# On-the-Fly Simulation of Two-Dimensional Fluorescence-Excitation Spectra

Sebastian V. Pios  Zhejiang Laboratory, Hangzhou 311100, China  Maxim F. Gelin
School of Science, Hangzhou Dianzi University, Hangzhou 310018, China  Luis
Vasquez  School of Science, Hangzhou Dianzi University, Hangzhou 310018, China
Jürgen Hauer  Department of Chemistry, Technical University of Munich, D-85747
Garching, Germany  Lipeng Chen  Zhejiang Laboratory, Hangzhou 311100, China
<EMAIL_ADDRESS>
###### Abstract

Two-dimensional (2D) fluorescence-excitation (2D-FLEX) spectroscopy is a
recently proposed nonlinear femtosecond technique for the detection of
photoinduced dynamics. The method records a time-resolved fluorescence signal
in its excitation- and detection-frequency dependence, and hence combines the
exclusive detection of excited state dynamics (fluorescence) with signals
resolved in both excitation and emission frequencies (2D electronic
spectroscopy). In this work, we develop an on-the-fly protocol for the
simulation of 2D-FLEX spectra of molecular systems, which is based on
interfacing the classical doorway-window representation of spectroscopic
responses with trajectory surface hopping simulations. Applying this
methodology to the gas-phase pyrazine, we show that femtosecond 2D-FLEX
spectra can deliver detailed information otherwise obtainable via attosecond
spectroscopy.

Two-dimensional electronic spectroscopy (2D-ES) was developed at the turn of
the millennia as the optical analogue to nuclear magnetic resonance (NMR)
spectroscopy. 1, 2, 3 Since then, 2D-ES has become one of the most powerful
and information-rich femtosecond four-wave-mixing techniques in the UV/Vis
regimes. It was implemented in collinear, 4 partially non-collinear,5, 6, 7
and fully non-collinear realizations 8, 9 and emerged as an efficient high-
precision technique for the comprehensive real-time characterization of a
large variety of material systems, from atoms 10 and polyatomic chromophores11
to molecular aggregates and solids. 12, 13, 14, 15, 16

2D-ES is a variant of the third-order heterodyne-detected four-wave-mixing
spectroscopy (see Figure 1(a)). The 2D-ES signal
$S^{(\sigma)}(\tau,T,\tau_{t})$ can be recorded in the rephasing ($\sigma=R$,
$\mathbf{k_{S}}=\mathbf{k_{1}}-\mathbf{k_{2}}+\mathbf{k_{3}}$) or non-
rephasing ($\sigma=NR$,
$\mathbf{k_{S}}=-\mathbf{k_{1}}+\mathbf{k_{2}}+\mathbf{k_{3}}$) phase-matching
direction as a function of the delay times between the first two pulses
(coherence time $\tau$), the second and the third pulses (population time
$T$), and the last two pulses (detection time $\tau_{t}$). The sketch of
Figure 1(a) indicates that 2D-ES delivers four-wave-mixing signal with maximal
information content as both excitation and emission frequencies are fully
resolved. This allows for an unprecedented level of scrutiny of material
systems, explaining why 2D-ES is so powerful and popular. 2D-ES signals are
usually Fourier transformed with respect to $\tau$ (excitation frequency
$\omega_{\tau}$) and $\tau_{t}$ (detection frequency $\omega_{t}$), and the
resultant 2D-ES spectra $S^{(\sigma)}(\omega_{\tau},T,\omega_{t})$ visualize
how correlation between the initial ($\omega_{\tau}$) and final ($\omega_{t}$)
states of the system evolves with the population time $T$.

Figure 1: Pulse sequences shaping 2D-ES (a), time- and frequency-resolved
fluorescence (b), and 2D-FLEX (c) signals.

2D-ES spectra consist of three contributions, which are customarily referred
to as ground-state bleach (GSB), stimulated emission (SE) and excited state
absorption (ESA)

$S^{(\sigma)}(\omega_{\tau},T,\omega_{t})=\sum_{j=\text{GSB, SE,
ESA}}S_{j}^{(\sigma)}(\omega_{\tau},T,\omega_{t}).$ (1)

However, the information content of these contributions differs substantially.
GSB monitors wavepacket motion on the electronic ground state, while SE and
ESA reflect wavepacket dynamics on the excited electronic states. Yet, SE and
ESA differ in their respective “spectator states” which are responsible for
encoding the dynamical information. In the SE, the wavepacket motion on the
excited electronic states is projected onto the electronic ground state, while
it is projected onto higher-lying excited electronic states in the ESA. For
complex material systems, it is difficult to decipher the intrinsic system
dynamics from GSB, SE, and notably ESA spectra. Furthermore, GSB, SE and ESA
spectra usually overlap, which leads to the partial cancellation of spectral
features and hinders proper interpretation of 2D-ES spectra. While the
techniques of polarization-sensitive detection 17, 18, 19 or beating maps 20,
21, 22 can partially solve the problem, they do not provide a universal
solution.

As has recently been proposed by Hauer and coworkers,23 this problem can be
addressed by a novel technique, 2D fluorescence-excitation (2D-FLEX)
spectroscopy, which combines the spectral resolution of excitation and
detection processes inherent in 2D-ES with the exclusive detection of the SE
contribution. The principle of 2D-FLEX can readily be grasped from the
following considerations. Time-resolved fluorescence spectroscopy 24 permits a
direct detection of wavepacket dynamics on the excited electronic states, but
cannot provide information on electronic coherences during the excitation of
the system with a single pump pulse (see Figure 1(b)). In 2D-FLEX, a single
pump pulse is replaced by a pump-pulse pair delayed by a time $\tau$, as
sketched in Figure 1(c). Hence, 2D-FLEX can be envisaged as a technique
combining the double-pulse excitation of 2D-ES with the broadband detection of
time- and frequency-resolved fluorescence.

