# Quantum decoherence from entanglement during inflation

Abasalt Rostami  Department of Physics, Sharif University of Technology,
Tehran, Iran  School of Physics, Institute for Research in Fundamental
Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran<EMAIL_ADDRESS>Javad
T. Firouzjaee  School of Astronomy, Institute for Research in Fundamental
Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran<EMAIL_ADDRESS>
###### Abstract

We study the primary entanglement effect on the decoherence of fields reduced
density matrix which are in interaction with another fields or independent
mode functions. We show that the primary entanglement has a significant role
in decoherence of the system quantum state. We find that the existence of
entanglement could couple dynamical equations coming from Schrödinger
equation. We show if one wants to see no effect of the entanglement parameter
in decoherence then interaction terms in Hamiltonian can not be independent
from each other. Generally, including the primary entanglement destroys the
independence of the interaction terms. Our results could be generalized to
every scalar quantum field theory with a well defined quantization in a given
curved space time.

###### Contents

  1. I Introduction
  2. II Decoherence in quantum physics
  3. III Schorodinge equation for entangled fields
  4. IV Decoherence from entanglement
    1. IV.1 Non-entangled Case
    2. IV.2 Entangled Case
  5. V Conclusion

##  I Introduction

Recent Cosmic Microwave Background (CMB) observations planck  of temperature
fluctuations agree well with the predictions made by inflation theory which
states the universe was in a accelerated phase after the Planck era. According
to the inflationary picture, not only primordial cosmological fluctuations
(CMB temperature) have quantum origin, but they are also created in a quantum
state. This quantum fluctuation provides an elegant description for the large
scale structure advent of our universe which explain how the density
perturbations which seeded structures in cosmos.

The basic point is that no noted cosmological data would present the actual
quantum state of primordial fluctuations because all known techniques of
observation focus on a restricted set of properties of those fluctuations.
Consequently, one has to consider not only the quantum aspects of the
cosmological fluctuations, but also the loss of quantum coherence prompted by
the partial description appropriate observation to study nontrivial quantum
behavior.

Along this way, main work on the quantum to classical transition of inflation
Polarski:1995jg ; Albrecht:1992kf ; Kiefer:2008ku ; Martin:2015qta  has
focused largely on the squeezing of the quantum state for each mode. This
squeezing, present on super-Hubble scales at the end of inflation, means that
the inflaton is effectively taking on different values at widely separated
points in space, which leads to an inhomogeneity in the temperature after
inflation which means quantum expectation values of products of quantities in
a highly squeezed state bring out to be identical to stochastic averages
calculated from a stochastic distribution of classical quantity
configurations.

However, this quantum to classical transition does not answer the question of
how we make a measurement. As is known, the inflaton field fluctuation is a
quantum field which is represented by Bunch-Davies vacuum state which is
completely homogeneous and isotropic. The inflaton dynamics preserves the
homogeneity and isotropy. Thus, we can not use inflaton state to explain the
observed inhomogeneous and anisotropic distribution of the primordial energy
density in our universe. As a result, the homogeneous quantum state which is a
coherent superposition of all field configurations collapses to a particular
stochastic realization of classical inhomogeneities Leon:2015eza ;
Leon:2016mdn . In this step, it is needed to have a mechanism of quantum
decoherence which necessitates the presence of additional environment degrees
of freedom that couple to quantum perturbations as a measuring device.
Decoherence describes the transition from pure to mixed state which arises
whenever the degrees of freedom of interest (quantum fluctuations) interact
with an environment involving other degrees of freedom whose properties are
not measured.

Decoherence is well-studied in the context of inflation Decoherence ;
Burgess:2014eoa ; Nelson:2016kjm . Nelson Nelson:2016kjm  argued that the
gravitational nonlinearities (from coupling between long-wavelength
fluctuations and an environmental sector in the interaction action) provide a
minimal mechanism for generating classical stochastic perturbations from
inflation via the decoherence. The best-suited framework to discuss about
decoherence of cosmological perturbations and the quantum to classical
transition is the Schrödinger field theory. This picture is the natural
framework to study the entanglement between the fields Martin-Martinez:2014gra
and can be used to study the entanglement effect on curvature power spectrum
Albrecht:2014aga  and bispectrum using the interaction picture Rostami:2016ulp
. There is no reason that one could not consider a more general initial state
such as an entangled one especially that inflation be an (low energy)
effective theory of a fundamental theory such as quantum gravity or has
multiple fields.

The purpose of the present article is to study the primary entanglement effect
on the decoherence of fields reduced density matrix which is in interaction
with another field or independent mode functions. It will be shown that the
primary entanglement has a significant role in decoherence of the system
quantum state. We discuss that the existence of entanglement could couple
dynamical equations coming from Schrödinger equation, and if someone wants to
see no effect of entanglement parameter in decoherence then interaction terms
in Hamiltonian can not be independent from each other. Generally, if we
include the primary entanglement the independence of the interaction terms
will be destroyed.

The article is organized as follows. Section II review the decoherence in the
quantum field theory. In the section III we present entanglement of fields in
the inflationary background and study the possible interactions in the third
order. Section IV is devoted to study the Schrödinger equation for the
entangled state and how the decoherence happen in this picture. Finally, we
conclude with a discussion in section V.

