get
$\mathbf{U}_{v}\in\mathbb{R}^{a_{v}\times
r},\hat{\mathbf{R}}_{u}\in\mathbb{R}^{r\times b_{v}}$, and 3) multiplying
$\hat{\mathbf{R}}_{u}$ with $\mathbf{M}_{u}$.

In Algorithm 4 with each partition contracted into a tensor, the steps include
1) forming the density matrix $\texttt{density\\_matrix}_{T}(u,v)$, 2) using
$\texttt{density\\_matrix}_{T}(u,v)$ and $\mathbf{M}_{v}$ to compute
$\mathbf{U}_{v}\in\mathbb{R}^{a_{v}\times r}$ and
$\mathbf{M}_{v}=\mathbf{U}_{v}^{T}\mathbf{M}_{v}$, and 3) multiplying
$\mathbf{M}_{v}\in\mathbb{R}^{r\times b_{v}}$ with $\mathbf{M}_{u}$.

The comparison between the two algorithms is visualized in Fig. 25. It can be
seen that the third step of both algorithms have the same asymptotic cost. For
the first step, we show in Lemma F.1 that both algorithms have the same
asymptotic cost. For the second step, the canonicalization-based algorithm
yields a cost of
$\Theta\left(a_{v}b_{v}^{2}+a_{v}b_{v}r\right)=\Theta\left(a_{v}b_{v}^{2}\right)$
using the cost model in Section 2.2. In addition, we show in Lemma F.2 that
the cost to compute $\mathbf{U}_{v}$ in the density matrix algorithm under the
assumption that each partition is contracted into a tensor is also
$\Theta\left(a_{v}b_{v}^{2}\right)$. Since the multiplication
$\mathbf{U}_{v}^{T}\mathbf{M}_{v}$ costs
$\Theta\left(a_{v}b_{v}r\right)=O\left(a_{v}b_{v}^{2}\right)$, the cost equals
the cost of the canonicalization-based algorithm, thus finishing the proof. ∎