The high information content of 2D-FLEX spectroscopy has been demonstrated by
simulating spectra for several model systems.23 Yet, there is still a lack of
practical tools to simulate 2D-FLEX spectra for realistic molecular systems.
The aim of this work is to develop an ab initio theoretical framework for the
simulation of 2D-FLEX spectra of molecular systems by interfacing the
classical doorway-window representation of spectroscopic responses with
trajectory surface hopping simulations. The method has been implemented for
the pyrazine molecule, a system known for its multiple conical intersections
of lower lying excited electronic states. The photophysics of pyrazine has
been comprehensively studied by ab initio quantum-chemistry methods,25, 26
dynamic calculations,27, 28 and simulations of nonlinear spectroscopic
signals. 29, 30, 31, 32, 33, 34, 35

The 2D-FLEX spectra (see Figure 1(c)) can be represented as a sum of rephasing
and nonrephasing contributions,

$S_{2DF}(\omega_{\tau},T,\omega_{t})\sim\mathrm{Re}\sum_{\sigma=R,\,NR}S_{2DF}^{(\sigma)}(\omega_{\tau},T,\omega_{t}),$
(2)

which can be further evaluated in terms of third-order response functions as
follows 23

$S_{2DF}^{(\sigma)}(\omega_{\tau},T,\omega_{t})\sim\,\int_{0}^{\infty}d\tau\,\int_{-\infty}^{\infty}dt\,\int_{0}^{\infty}dt_{3}\,\int_{0}^{\infty}dt_{2}\,\int_{0}^{\infty}dt_{1}\,R_{\sigma}(t_{3},t_{2},t_{1})e^{i\eta_{\sigma}\omega_{\tau}\tau}\times$
(3)
$e^{i\eta_{\sigma}\omega_{pu}(t_{1}-\tau)}e^{-(\zeta-i\omega_{t})t_{3}}E_{t}(t)E_{t}(t-t_{3})E_{pu}(t+T-t_{3}-t_{2})E_{pu}(t+\tau+T-t_{3}-t_{2}-t_{1}).$

Here $\sigma=R,NR$, $\eta_{NR}=1$, $\eta_{R}=-1$; $\omega_{pu}$ and
$E_{pu}(t)$ represent the carrier frequency and the dimensionless envelope of
the pump pulse; $\omega_{t}$ is the fluorescence frequency, and $E_{t}(t)$ is
the envelope of the up-conversion (gate) pulse which is responsible for the
time resolution of the emitted fluorescence; $t_{1}$, $t_{2}$, and $t_{3}$ are
the time intervals between sequential system-field interactions;
$R_{NR}(t_{3},t_{2},t_{1})$ and $R_{R}(t_{3},t_{2},t_{1})$ are the non-
rephasing and rephasing response functions which correspond, respectively, to
the response functions $R_{1}(t_{3},t_{2},t_{1})$ and
$R_{2}(t_{3},t_{2},t_{1})$ of Mukamel’s monograph.36

2D-FLEX spectra $S_{2DF}(\omega_{\tau},T,\omega_{t})$ of Eq. (2) and SE
contributions to the total (rephasing + nonrephasing) 2D-ES spectra

$S_{SE}(\omega_{\tau},T,\omega_{t})\sim\mathrm{Re}\sum_{\sigma=R,\,NR}S_{SE}^{(\sigma)}(\omega_{\tau},T,\omega_{t}),$
(4)

are determined by the same response functions and are therefore quite similar.
Yet, there is a fundamental difference between these two spectra.23 The time
($T$) – frequency ($\omega_{t}$) resolution of 2D-FLEX spectra is Fourier-
limited ($\delta_{T}\delta_{\omega_{t}}\sim 1$) because it is governed by the
same gate pulse (Figure 1(c)), while the time-frequency resolution of 2D-ES
spectra is not Fourier limited because it is determined by time delays between
different pairs of pulses (Figure 1(a)). This observation has profound effect
on the peak shapes and time-frequency resolution of 2D-FLEX and 2D-ES SE
spectra. While it is instructive to compare 2D-FLEX and 2D-ES SE spectra, we
note that only 2D-FLEX spectra can be measured directly. There is no
experimentally feasible way to measure the SE-part of a 2D-ES spectrum.

Interfacing dynamical trajectory methods with ab initio electronic-structure
methods into on-the-fly simulation protocols is a flourishing field of
research in theoretical spectroscopy (see recent reviews 37, 38, 39, 40, 41,
42, 43, 44). In the context of the present work, we require a simulation
protocol that can account for finite pulse durations, which, as explained
above, determine the time-frequency resolution of 2D-FLEX spectra. The
protocol which is based on combining trajectory surface hopping (SH)
simulations 45, 46 with the classical doorway-window (DW) representation of
four-wave-mixing signals excellently meets this criterion.43 This protocol was
developed and applied for the evaluation of transient-absorption pump-probe,29
2D-ES, 30 and time-resolved fluorescence spectra.47 Its validity to predict
pump-probe and 2D-ES spectra of pyrazine has recently been proved by comparing
with full quantum simulations of the models of reduced dimensionality. 35

Since 2D-FLEX spectra combine a double-pulse excitation of 2D-ES with the
detection of time- and frequency-resolved fluorescence, the DW-SH simulation
protocol for 2D-FLEX spectra can be immediately constructed by combining the
doorway function of 2D-ES spectra 30 with the window function of time-resolved
fluorescence spectra.47 The computational details are presented in Supporting
Information, while the flowchart clarifying the simulation procedure is
depicted in Figure 2. In the DW-SH simulation protocol, 2D-FLEX spectra can be
evaluated as a product of the doorway and window functions averaged over SH
trajectories

$S_{2DF}(\omega_{\tau},T,\omega_{t})\sim\langle
D_{2DF}(\omega_{\tau},\boldsymbol{R}_{g},\boldsymbol{P}_{g};e)W_{2DF}(\omega_{t},\boldsymbol{R}_{e}(T),\boldsymbol{P}_{e}(T);e(T))\rangle.$
(5)