##  II Decoherence in quantum physics

One of the most important problems in quantum mechanics is the
classicalization problem. The classicalization means a process which
transforms a quantum system to a classical one. The straightforward (and of
course the most difficult) idea to unlock this problem is to replace the wave
collapse assumption with a deterministic dynamical process which can describe
how the collapse happens. This idea entails a departure during measurement
from Born rule for instance collapse models Collapse . One could try to find a
non-fundamental solution for the case of statistical quantum systems. This
method is not fundamental because implicitly includes Born rule. In despite it
can describe how we find a statistical quantum system in a statistical
classical system. In other words, it means how quantum probabilities change to
classical probabilities.

What happens in relation between system and its environment has been emerged
in recent years a dramatic picture (which people like to call it measurement).
This has been widely due to the attention of the phenomenon of decoherence. In
this section, we review the decoherence concept.

It is clear that the first requirement in the effect of environment on the
system under study is an evolution of the state vector in the Schrödinger
picture, which creates a correlation between the system (like inflaton at the
early universe) and states of environment (like other fields which affected on
inflaton during inflation). Suppose that the system can be in various states
labeled with an index $s$, while the environment can be in states labeled with
an index $e$, such that the states of the total system in Hilbert space can be
written in terms of a complete orthonormal basis of state vectors presented as
$\Psi_{se}$. We assume that at $t=0$ environment is placed in a initial state
denoted $e=0$, with the system in a general superposition of its states (in a
subspace of the total Hilbert space ), so that the combined system would have
an initial wave function as following

$\Psi(0)=\sum_{s}c_{s}\Psi_{s0}.$ (1)

When we turn on an interaction between the system under study and the its
correspond environment, the combined system evolves in a time $t$ to
$U\Psi(0)$, where $U$ is the time evolution unitary operator $U=e^{-itH}$. To
have an ideal decoherence, we need to choose the Hamiltonian $H$ to be in a
way that the basis states $\Psi_{s0}$ should evolve into states
$U\Psi_{s0}=\Psi_{se_{s}}$ , with the index $s$ unchanged, and with $e_{s}$
labeling some definite state of the environment in a one to one correspondence
with the state of the system under study, such that $e_{s}\neq e_{s^{\prime}}$
if $s\neq s^{\prime}$. For this, we just need

$U_{s^{\prime}e^{\prime},s0}=\delta_{ss^{\prime}}\delta_{e^{\prime}e_{s}}.$
(2)

It is possible always to choose the other elements of
$U_{s^{\prime}e^{\prime},se}$ with $e\neq 0$, to make the whole transformation
unitary. For instance, in the case of $e\neq 0$, we can take this
transformation as following

$U_{s^{\prime}e^{\prime},se}=\left\\{\begin{array}[]{rcl}\delta_{ss^{\prime}}\mathcal{U}^{(s)}_{ee^{\prime}},&\mbox{for}&e^{\prime}\neq
e_{s^{\prime}}\\\ 0,&\mbox{for}&e^{\prime}=e_{s^{\prime}}\end{array}\right.$
(3)

where the matrix $\mathcal{U}^{(s)}$ has been constrained by the condition
that, for all $e\neq 0$ and $e^{\prime}\neq 0$,

$\delta_{ee^{\prime}}=\sum_{e^{\prime\prime}\neq
e_{s}}{\mathcal{U}^{(s)}}^{*}_{e^{\prime\prime}e^{\prime}}\mathcal{U}^{(s)}_{e^{\prime\prime}e}.$
(4)

These conditions thus simply require that $\mathcal{U}^{(s)}$ are unitary
matrices, and since they are not subject to any other constraints, one can
establish any number of matrices which satisfy this condition.

After the system under study and the environment have interacted, the total
system would be found in following superposition

$U\Psi(0)=\sum_{s}c_{s}\Psi_{se_{s}},$ (5)

which is an entangled state of system and environment created by interaction.
We have not yet a decoherence, because the combined system is still in a pure
state and we just see a definite superposition of the basis. Based on Born
rule the system must make a transition during the measurement to one or other
of these states, with probabilities ${|c_{s}|}^{2}$. Here by classical state
we mean the favored states produced by measurement (interaction between system
and environment) which the system under study goes to them. Zurek identified
such sates with the name of “pointer states.”

After this introduction we are ready to ask why we see most of systems around
us classical? The answer has to do with the phenomenon of decoherence. This
happens because any specific environment will always be subject to tiny noises
which could rises the environment number of degrees of freedom extremely.
These perturbations could not by themselves change one classical state into
another one. We can investigate this issue in two equivalent looks. The
decoherence converts Eq. (5) as following

$\sum_{s}c_{s}\Psi_{se_{s}}\longrightarrow\sum_{s}\exp(i\phi_{s})\
c_{s}\Psi_{se_{s}},$ (6)

where the $\phi_{s}$ are randomly fluctuating phases. Consequently, when we
take account the expectation values the interferences between different terms
in the above superposition would average to zero, and the expectation value of
any observer operator $A$ gives

$\overline{\langle
A\rangle}=\sum_{s}{|c_{s}|}^{2}\left(\Psi_{se_{s}},A\Psi_{se_{s}}\right),$ (7)

with the bar over the expectation value indicating that it is averaged over
the phases $\phi_{s}$. Here we see that the expectation value of $A$ is just
given by a classical distribution. One may note that this is not really a
solution for measurement problem because we have used Born rule in (7).

One can also indicate this phenomenon in another way (and equivalent to the
former). To see it better, we go to the usual ket-bra notation. Suppose that
$|E\rangle$ and $|S_{i}\rangle$ be states of environment and system
respectively. Here we have assumed that $|S_{i}\rangle$ states establish a
orthonormal subspace. It is clear that the interaction defined in Eq. (2) and
(3) takes the combined system at $t_{0}$ to any later time as following

$|E(t_{0})\rangle|S_{i}(t_{0})\rangle\longrightarrow|E_{i}(t)\rangle|S_{i}(t)\rangle.$
(8)

Then if the environment acts as during an ideal measurement we will have

$\langle E_{i}(t)|E_{j}(t)\rangle\approx\delta_{ij}.$ (9)

Note that this would happen when there are many number of degrees of freedom
for the environment. Now an initially coherent superposition of system goes to
an entanglement state when time pasts as (5)

$|E(t_{0})\rangle\left(\sum_{i}c_{i}|S_{i}(t_{0})\rangle\right)\longrightarrow\sum_{i}c_{i}|E_{i}(t)\rangle|S_{i}(t)\rangle.$
(10)

Therefore, to see how decoherence comes across it is enough to find reduced
density matrix of system under study. If this matrix be respect
$|S_{i}\rangle$ then we have a classical distribution. Using of Eq. (9), the
components of reduced density matrix become

$\rho_{R}(S_{i},S_{j})\approx{|c_{i}|}^{2}\delta_{ij}.$ (11)

Then the effect of such kind of interactions is to eliminate off diagonal
components of a density matrix.

It would be useful to translate above discussion to a scalar field theory in
Schrödinger picture. Against Heisenberg picture for a field theory in which
one works with a specific Fock space, in Schrödinger picture we use wave
functional to describe what are happening in a quantum system. To do that,
suppose we have a scalar theory for $\phi(x)$. Then in Schrödinger picture, we
have to find a basis for this theory. Thus, this is natural to choose the
eigenstates of the operator $\hat{\phi}(x)$ as a suitable basis which has been
defined as following

$\hat{\phi}(x)|\phi(x)\rangle=\phi(x)|\phi(x)\rangle.$ (12)

Now an arbitrary state could be represented as a superposition of the field
eigenstates,

$|\Psi_{\phi}\rangle=\sum_{\phi(x)}\Psi[\phi(x)]\ |\phi(x)\rangle,$ (13)

which $\Psi[\phi(x)]$ has the role of wave functional. Note that the above
summation is functional integration and we write down it formally. To
establish a decohering system it’s enough to treat like what we did in Eq.
(10),

$|\Psi_{E}\rangle|\phi(x)\rangle\longrightarrow\left(|\Psi_{E}|_{\phi(x)}\rangle\right)|\phi(x)\rangle.$
(14)

Therefore, if one establish a simple combined system at the initial state
$|\Psi_{E}\rangle|\Psi_{\phi}\rangle$, the one could easily show that the
correspond reduced density matrix of $\phi(x)$ becomes

$\rho_{R}[\phi(x),\phi^{{}^{\prime}}(x)]=\Psi_{\phi}[\phi(x)]\Psi_{\phi}^{*}[\phi^{{}^{\prime}}(x)]\sum_{E}\left(\Psi_{E}[E]|_{\phi(x)}\right)\left(\Psi^{*}_{E}[E]|_{\phi^{{}^{\prime}}(x)}\right).$
(15)

Now if the interaction leads the summation term in the above equality to zero,
then we will have decoherence. To this end, we are tracking some interaction
terms such that satisfy this condition.

##  III Schorodinge equation for entangled fields

In this section we expand the Schrödinger field theory during inflation for a
combined system including scalar field $\phi(x)$ or a special degree of
freedom (as the main system) and another field like $\chi(x)$ or the rest of
degrees of freedom (as the environment). What we are looking for is wave
function of the combined system. Once we find it, then all information of
environment and system under study would be obtained and then we would be able
to see if the system can experience decoherence or not. Here the field
$\phi(x)$ has the role of fluctuation of inflaton and the field $\chi(x)$ is
another field which could exist during inflation but has no role in dynamics
of inflation. Nevertheless, depending on initial state, the entanglement
between this field and inflaton could appear in power spectrum and bispectrum
of the inflationary universe Bolis:2016vas . We are interested in to know,
what’s happened for the wave function when one start from an entangled state
of system and environment. We would like to emphasis that the following is not
just for two fields or inflation theory or different length modes but could
even use for scalar and tensor modes interactions.