Here, $\boldsymbol{R}_{g}$ and $\boldsymbol{P}_{g}$ are initial nuclear
coordinates and momenta, which can be sampled from the Wigner distribution in
the electronic ground state,
$\rho_{g}^{\mathrm{Wig}}(\boldsymbol{R}_{g},\boldsymbol{P}_{g})$; 48
$D_{2DF}(\omega_{\tau},\boldsymbol{R}_{g},\boldsymbol{P}_{g};e)$ is the
doorway function of Eq. (S3) in the initial electronic state $e$ sampled
according to its oscillator strength and pump-frequency detuning; 43
$W_{2DF}(\omega_{t},\boldsymbol{R}_{e}(T),\boldsymbol{P}_{e}(T);e(T))$ is the
window function of Eq. (S4) in which
$\boldsymbol{R}_{e}(T),\,\boldsymbol{P}_{e}(T)$ denote nuclear positions and
momenta in the electronic state $e(T)$ at the time $T$, and the notation
$e(T)$ emphasizes that a trajectory initiated at T=0 fs in an excited state
$e$ may end up at time $T$ in another excited state $e(T)$ due to nonadiabatic
transitions; $\langle...\rangle$ denotes averaging over SH trajectories. The
doorway-window expressions for the evaluation of 2D-ES SE spectra 30 are also
presented in Supporting Information. It should be noted that the doorway and
window functions are fully specified by electronic energies and transition
dipole moments along SH trajectories, as well as by the carrier frequencies
and envelops of the pump and gate pulses.

In all simulations of the present work, the pump pulse is tuned into resonance
with the bright state ($B_{2u}$) of pyrazine ($\omega_{p}=5.2$ eV), and the
envelopes of the pump and gate (for 2D-FLEX) and probe (for 2D-ES SE) pulses
are modeled by Gaussian functions, $E_{a}(t)=\exp(-(t/\tau_{a})^{2})$,
$E_{a}(\omega)=\exp(-(\omega\tau_{a})^{2}/4)$, where $\tau_{a}$ is the pulse
duration ($a=pu,\,t,\,pr$ correspond to pump, gate and probe pulse,
respectively). The widths of the spectral features of 2D-FLEX and 2D-ES SE
spectra are determined by the spectral widths of the laser pulses and by the
peak shape broadening (see Eq. (S5)), in which the inhomogeneous broadening
parameter is fixed at $\tau_{\nu}^{-1}=0.05$ eV in all our calculations.

Before we delve into discussion of the simulated 2D-FLEX and 2D-ES SE spectra,
it is instructive to briefly characterize the main photophysical processes in
pyrazine and determine their spectroscopic manifestations. Pyrazine possesses
three low-lying excited electronic states in the Franck-Condon region, the
$B_{2u}(\pi\pi^{*})$, $B_{3u}(n\pi^{*})$ and $A_{u}(n\pi^{*})$, which are
nonadiabatically coupled through a series of conical intersections.25, 26 The
experimental gas-phase UV linear absorption spectrum of pyrazine at 353$\pm$3
K is shown in Figure 3 (black, data is taken from Ref. 49). The spectrum
consists of two bands, an intense band centered around 4.9 eV and a weaker
band centered around 4.0 eV. These bands are attributed to the bright $B_{2u}$
state and the weakly absorbing $B_{3u}/A_{u}$ states, respectively. It is
essential that the shape and intensity of the $B_{3u}/A_{u}$ band is mostly
determined by the nonadiabatic coupling between the bright $B_{2u}$ state and
almost dark $B_{3u}/A_{u}$ states (the so-called intensity borrowing effect).
As revealed by the femtosecond photoelectron spectroscopy measurements, the
$B_{2u}-B_{3u}$ coupling results in the $\sim 20$ fs conical-intersection-
driven $B_{2u}\rightarrow B_{3u}$ population transfer.50, 51, 52 Internal
conversion from the coupled $B_{2u}/B_{3u}$ states to the electronic ground
state occurs on a much longer timescale of $\sim 20$ ps.50, 51, 52 Hence, only
$B_{2u}/B_{3u}$ steady-state fluorescence of pyrazine in the gas phase was
measured 53 and simulated.54, 55 On the other hand, the previous simulations
of time-resolved fluorescence of pyrazine in various reduced-dimensional
models 31, 32, 56 as well as recent ab initio simulations of the SE
contribution to pyrazine’s transient-absorption pump-probe spectra 29, 35
reveal that the $B_{2u}/B_{3u}$ fluorescence of pyrazine survives on a
timescale of at least 150 fs, but experiences a fivefold intensity reduction
caused by the $B_{2u}/B_{3u}$ conical intersection.

Sample the initial $\boldsymbol{R}_{g}$, $\boldsymbol{P}_{g}$, $e$ and
evaluate the doorway function
$D_{2DF}(\omega_{\tau},\boldsymbol{R}_{g},\boldsymbol{P}_{g};e)$Propagate and
store the stochastically selected classicaltrajectories using a surface-
hopping algorithmfor a selected simulation timeEvaluate the window function
$W_{2DF}(\omega_{t},\boldsymbol{R}_{e}(T),\boldsymbol{P}_{e}(T);e(T))$for the
desired population times T Convergence of
$S_{2DF}(\omega_{\tau},T,\omega_{t})$ achieved? End simulationNoYes

Figure 2: Flowchart depicting the simulation protocol for obtaining 2D-FLEX
spectra as proposed in this work. Figure 3: Black: experimental absorption
spectrum of pyrazine molecule in the gas-phase at 353$\pm$3 K (data is taken
from Ref. 49); Red: energy profile of the Gaussian laser pulse centered at 5.2
eV with width of $\tau=$ 5 fs. Figure 4: 2D-FLEX spectra
$S_{\mathrm{2DF}}(\omega_{\tau},T,\omega_{t})$ of pyrazine as a function of
the excitation frequency $\omega_{\tau}$ and the detection frequency
$\omega_{\mathrm{t}}$ at population times T = 0 (a), 20 (b), 35 (c), 60 (d),
100 (e), 130 (f), 160 (g), 190 fs (h) for $\tau_{pu}$ = 5 fs and $\tau_{t}$ =
30 fs. For each $T$, the intensities of the spectra are scaled for better
visibility.