The total wave function would evolve according to the Schrödinger equation

$i\frac{d}{dt}\Psi[E,S]=H[E,S;t]\Psi[E,S]$ (16)

with the time-depended Hamiltonian $H(t)$. We assume that the interaction
between system and environment could be treated perturbatively. In our case,
this is totally reasonable because we are working with fluctuations of fields.
The accuracy of such calculation would be valid up to the third order of
fluctuations. This Hamiltonian includes free terms of fields and the
interaction part

$H[E,S]=H_{0}[E,S]+H_{int}[E,S],$ (17)

which $H_{int}$ is the interaction between system and environment 111For
instance in the case which the long wavelength modes and short ones have the
role of system and environment respectively, this interaction comes from the
cubic terms in the perturbed action Maldacena:2002vr ; Nelson:2016kjm .
Moreover, these interaction terms can come from the extension of the standard
model of particle physics or moduli of compactification in string theory..
Here the free Hamiltonian $H_{0}$ includes a kinetic term with Fourier
transformation

$H_{k}[S]=\frac{1}{2}\int\frac{d^{3}p}{(2\pi)^{3}}f_{s}(\tau)\pi_{p}[S]\pi_{p}^{*}[S]$
(18)

which the conjugate momentum is defined by

$\pi_{p}[S]=-i\frac{\delta}{\delta\phi_{p}}.$ (19)

Here $f_{s}$ depends on the geometry of space time and the kind of fields. For
example in case of de Sitter space time for a free scalar field action one
find it $f_{s}=\frac{1}{2a(\tau)}$ and it is independent of fields although in
general hyper-globally space time it couldn’t be true Townsend:1997ku . To
solve Eq. (16) perturbatively, one needs to know the solution of the free part
in Eq. (17). The general solution $\Psi_{en}[E,S]$ for two independent free
scalar fields, should satisfy

$i\frac{d}{dt}\Psi_{en}[E,S]=H_{0}[E,S;t]\Psi_{en}[E,S].$ (20)

One could do it just by an anzats for ground state

$\Psi_{en}=N_{en}(\tau)\exp\left[-\int\frac{d^{3}k}{(2\pi)^{3}}\left(A_{k}(\tau)\phi_{k}\phi_{-k}+B_{k}(\tau)\chi_{k}\chi_{-k}+2C_{k}(\tau)\phi_{k}\chi_{-k}\right)\right],$
(21)

and finds some definite differential equations for $A$, $B$, and entangled
parameter $C$ and finally solve them with a initial condition, suitable for a
inflation theory. One notes that, this form of wave function222The
entanglement is put in here when $C(\tau_{0})\neq 0$ at the beginning. In
fact, using Schrödinger equation for free part of combined Hamiltonian, one
could find the simple equation
$\frac{C_{k}^{{}^{\prime}}}{C_{k}}=\frac{(A_{k}+B_{k})}{a^{2}(\tau)}$. The
non-vanishing value of entangled parameter $C_{k}$ in the start of inflation,
gives non-zero value of it when inflation goes on Bolis:2016vas . It maybe
useful to know that, the existence of $C_{k}$ in Schrödinger picture is
equivalence to making an entanglement with Bogoliubov transformation of vacuum
in Heisenberg picture Kanno:2015ewa .  is invariant under rotations and
spatial translations. This solution would be Gaussian because of quadratic
form of the free Hamiltonian. Now, to solve Hamiltonian including interactions
of cubic terms, we suppose there is solution as following

$i\Psi_{en}\dot{\Psi}_{ng}=\left(H_{k}[S]+H_{k}[E]+H_{int}[E,S]\right)\Psi_{en}\Psi_{ng}.$
(22)

Here we see that the effect of interactions appear as a non-Gaussian part
$\Psi_{ng}$ in the wave function at the ground state. Note that we have just
used kinetic parts of Hamiltonian. When one uses Eq. (20), the potential terms
of free Hamiltonian would be canceled from RHS of Eq. (22). Using of Eq. (18),
one can find the effective term of kinetic parts in Eq. (22)

$H_{k}[S]\Psi_{en}\Psi_{ng}\rightarrow\frac{1}{2}\int\frac{d^{3}p}{(2\pi)^{3}}f_{s}(\tau)\\{\frac{\delta\Psi_{en}}{\delta\phi_{-p}}\frac{\delta\Psi_{ng}}{\delta\phi_{p}}\\}$
(23)

and the same for environment. One can again find a good anzats for the non-
Gaussian part of wave function as

$\Psi_{ng}=\exp\int_{k^{{}^{\prime}},k,p}\left(\phi_{p}\chi_{k}\chi_{k^{{}^{\prime}}}F_{kk^{{}^{\prime}}p}+\phi_{p}\phi_{k}\phi_{k^{{}^{\prime}}}M_{kk^{{}^{\prime}}p}+\chi_{p}\phi_{k}\phi_{k^{{}^{\prime}}}N_{kk^{{}^{\prime}}p}+\chi_{p}\chi_{k}\chi_{k^{{}^{\prime}}}Q_{kk^{{}^{\prime}}p}\right)$
(24)

which the dynamical coefficients $N,M,F$ and $Q$ must be find by Schrödinger
equation 333One may want to know why the normalization factor has not been
considered in (24). The reason is that the dynamical equation for
normalization is related to zero order of perturbation and so is irrelevant
here. In other words, this consideration has not any effect in derivation of
(III\- III). In the above integral we use the usual convention
$\int_{k,k^{{}^{\prime}},p}\equiv\int\frac{dk^{3}}{(2\pi)^{3}}\frac{{dk^{\prime}}^{3}}{(2\pi)^{3}}\frac{dp^{3}}{(2\pi)^{3}}\delta(k+k^{{}^{\prime}}+p)$.
Also it is convenient to write down action of the cubic interaction on the
solution in Fourier space like