Figure 4 depicts 2D-FLEX spectra $S_{2DF}(\omega_{\tau},T,\omega_{t})$ of
pyrazine as a function of the excitation frequency $\omega_{\tau}$ and the
fluorescence frequency $\omega_{t}$ at population times T = 0 (a), 20 (b), 35
(c), 60 (d), 100 (e), 130 (f), 160 (g) and 190 fs (h). The pulse durations for
the pump and gate pulses are set to $\tau_{pu}=5$ fs (with $E_{pu}(\omega)$
plotted in red in Figure 3) and $\tau_{t}=30$ fs, respectively, which
corresponds to the achievable time resolution in fluorescence up-conversion
experiments. 57, 58 2D-FLEX spectra excited with longer pump pulses
$\tau_{pu}=15$ fs exhibit similar structure, but the peaks are narrower along
$\omega_{\tau}$ (see Figure S1 in Supporting Information). The spectra in
Figure 4 illustrate the ability of 2D-FLEX to directly monitor the downhill
(conical-intersection-mediated) population transfer. Namely, the
$\omega_{\tau}$-position of all peaks reveals the initially excited bright
$B_{2u}$ state, while the $\omega_{t}$-positions of the peaks follow the
$B_{2u}\rightarrow B_{3u}$ population transfer and subsequent energy
redistribution in the coupled $B_{3u}/A_{u}$ states.

At T = 0 fs (Figure 4a), a single peak around $\omega_{\tau}$ = $\omega_{t}$ =
5.2 eV corresponds to the emission from the initially populated bright
$B_{2u}$ state. The peak is elongated along the main diagonal, which signifies
a substantial inhomogeneous broadening.59 As T increases, the wavepacket moves
down to $\omega_{t}\approx 4$ eV at $T=20$ fs (Figure 4b), monitoring the
conical-intersection-driven $B_{2u}\rightarrow B_{3u}$ internal conversion
process. The spectrum splits into two branches at T = 35 fs with stronger
peaks in the upper branch and weaker peaks in the lower branch (Figure 4c). At
T = 60 fs (Figure 4d), the separation between those two branches disappears,
and the spectrum exhibits pronounced doublets at $\omega_{t}\approx$ 4.4 and
3.2 eV as well as numerous weaker peaks in the entire frequency range
$2\,\,\text{eV}<\omega_{t}<4.5\,\,\text{eV}$. These peaks reveal the lower
vibronic levels of the $B_{2u}$ state and higher vibronic levels of the
$B_{3u}/A_{u}$ states. For $T\geq 100$ fs, the spectrum splits again into
upper ($\omega_{t}>4.3$ eV) and lower ($\omega_{t}<4$ eV) branches. At T = 100
fs, the upper branch is dominated by a relatively pronounced peak around
$\omega_{t}=4.7$ eV (Figure 4e), and the lower branch is composed of several
well-resolved peaks in the frequency range of $\omega_{t}$ = 2.0 - 4.0 eV
which mirror the wavepacket motion in the coupled $B_{3u}/A_{u}$ states. The
intensities of peaks in both upper and lower branches decrease with $T$,
indicating the irreversible $B_{2u}\rightarrow B_{3u}/A_{u}$ population
transfer (see Figures 4e-h). From T = 0 fs to T = 190 fs, the maximum
intensities of the 2D-FLEX spectra decrease by a factor of about 36. The
complete time evolution of 2D-FLEX spectra with a time step of 5 fs for T are
shown in Figures S2-S6 in the Supporting Information.

Figure 5: 2D-ES SE spectra $S_{SE}(\omega_{\tau},T,\omega_{\mathrm{t}})$ of
pyrazine as a function of the excitation frequency $\omega_{\tau}$ and the
detection frequency $\omega_{\mathrm{t}}$ at population times T = 0 (a), 20
(b), 35 (c), 60 (d), 100 (e), 130 (f), 160 (g), 190 fs (h) for $\tau_{pu}$ =
$\tau_{pr}$ = 0.1 fs. For each $T$, the intensities of the spectra are scaled
for better visibility.

The 2D-ES SE spectra simulated with pulse durations of $\tau_{pu}=\tau_{pr}=$
5 fs at different population times T monitor exclusively the time evolution of
a single peak around $\omega_{\tau}=\omega_{t}=$ 5.2 eV, while the peaks for
lower $\omega_{t}$ completely vanish (see Figure S7 in the Supporting
Information). This is because the laser pulse with duration of $\tau=$ 5 fs
only covers the $B_{2u}$ band of absorption spectrum of pyrazine, but does not
overlap with the $B_{3u}/A_{u}$ band (see the energy profile of the laser
pulse in Figure 3). To cover the entire frequency domain of the absorption
spectrum of pyrazine, one needs to use nowadays unavailable attosecond pulses.
The 2D-ES SE spectra simulated with such pulses ($\tau_{pu}$ = $\tau_{pr}$ =
0.1 fs, almost impulsive limit) at population times $T=0$ (a), 20 (b), 35 (c),
60 (d), 100 (e), 130 (f), 160 (g), 190 fs (h) are displayed in Figure 5.
Comparison of Figures 4 and 5 shows that positions of the peaks of 2D-ES SE
and 2D-FLEX spectra coincide, as expected. Furthermore, intensities of the
peaks in 2D-ES SE spectra decrease by a factor of 44 from $T=0$ to $T=190$ fs,
which is only slightly larger than a factor of 36 for 2D-FLEX spectra.
However, different detection schemes affect significantly the spectral
resolution and peak shapes of the two signals. While 2D-FLEX spectra in Figure
4 exhibit distinct peaks along $\omega_{t}$, the peaks of 2D-ES SE spectra in
Figure 5 are much broader. The reason for this difference is as follows. The
widths of peaks of 2D-ES SE spectra in Figure 5 are primarily determined by
the optical dephasing parameter of 0.05 eV and are almost independent of the
spectral widths of the laser pulses (impulsive limit). The widths and shapes
of peaks of 2D-FLEX spectra in Figure 4 are predominantly determined by the
spectral widths of the pump and gate pulses.

Comparison of simulated 2D-FLEX and 2D-ES SE spectra demonstrates that
different detection schemes can affect the signal dramatically. Detection of
2D-ES SE spectra is determined by the carrier frequencies and bandwidths of
the pulses (cf. Ref. 60). As a result, obtaining 2D-ES SE spectra of pyrazine
in the entire energy domain of $2\,\text{eV}\leq\omega_{t}\leq 5.2\,\text{eV}$
requires attosecond pulses. On the other hand, detection of 2D-FLEX spectra in
a broad range of wavelengths of spontaneously emitted photons allows fine-
tuning of the simultaneous time- and frequency-resolution,61, 62 which in turn
yields a highly informative and detailed view of the intramolecular energy
transfer and redistribution. Hence, 2D-FLEX can become a method of choice for
studying complex nonadiabatic dynamics which could only be monitored by 2D-ES
with still unavailable attosecond pulses.