$H_{int}\Psi_{en}\Psi_{ng}=\int_{k^{{}^{\prime}},k,p}\left[\mathcal{H}^{(1)}_{kk^{{}^{\prime}}p}\phi_{k}\phi_{k^{{}^{\prime}}}\phi_{p}+\mathcal{H}^{(2)}_{kk^{{}^{\prime}}p}\chi_{k}\chi_{k^{{}^{\prime}}}\phi_{p}+\mathcal{H}^{(3)}_{kk^{{}^{\prime}}p}\phi_{k}\phi_{k^{{}^{\prime}}}\chi_{p}+\mathcal{H}^{(4)}_{kk^{{}^{\prime}}p}\chi_{k}\chi_{k^{{}^{\prime}}}\chi_{p}\right]\Psi_{en}\Psi_{ng},$
(25)

which the integration is just eigenvalue of $H_{int}$ operator. Substituting
Eq. (24), (25) and (21) in (22) and using this fact that all cubic
multiplication of fields in Schrödinger equation are independent, one would
get four coupled first order differential equations for unknown dynamical
variables in non-Gaussian part of wave function 444Here we have used conformal
time instead of cosmological time by substituting
$\frac{d}{dt}=-H\frac{d}{d\tau}$ Eq. (24)

$\displaystyle-iH\tau\dot{F}_{kk^{{}^{\prime}}p}=$ $\displaystyle
f_{s}(p,\tau)A_{p}(\tau)F_{kk^{{}^{\prime}}p}+2f_{s}(k,\tau)C_{k}(\tau)N_{kpk^{{}^{\prime}}}+2f_{e}(k^{{}^{\prime}},\tau)B_{k^{{}^{\prime}}}(\tau)F_{kk^{{}^{\prime}}p}+3f_{e}(p,\tau)C_{p}(\tau)Q_{kk^{{}^{\prime}}p}+\mathcal{H}^{(2)}_{kk^{{}^{\prime}}p}$
(26)

which comes from $\chi^{2}\phi$ coefficients in Schrödinger equation, and

$-iH\tau\dot{M}_{kk^{{}^{\prime}}p}=3f_{s}(p,\tau)A_{p}(\tau)M_{kk^{{}^{\prime}}p}+f_{e}(p,\tau)C_{p}(\tau)N_{kk^{{}^{\prime}}p}+\mathcal{H}^{(1)}_{kk^{{}^{\prime}}p}$
(27)

is the coefficient of $\phi^{3}$, and

$-iH\tau\dot{Q}_{kk^{{}^{\prime}}p}=f_{s}(p,\tau)C_{p}(\tau)F_{kk^{{}^{\prime}}p}+3f_{e}(p,\tau)B_{p}(\tau)Q_{kk^{{}^{\prime}}p}+\mathcal{H}^{(4)}_{kk^{{}^{\prime}}p}$
(28)

is related to the term of $\chi^{3}$, and finally

$\displaystyle-iH\tau\dot{N}_{kk^{{}^{\prime}}p}=$ $\displaystyle
3f_{s}(p,\tau)C_{p}(\tau)M_{kk^{{}^{\prime}}p}+2f_{s}(k^{{}^{\prime}},\tau)A_{k^{{}^{\prime}}}(\tau)N_{kk^{{}^{\prime}}p}+2f_{e}(k,\tau)C_{k}(\tau)F_{kpk^{{}^{\prime}}}+f_{e}(p,\tau)B_{p}(\tau)N_{kk^{{}^{\prime}}p}+\mathcal{H}^{(3)}_{kk^{{}^{\prime}}p},$
(29)

which is the coefficient of $\phi^{2}\chi$. In Eq. (III-III) we’ve used dote
as conformal time derivation. Solving these equations we able to talk about
all quantum effect on inflaton perturbations like decoherence. In the next
section, we will show that just Eq. (III) is related to decoherence and try to
find some exact solutions for these equations. In general one has to solve
these kind of equations numerically which we left it for next papers.

We end this section with understanding this question that, why the
entanglement parameter is related to interaction parts. To answer this
question one may look at the path integral method. We know that one could
relate the propagator to wave function from path integral method to
Schrödinger picture version of quantum mechanics

$\Psi\leftrightarrow\int\mathcal{D}X\ \exp\left(iS\right).$ (30)

Now, if we insert the interaction part to the exponent in RHS, then the LHS
should be modified and vice versa. Because the RHS would be changed
exponentially, it is reasonable, the LHS be modified as the same.

##  IV Decoherence from entanglement

In this section we want to find some exact solution of Eq. (III-III) and
investigate the phenomenon of decoherence related to the form of interactions.
As mentioned before, it seems difficult to solve this coupled system of
differential equations although one could try for numerical ones. Here we
shall check two especial exact solutions of them.