In summary, we develop an on-the-fly SH protocol for the simulation of 2D-FLEX
spectra of polyatomic chromophores. If necessary, the developed simulation
routine can be further enhanced by employing the machine-learning technique
following the recipes of Ref. 35. The simulated 2D-FLEX spectra of pyrazine
demonstrate that three facets ensure high predictive power and information
content of this novel spectroscopic technique. (i) Similar to 2D-ES, 2D-FLEX
spectra monitor wavepacket motions of molecular systems excited at frequency
$\omega_{\tau}$ and probed at frequency $\omega_{t}$. (ii) In contrast to
2D-ES, 2D-FLEX spectra contain exclusively the SE contribution. (iii) 2D-FLEX
spectra provide sufficient temporal and spectral resolution by employing the
experimentally available pump ($\tau_{pu}\sim 5$ fs) and gate ($\tau_{t}\sim
30$ fs) pulses, as well as cover the spectral range of several eV along
$\omega_{t}$. Separately, the facets (i)-(iii) are inherent in a number of
spectroscopic techniques. For example, time-resolved fluorescence detects
solely SE contribution with sufficient temporal ($T$) and frequency
($\omega_{t}$) resolution,57, 58 and UV/Vis 2D-ES SE spectra permit broadband
detection.6, 63 It is the combination of all three facets that makes 2D-FLEX
spectroscopy a highly promising tool for scrutinizing nonadiabatic dynamics in
a number of molecular systems. It is expected that the present work will
promote practical implementation of 2D-FLEX spectroscopy and facilitate
interpretation of the forthcoming 2D-FLEX experiments.

##  1 Supporting Information

Doorway-Window functions in 2D-FLEX, Computational details, 2D-FLEX spectra
for $\tau_{\mathrm{pu}}$ = 15 fs, the complete time evolution of 2D-FLEX
spectra for $\tau_{\mathrm{pu}}$ = 5 fs, 2D-ES SE spectra for
$\tau_{pu}=\tau_{pr}$=5 fs.

S.V.P. and L.P.C. acknowledge support from the starting grant of research
center of new materials computing of Zhejiang Lab (No. 3700-32601). M.F.G.
acknowledges support from the National Natural Science Foundation of China
(No. 22373028). L.V. acknowledges postdoctoral fellowship of Hangzhou Dianzi
University. J.H. acknowledges support from the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation), Project No. 514636421.

##  2 Data availability statement

The data is available from the authors upon reasonable request.

##  3 Code availability statement

The code for the generation of 2D-FLEX spectra from ab initio data is
available at https://github.com/psebastianzjl/2d_flex_spectra_generation.

##  4 Competing interests

The authors declare that they have no competing interests.

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# Constructing an entangled state in Heisenberg picture for inflationary
cosmology

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

The effects of the entanglement of the inflaton field which initially
entangled with those of another field on observables like power spectrum are
known in the context of the Schrödinger field theory. To clarify this effect
in Heisenberg picture, there were some attempts to construct the initial
entangled state by making use of an entangled transformation (like Bogoliubov
transformation) between the Bunch-Davies vacuums and squeezed states. We study
the role of the time-dependent entangled transformation in the Schrödinger
field theory. We derive the relation between two vacuum states which their
mode functions are transformed by the entangled transformation in Schrödinger
picture. We discuss that the time-dependency of the entanglement parameter is
inevitable and only in the first order of the entanglement parameter
perturbation this time-dependency vanishes. We study the entangled
transformation in the Heisenberg picture in term of the entangled parameter
which appears in the entangled state in Schrödinger picture.

###### Contents

  1. I Introduction
  2. II A Short Review On Mono-field Quantization
  3. III Bogoliubov transformation
  4. IV ABH State
  5. V Entanglement Transformation
  6. VI Conclusion

##  I Introduction

The main experimental observation of the early universe, Cosmic Microwave
Background (CMB), have become very accurate and have allowed us to deduce the
details of the even earlier phase of the cosmos Ade:2015xua . It is known that
the temperature fluctuation in the CMB are originated from the early quantum
fluctuation which becomes classical as the universe expand. Therefore,
studying the quantum nature of the inflationary background can lead to better
understanding of the observables like power spectrum and bispectrum which can
have quantum correction.

An important aspect of the quantum physics is the entanglement which has been
one of the fascinating features predicted by quantum mechanics since Einstein-
Podolsky-Rosen (EPR) EPR . Although more attention has been paid to how to
make use of quantum entanglement of EPR pairs in quantum cryptography and
quantum information review-en , the entanglement entropy plays an important
role in studying the black hole information problem and holography which we
can count of the degrees of entanglement of a quantum system. In the
cosmology, Maldacena Maldacena:2012xp  developed an explicit method to
calculate the entanglement entropy in a quantum field theory in the Bunch-
Davies vacuum of the de Sitter space and showed that there can be the quantum
entanglement between two causally disconnected open charts in de Sitter space
and in this way more ideas presented to detect these quantum effects En-
Observational . Along the way, one can study the quantum entanglement in the
Schrödinger field theory which the operators are time independent. The
Schrödinger picture is the natural framework to study the entanglement between
the fields Holman-entanglement . The Schrödinger representation is also best
suited to discuss the decoherence of the cosmological perturbations and the
quantum to classical transition in inflationary models Polarski:1995jg  and
one can develop in-in formalism in the Schrödinger picture for curvature
perturbations Rostami:2016ulp  to study N-point function in the observation.
In the way, many works have been done to study the entanglement effect and the
quantum to classical transition in the inflationary models Martin:2015qta . On
the other hand, apart from this fact that the Heisenberg picture field theory
is more common and easier to study the inflationary models (for example the
calculation of the Power spectrum and non-gaussianity in Schrödinger picture
needs the heavy calculation on the functional space), this picture has better
description for particle representation of each state with explicit
presentation of annihilation and creation process. Moreover, constructing the
entangled states using the entanglement transformation for fermions is easier
in the Heisenberg picture than Schrödinger picture kanno-sasaki . There is an
instruction to built the entangled state in Heisenberg picture.

Recently, in Kanno:2015ewa  an initially entangled state between two free
massive scalar fields in de Sitter space was studied and the entangled state
using a Bogoliubov transformation between the Bunch-Davies vacuum and a four-
mode squeezed state was constructed. In this paper, we show that though Kanno
Kanno:2015ewa  presented a novel way to study the entangled state in the
Heisenberg picture, her calculation and analysis to introduce the entanglement
was not correct in some parts. We follow her way to study the role of the
time-dependent entangled transformation in the Schrödinger field theory. We
derive the relation between two vacuum states which their mode functions are
transformed by the entangled transformation and also Bogoliubov transformation
in Schrödinger picture. We calculate the entangled parameter in Schrödinger
picture in term of the time-dependent entangled transformation in the
Heisenberg picture, and the inflation power spectrum will be calculated in
term of the time-dependent entangled parameter in Heisenberg picture. It is
discussed that the time-dependency of the entanglement parameter is inevitable
and only in the first order of the entanglement parameter perturbation this
time-dependency vanishes. We will discuss the limitation that the current
definition of the entanglement.

The paper is organized as follows. First, we review the quantization of a
mono-field for Friedmann-Lemaitre-Robertson-Walker (FLRW) background in
Section II. Section III is devoted to introduce and use the Bogoliubov
transformation to get the vacuum state in the Schrödinger picture. In Section
IV we introduce the ABH entangled state in the Schrödinger picture of the
inflationary cosmology. Next, after discussing the relation between the ABH
entangled state and the entangled stated which built of the entangled
transformation, the inflationary power spectrum will be calculated in Section
V. We end with a discussion in Section VI.

##  II A Short Review On Mono-field Quantization

The main goal of this section is to study a free scalar field in time-
dependent gravitational backgrounds. At first step, we shall consider a free
scalar field in a homogeneous and isotropic universe and then try to quantize
it. In this way, we follow Mukhanov and Winitzki book notation mukhanov-book .

A minimally coupled real scalar field φ(x) in a curved spacetime is described
by the action

$S[\phi]=\int
d^{4}x\sqrt{-g}\left(\frac{1}{2}g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi-\frac{1}{2}m^{2}\phi^{2}\right).$
(1)

Now, we consider an important class of metrics which is used for the
homogeneous and isotropic space-time namely FLRW. This is characterized by the
following metric

$ds^{2}=-dt^{2}+a^{2}(t)d\vec{x}^{2}.$ (2)

where $a(t)$ is the scale factor. A flat FRW space-time is a conformally flat
space-time. If one replaces the coordinate $t$ by the conformal time $\eta$,
the metric would be transformed into a conformally flat form

$\eta(t)\equiv\int\limits_{{t_{0}}}^{t}{\frac{{dt}}{{a(t)}}}.$ (3)

By this transformation the action (1) is written as

$S[\phi]=\frac{1}{2}\int d^{3}xd\eta\
a^{2}\left({\phi^{\prime}}^{2}-(\nabla\phi)^{2}-m^{2}a^{2}\phi^{2}\right).$
(4)

where a prime denotes the derivative with respect to the conformal time
$\eta$. This action gives the following equation of motion for our scalar
field

${\phi^{{}^{\prime\prime}}}^{2}-\nabla^{2}\phi+2\frac{a^{{}^{\prime}}}{a}\phi^{{}^{\prime}}+m^{2}a^{2}\phi=0.$
(5)

Although this equation seems a bit different from the equation of motion which
one derives for a scalar field in flat space-time but by using of the field
redefinition $\tilde{\phi}=a\phi$, we could make it more similar to the
equation of motion a scalar field in flat background. In fact with this
redefinition the action for $\tilde{\phi}$ is

$S[\tilde{\phi}]=\frac{1}{2}\int
d^{3}xd\eta\left({\tilde{\phi^{{}^{\prime}}}}^{2}-(\nabla\tilde{\phi})^{2}-m^{2}(\eta)\
\tilde{\phi}^{2}\right)$ (6)

where, we have defined

$m^{2}(\eta)=m^{2}a^{2}-\frac{a^{{}^{\prime\prime}}}{a}.$ (7)

Now, the equation of motion for the new field is

${\tilde{\phi^{{}^{\prime\prime}}}}-\nabla^{2}\tilde{\phi}+m^{2}(\eta)\tilde{\phi}=0$
(8)

this equation is like what we had in flat space-time for a free field with
this difference that here the mass term is time dependent. All information
about the influence of the gravitational field on $\tilde{\phi}$ is
encapsulated in the time-dependent mass $m(\eta)$. This would lead to this
fact that the energy of the field $m(\eta)$ is generally not conserved. In
quantum physics language, this leads to the possibility of particle creation
which the energy for new particles is supplied by the gravitational field. If
we expand the field $\tilde{\phi}$ in Fourier modes, we have

$\tilde{\phi}(\eta,x)=\int\frac{d^{3}k}{(2\pi)^{3}}\tilde{\phi}_{k}(\eta)e^{ik.x}.$
(9)

Thus, reach to the the decoupled equations of motion for the modes
$\tilde{\phi}_{k}$

${\tilde{\phi^{{}^{\prime\prime}}}}_{k}^{2}+\omega_{k}^{2}(\eta)\tilde{\phi}_{k}=0,$
(10)

where we have defined

$\omega_{k}^{2}(\eta)\equiv k^{2}+m^{2}(\eta).$ (11)

This equation has two linearly independent solutions for each mode which these
two solutions span the two-dimensional space of all solutions. If
$v_{k}(\eta)$ is a complex solution then $v_{k}^{*}(\eta)$ would be another
independent solution. Any two linearly independent solutions have to have a
non-vanishing Wronskian. One can easily show the Wronskian of any two
independent solutions of (10) is constant in conformal time. We choose this
constant as following