###  IV.1 Non-entangled Case

One of the interesting case is non-entangled states. In this case since $C=0$
for all modes, system and environment are not correlated to each other at
early time. Now, we analyze the above coupled system of differential equations
and understand more about decoherence in such theories. When we justify
entangled parameter to zero, this coupled system of equations is transformed
to decoupled one and so the solution would be pretty easy. At first, we would
like to focus on Eq. (III)

$iH\tau\dot{F}_{kk^{{}^{\prime}}p}+g(\tau;k,k^{{}^{\prime}},p)H\tau
F_{kk^{{}^{\prime}}p}+\mathcal{H}^{(2)}_{kk^{{}^{\prime}}p}=0$ (31)

where $g\equiv f_{s}A+2f_{e}B$. To solve it, we need an initial condition.
Actually we are interested in theories which interactions are active during
inflation and have no effect at early time. With this assumption, it would be
reasonable to assume $F(\tau_{0})=0$. Therefore, the solution is

$F_{kk^{{}^{\prime}}p}(\tau)=i\int_{\tau_{0}}^{\tau}\frac{d\tau^{{}^{\prime}}}{H\tau^{{}^{\prime}}}\mathcal{H}^{(2)}_{kk^{{}^{\prime}}p}(\tau^{{}^{\prime}})\exp\left[i\int_{\tau^{{}^{\prime}}}^{\tau}d\tau^{{}^{\prime\prime}}g(\tau^{{}^{\prime\prime}};k,k^{{}^{\prime}},p)\right].$
(32)

This relation would be more simple in some cases. For example, in de Sitter
space suppose the theory in which $\mathcal{H}^{(2)}$ is proportional to
$a^{n}$ with $n>0$ for scale factor Nelson:2016kjm . If the coupling between
system and environment is weak enough such that the density matrix remains
close to Gaussian, then the real part of $F$ doesn’t grow at late time. In
other words, when $\tau^{{}^{\prime}}$ is close to $\tau$, the exponent part
in Eq. (33) is negligible. So for late time we have just imaginary part of
this as following

$\lim_{\tau\rightarrow
0}ImF_{kk^{{}^{\prime}}p}(\tau)=\int_{\tau_{0}}^{\tau}\frac{d\tau^{{}^{\prime}}}{H\tau^{{}^{\prime}}}\mathcal{H}^{(2)}_{kk^{{}^{\prime}}p}(\tau^{{}^{\prime}}).$
(33)

Now, let’s come back to decoherence phenomenon. The summation in the reduced
density matrix Eq. (15) is proportional to

$\displaystyle\sum_{E}\left(\Psi_{E}[E]|_{\phi(x)}\right)\left(\Psi^{*}_{E}[E]|_{\phi^{{}^{\prime}}(x)}\right)\propto\int\mathcal{D}\chi|\Psi_{en}[E,S]|^{2}\exp\left[\int_{k,k^{{}^{\prime}},p}\chi_{k}\chi_{k^{{}^{\prime}}}\left(\phi_{p}F_{kk^{{}^{\prime}}p}+\phi^{{}^{\prime}}_{p}F^{*}_{kk^{{}^{\prime}}p}\right)\right]=$
$\displaystyle\langle\exp\left[i\int_{k,k^{{}^{\prime}},p}\chi_{k}\chi_{k^{{}^{\prime}}}\Delta\phi_{p}Im(F_{kk^{{}^{\prime}}p})\right]\rangle$
(34)

where $\Delta\phi_{p}=\phi_{p}-\phi^{\prime}_{p}$, and this equation appears
as a average value of the exponential on environment degrees of freedom. Thus,
based on Riemann’s integration theorem, if imaginary part of $F$ be large then
the off-diagonal elements of density matrix go to zero and decoherence would
happen. From here, we can see $F$ is the most important term which decoherence
phenomenon is related to. This happens because $\Delta\phi$ has been coupled
only with $F$.

Although the dynamics of $Q$, $N$ and $M$ have no effect in decoherence but
for a complete description, we shall solve them here. Because all these
differential equations are decoupled and the same, the solutions are like the
solution of $F$ if we choose the same initial conditions:

$M_{kk^{{}^{\prime}}p}(\tau)=i\int_{\tau_{0}}^{\tau}\frac{d\tau^{{}^{\prime}}}{H\tau^{{}^{\prime}}}\mathcal{H}^{(1)}_{kk^{{}^{\prime}}p}(\tau^{{}^{\prime}})\exp\left[i\int_{\tau^{{}^{\prime}}}^{\tau}d\tau^{{}^{\prime\prime}}m(\tau^{{}^{\prime\prime}};k,k^{{}^{\prime}},p)\right],$
(35)

where $m\equiv 3f_{s}A$. For $N$ we have

$N_{kk^{{}^{\prime}}p}(\tau)=i\int_{\tau_{0}}^{\tau}\frac{d\tau^{{}^{\prime}}}{H\tau^{{}^{\prime}}}\mathcal{H}^{(3)}_{kk^{{}^{\prime}}p}(\tau^{{}^{\prime}})\exp\left[i\int_{\tau^{{}^{\prime}}}^{\tau}d\tau^{{}^{\prime\prime}}n(\tau^{{}^{\prime\prime}};k,k^{{}^{\prime}},p)\right],$
(36)