$W(v_{k},v^{*}_{k})=v_{k}^{{}^{\prime}}v^{*}_{k}-v_{k}{v^{{}^{\prime}}}^{*}_{k}=2i.$
(12)

The general solution $\tilde{\phi}_{k}$ can be expressed as a linear
combination of $v_{k}^{*}(\eta)$ and $v_{k}(\eta)$ as

$\tilde{\phi}_{k}(\eta)=\frac{1}{\sqrt{2}}[a_{k}v_{k}^{*}(\eta)+a^{*}_{-k}v_{k}(\eta)],$
(13)

where the $v_{k}^{*}(\eta)$ and $v_{k}(\eta)$ are called mode functions. Now,
by replacing (13) in (9), we obtain $\tilde{\phi}(\eta,x)$

$\tilde{\phi}(\eta,x)=\int{\frac{{{d^{3}}k}}{{{{(2\pi)}^{\frac{3}{2}}}}}\frac{1}{{\sqrt{2}}}\left[{{a_{k}}v_{k}^{*}(\eta){e^{ik.x}}+a_{k}^{*}{v_{k}}(\eta){e^{-ik.x}}}\right]}.$
(14)

Note that, the reality condition for $\phi(\eta,x)$ entails
$\tilde{\phi}_{k}=\tilde{\phi}^{*}_{-k}$. To obtain a quantum picture of such
theory, one has to consider $a_{k}$ and its conjugate $a_{k}^{\dagger}$ as a
operator in given Hilbert space so at first we should write down $a_{k}$ as a
combination of $\tilde{\phi}_{k}$ and its conjugate momentum. Using the
equation (13) and (12) we find $a_{k}$

$a_{k}=\frac{1}{i\sqrt{2}}(v_{k}^{{}^{\prime}}(\eta)\tilde{\phi}_{k}(\eta)-\tilde{\phi}_{k}^{{}^{\prime}}(\eta)v_{k}(\eta)),$
(15)

and its conjugate

$a_{k}^{*}=-\frac{1}{i\sqrt{2}}({v_{k}^{{}^{\prime}}}^{*}(\eta)\tilde{\phi}_{-k}(\eta)-\tilde{\phi}_{-k}^{{}^{\prime}}(\eta)v_{k}^{*}(\eta)).$
(16)

Since the conjugate momentum of $\tilde{\phi}$ is

$\tilde{\Pi}(\eta,x)=\tilde{\phi}^{{}^{\prime}}(\eta,x),$ (17)

we have

$a_{k}=\frac{1}{i\sqrt{2}}(v_{k}^{{}^{\prime}}(\eta)\tilde{\phi}_{k}(\eta)-\tilde{\Pi}_{k}(\eta)v_{k}(\eta))$
(18)

and

$a_{k}^{*}=-\frac{1}{i\sqrt{2}}({v_{k}^{{}^{\prime}}}^{*}(\eta)\tilde{\phi}_{-k}(\eta)-\tilde{\Pi}_{-k}(\eta)v_{k}^{*}(\eta)).$
(19)

Now, to reach a quantum picture, it is enough to define the usual following
commutator

$\left[\tilde{\phi}(\eta,x),\tilde{\Pi}(\eta,y)\right]=i\delta(x-y).$ (20)

This gives a representation for momentum as

$\tilde{\Pi}_{k}(\eta)=-i(2\pi)^{3}\frac{\delta}{\delta\tilde{\phi}_{-k}(\eta)}.$
(21)

The above equation leads to the usual commutation relation for creation and
annihilation operators

$\left[a_{k},a^{\dagger}_{p}\right]=(2\pi)^{3}\delta(k-p).$ (22)

Once the operators $a_{k}$ and $a_{k}^{\dagger}$ are determined, the vacuum
state $|0\rangle$ is defined as the eigenstate of all annihilation operators
$a_{k}$ with eigenvalue $0$, i.e. $a_{k}|0\rangle=0$ for all k. An excited
state $|n_{k_{1}},m_{k_{2}},\ldots\rangle$ with the occupation numbers
$m,n,...$ in the modes $\tilde{\phi}_{k_{1}},\tilde{\phi}_{k_{2}},\ldots$ is
then given by

$|{n_{{k_{1}}}},{m_{{k_{2}}}},\cdots\rangle=\frac{1}{{\sqrt{n!m!\ldots}}}\left[{{{({a_{{k_{1}}}}^{\dagger})}^{n}}{{({a_{{k_{2}}}}^{\dagger})}^{m}}\cdots}\right]|{0_{{k_{1}}}},{0_{{k_{2}}}},\cdots\rangle$
(23)

It is interesting to know what is Schrödinger picture of vacuum state. After
determining this state one could use (19) and (22) to derive exited states in
this picture. To derive the explicit form of vacuum, we use this fact that

$\langle\tilde{\phi_{k}}|a_{k}|0\rangle=0,$ (24)

where $|\tilde{\phi_{k}}\rangle$ is eigenstate of $\tilde{\phi_{k}}$. Now
using (18), we will get a simple functional differential equation with the
following solution

$\Psi[\tilde{\phi}]=\langle\tilde{\phi_{k}}|0\rangle=N(\eta)\exp\left[\frac{i}{2}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{v^{{}^{\prime}}_{k}}{v_{k}}(\eta)\
\tilde{\phi}_{k}\tilde{\phi}_{-k}\right]$ (25)

where $N(\eta)$ is the normalization factor and can be calculated by using of

$\int{D\tilde{\phi}}\ \Psi[\tilde{\phi}]{\Psi^{*}}[\tilde{\phi}]=1$ (26)

for the measure
$D\tilde{\phi}=d\tilde{\phi}_{k_{1}}d\tilde{\phi}_{{-k}_{1}}d\tilde{\phi}_{k_{2}}d\tilde{\phi}_{{-k}_{2}}\ldots$.
It is easy to show that apart of a constant for a given $k$-mode, we have
$N_{k}\sim\frac{1}{|v_{k}(\eta)|}$. In the mathematical point of view, the
wave function $\Psi[\tilde{\phi}]$ is defined on functional space of an
infinite dimensional manifold. People rarely use this picture of quantum field
theory because occupation number of each state is not manifest in this
picture. However, some kind of calculations is easier than Heisenberg picture.
To end up this section for future advantages, we derive wave function of the
main field i.e. $\phi$. One can check out the annihilation and creation
operators for this field would be