which $n\equiv 2f_{s}A+f_{e}B$, and finally

$Q_{kk^{{}^{\prime}}p}(\tau)=i\int_{\tau_{0}}^{\tau}\frac{d\tau^{{}^{\prime}}}{H\tau^{{}^{\prime}}}\mathcal{H}^{(4)}_{kk^{{}^{\prime}}p}(\tau^{{}^{\prime}})\exp\left[i\int_{\tau^{{}^{\prime}}}^{\tau}d\tau^{{}^{\prime\prime}}q(\tau^{{}^{\prime\prime}};k,k^{{}^{\prime}},p)\right],$
(37)

where $q\equiv 3f_{e}B$. The significance of these terms is in calculation
related to bi-spectrum but not used in the decoherence of density matrix or
spectrum of CMB. In fact one could make an standard theory for inflation just
by choosing $\mathcal{H}^{(1)}=\mathcal{H}^{(3)}=\mathcal{H}^{(4)}=0$. In such
theory, $N$, $Q$ and $M$ are equal to zero and $F$ has a nonzero value which
can give a contribution to bispectrum of CMB.

###  IV.2 Entangled Case

In this part of paper, we investigate a theory in which decoherence happens as
before case with this difference that the theory has the entangled initial
state. In the previous case we saw that if there is no entanglement in
combined system then there wouldn’t be any correlation between other
interaction terms. Here we will find a solution in the presence of
entanglement such that interaction terms are related to entanglement variable
$C$.

To find such solution we choose $N=Q=0$. Therefore, Eq. (III) which is
responsible of decoherence phenomenon would be the same as before and
decoherence will be happened like the case $C=0$. One can easily see that the
solution for $M$ is the same as Eq. (35) but there are two consistency
relations for $\mathcal{H}^{(3)}$ and $\mathcal{H}^{(4)}$ as following

$\displaystyle
3if_{s}(p,\tau)C_{p}(\tau)\int_{\tau_{0}}^{\tau}\frac{d\tau^{{}^{\prime}}}{H\tau^{{}^{\prime}}}\mathcal{H}^{(1)}_{kk^{{}^{\prime}}p}(\tau^{{}^{\prime}})\exp\left[i\int_{\tau^{{}^{\prime}}}^{\tau}d\tau^{{}^{\prime\prime}}m(\tau^{{}^{\prime\prime}};k,k^{{}^{\prime}},p)\right]+$
$\displaystyle
2if_{e}(k,\tau)C_{k}(\tau)\int_{\tau_{0}}^{\tau}\frac{d\tau^{{}^{\prime}}}{H\tau^{{}^{\prime}}}\mathcal{H}^{(2)}_{kk^{{}^{\prime}}p}(\tau^{{}^{\prime}})\exp\left[i\int_{\tau^{{}^{\prime}}}^{\tau}d\tau^{{}^{\prime\prime}}g(\tau^{{}^{\prime\prime}};k,k^{{}^{\prime}},p)\right]=-\mathcal{H}^{(3)}_{kk^{{}^{\prime}}p}$
(38)

and

$if_{s}(p,\tau)C_{p}(\tau)\int_{\tau_{0}}^{\tau}\frac{d\tau^{{}^{\prime}}}{H\tau^{{}^{\prime}}}\mathcal{H}^{(2)}_{kk^{{}^{\prime}}p}(\tau^{{}^{\prime}})\exp\left[i\int_{\tau^{{}^{\prime}}}^{\tau}d\tau^{{}^{\prime\prime}}g(\tau^{{}^{\prime\prime}};k,k^{{}^{\prime}},p)\right]=-\mathcal{H}^{(4)}_{kk^{{}^{\prime}}p}.$
(39)

These equations imply that two of four interactions are not independent and
for example once $\mathcal{H}^{(1)}$ and $\mathcal{H}^{(2)}$ are given, the
others would be completely defined. One may know if Eq. (III-III) have
solutions such that affect on decoherence process. In other words, the
question arise as to whether there is a solution in which decoherence depends
on entanglement. Generally the answer is positive and this can be seen by
choosing $N=0$ and $Q\neq 0$. Here, one should note that the entanglement
parameter $C$ is independent from the magnitude of slow-roll parameters in
case of inflationary universe example. This happens because $C$ just depends
on ratio of interactions. Now, Eq. (III) is no longer independent of other
equations and should be solved again with entanglement variable $C$. Here we
can consider $\mathcal{H}^{(4)}$ as a independent term and use equation (III)
to find $F$ as

$\displaystyle F_{kpk^{{}^{\prime}}}=$
$\displaystyle-\frac{i}{2f_{e}(k,\tau)C_{k}(\tau)}3f_{s}(p,\tau)C_{p}(\tau)\int_{\tau_{0}}^{\tau}\frac{d\tau^{{}^{\prime}}}{H\tau^{{}^{\prime}}}\mathcal{H}^{(1)}_{kk^{{}^{\prime}}p}(\tau^{{}^{\prime}})\exp\left[i\int_{\tau^{{}^{\prime}}}^{\tau}d\tau^{{}^{\prime\prime}}m(\tau^{{}^{\prime\prime}};k,k^{{}^{\prime}},p)\right]-\frac{1}{2f_{e}(k,\tau)C_{k}(\tau)}\mathcal{H}^{(3)}_{kk^{{}^{\prime}}p}.$
(40)

We see here in this case decoherence is completely affected by
$\lambda_{k,p}\equiv\frac{C_{p}(\tau)}{C_{k}(\tau)}$. In fact if $\lambda$ is
very small then decoherence happens weakly but if this is very large then
decoherence happens rapidly based on how the behavior of $\mathcal{H}^{(1)}$
is. There are two important key points: first, the last term in the right hand
side Eq. (IV.2) has no effect on decoherence because the imaginary part of $F$
plays role in the decoherence; second, in contrast to the Eq. (33) which the
imaginary part of $F$ is proportional to the integration of
$\mathcal{H}^{(2)}$, the imaginary part of $F$ is related to the integration
of $\mathcal{H}^{(1)}$. As a result, $\mathcal{H}^{(1)}$ has to satisfy the
same condition needed for $\mathcal{H}^{(2)}$ to have decoherence
Burgess:2014eoa ; Nelson:2016kjm .

##  V Conclusion

Our goal of this work was to investigate the decoherence of a field (inflation
in cosmology) which is not alone in the universe and it can be in interaction
with environment or any other independent modes function. There is no reason
one could not consider a more general initial state with including the
entanglement between field and modes. Especially that inflation be an (low
energy) effective theory of a fundamental theory such as quantum gravity or
has multiple fields.

We have two types of entanglements: first, primary entanglement which comes
from initial state of combined system including environment and the system
under study; second, late entanglement come cross during interaction system
and environment. To have decoherence during the interaction the late
entanglement is necessary.

Let us conclude with some remarks:

  * •

In this paper we showed that the primary entanglement has a significant role
in decoherence of the system quantum state.

  * •

It was shown that if there is no primary entanglement in combined system then
the interaction terms which can be responsible for late entanglement are
independent. In this case, the interaction term,
$H_{int}\sim\int_{k^{{}^{\prime}},k,p}\mathcal{H}^{(2)}_{kk^{{}^{\prime}}p}\chi_{k}\chi_{k^{{}^{\prime}}}\phi_{p}$,
contribute to the decoherence.

  * •

If we have primary entanglement $C\neq 0$ there is a solution of the
Schrödinger wave equation, $N=Q=0$, the interaction terms are not independent.
If we have two of the interaction terms then we can find the others. It is
commonly believed that we can add any interaction term between combined
system, but this result show that the primary entanglement limit us to choose
the interaction term. In this case the decoherence happens like the case of no
primary entanglement, $C=0$.

  * •

We also find another exact solution $C,Q\neq 0$, $N=0$, in which the
decoherence process is directly affected by entanglement parameter $C$. There
are two important key points here: first, the last term in the right hand side
Eq. (IV.2) has no effect on decoherence because the imaginary part of $F$
plays role in the decoherence; second, in contrast to the Eq. (33) which the
imaginary part of $F$ is proportional to the integration of
$H_{int}\sim\int_{k^{{}^{\prime}},k,p}\mathcal{H}^{(2)}_{kk^{{}^{\prime}}p}\chi_{k}\chi_{k^{{}^{\prime}}}\phi_{p}$,
the imaginary part of $F$ is related to the integration of
$H_{int}\sim\int_{k^{{}^{\prime}},k,p}\mathcal{H}^{(1)}_{kk^{{}^{\prime}}p}\phi_{k}\phi_{k^{{}^{\prime}}}\phi_{p}$.
As a result, $\mathcal{H}^{(1)}$ has to satisfy the same condition needed for
$\mathcal{H}^{(2)}$ to have decoherence Burgess:2014eoa ; Nelson:2016kjm .

  * •

The dependency between the primary entanglement and the interaction terms can
have can have a teleological interpretation. Suppose that the semi-classical
picture of inflation theory is an effective low energy theory of a universal
quantum gravity theory (UQGT). Therefore, both primary entangled states and
the interaction terms emerge from the low energy limitation from UQGT. From
this prospective, the entanglement and the interaction terms can not be
independent.

  * •

And at the end we should to emphasis that the difference between theories with
different interactions could appear not only their two-points correlation
functions at early time but also three point functions are needed. The
contributions of these three-points functions come from non-Gaussian part of
theories which now is related to entanglement parameter $C_{k}$ in the general
solution.

Several directions for future research exist; One can use a multi fields model
(for example two scalar fields or tensor-scalar field models) like with
primary entanglement, to look at the decoherence rate of wave function in the
super horizon and verify whether this entanglement delay the classicalization
or not. Even for single field inflation, one can look at third order action
with different coupling between independent modes and look at wave density
matrix decoherence in presence of primary entanglement. It would be also
interesting to study the dynamics of the entangled state in phase space. With
calculating the wave function’s Wigner function, we can understand the
coherence lengths and squeezing at late times, and whether the diagonal matrix
elements evolve according to the standard Fokker-Planck equation of
Starobinsky’s stochastic inflation. One can also study the entanglement
effects on the redundant records of long wavelength perturbation during
inflation to study the squeezing of the quantum stats Nelson:2017pmc .

Acknowledgments:

We would like to thank M.M. Sheikh-Jabbari and Hassan Firouzjahi for useful
comments.

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