$a_{k}=\frac{1}{i\sqrt{2}}\left[(av_{k}^{{}^{\prime}}(\eta)-a^{{}^{\prime}}v_{k}(\eta))\phi_{k}(\eta)+i(2\pi)^{3}\frac{v_{k}}{a}(\eta)\frac{\delta}{\delta\phi_{-k}}\right]$
(27)

and

${a}^{\dagger}_{k}=-\frac{1}{i\sqrt{2}}\left[(a{v_{k}^{*}}^{{}^{\prime}}(\eta)-a^{{}^{\prime}}v_{k}^{*}(\eta))\phi_{-k}(\eta)+i(2\pi)^{3}\frac{{v_{k}}^{*}}{a}(\eta)\frac{\delta}{\delta\phi_{k}}\right].$
(28)

Now, we could find the wave function as

$\Psi[\phi]=\langle\phi_{k}|0\rangle=N(\eta)\exp\left[\frac{i}{2}\int\frac{d^{3}k}{(2\pi)^{3}}(\frac{v^{{}^{\prime}}_{k}}{v_{k}}(\eta)-\frac{a^{{}^{\prime}}}{a}(\eta))\
\phi_{k}\phi_{-k}\right]$ (29)

and we see that the difference is just $\frac{a^{{}^{\prime}}}{a}$ in the
exponent which called Hubble factor. Note that one could use the Schrödinger
equation in functional space to reach this wave function. We shall do it for
two entangled scalar field later on.

##  III Bogoliubov transformation

Consider two sets of isotropic mode functions $u_{k}(\eta)$ and $v_{k}(\eta)$
are chosen. Since $v_{k}$ and $u_{k}^{*}$ are a basis, the function $u_{k}$
must be a linear combination of $v_{k}$ and $u_{k}^{*}$,

$u_{k}(\eta)={\alpha}_{k}v_{k}(\eta)-{\beta}^{*}_{k}v_{k}^{*}(\eta)$ (30)

with time-independent complex coefficients $\alpha_{k}$ and $\beta_{k}$. If
both sets $v_{k}$ and $u_{k}$ are normalized by (12), it follows that the
coefficients $\alpha_{k}$ and $\beta_{k}$ satisfy

$|{\alpha}_{k}|^{2}-|{\beta}_{k}|^{2}=1.$ (31)

Using the mode functions $u_{k}$ instead of $v_{k}$, one obtains an
alternative mode expansion which defines another set $\tilde{a}_{k}$ and
$\tilde{a}_{k}^{\dagger}$ of creation and annihilation operators

$\tilde{\phi}(\eta,x)=\int{\frac{{{d^{3}}k}}{{{{(2\pi)}^{\frac{3}{2}}}}}\frac{1}{{\sqrt{2}}}\left[{{\tilde{a}_{k}}u_{k}^{*}(\eta){e^{ik.x}}+\tilde{a}_{k}^{*}{u_{k}}(\eta){e^{-ik.x}}}\right]},$
(32)

in which we have defined

$\tilde{a}_{k}={\alpha}_{k}a_{k}+{\beta}^{*}_{k}a_{-k}^{*}.$ (33)

Now one can refer a new vacuum state $|\tilde{0}\rangle$ respect to
$\tilde{a}_{k}$ which annihilated by this operator and then by expanding the
new vacuum in term of previous basis (23) to reach

$|\tilde{0}\rangle=\Pi_{k}\frac{1}{\sqrt{|\alpha_{k}|}}\exp\left(-\frac{\lambda_{k}}{2}a_{k}^{\dagger}a^{\dagger}_{-k}\right)\
|0\rangle$ (34)

where

$\lambda_{k}=\frac{\beta^{*}_{k}}{\alpha_{k}}.$ (35)

This is an infinite series which can be express explicitly for each mode as

$|{0_{k}},{0_{-k}}\rangle=\frac{1}{{|{\alpha_{k}}|}}\sum\limits_{n=0}^{\infty}{{{\left({-\frac{{\beta_{k}^{*}}}{{{\alpha_{k}}}}}\right)}^{n}}}|{n_{k}},{n_{-k}}\rangle.$
(36)

The $\alpha_{k}$ and $\beta_{k}$ constants are called Bogoliubov constants. It
is interesting to know the closed exact form of the Bogoliubov transformation
in the Schrödinger picture. To do it, we need to have the new vacuum in the
Schrödinger picture. It is obvious that

$\langle\tilde{\phi_{k}}|\tilde{0}\rangle=\tilde{N}\exp\left[\frac{i}{2}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{u^{{}^{\prime}}_{k}}{u_{k}}(\eta)\
\tilde{\phi}_{k}\tilde{\phi}_{-k}\right]$ (37)

because the difference between two vacuums is in mode functions and it is
enough to replace $v_{k}$ with $u_{k}$. Using this fact that
$|\lambda_{k}\frac{v^{*}_{k}}{v_{k}}|<1$ and expanding
$\frac{u_{k}^{{}^{\prime}}}{u_{k}}$ in term of
$\frac{v_{k}^{{}^{\prime}}}{v_{k}}$, one obtains

$\langle\tilde{\phi_{k}}|\tilde{0}\rangle=N\exp\left[\frac{i}{2}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{v^{{}^{\prime}}_{k}}{v_{k}}(\eta)\
\tilde{\phi}_{k}\tilde{\phi}_{-k}\right]\left(\prod_{k}\frac{1}{|\alpha_{k}|}\frac{1}{|1-\lambda_{k}\frac{v_{k}^{*}}{v_{k}}|}\right)\
exp\left(-\sum_{n=0}^{\infty}\int\frac{d^{3}k}{(2\pi)^{3}}\lambda_{k}^{n+1}\frac{1}{v_{k}^{2}}(\frac{v_{k}^{*}}{v_{k}})^{n}\tilde{\phi}_{k}\tilde{\phi}_{-k}\right)$
(38)

This is a magic relation because not only expresses the new vacuum in a closed
form of previous basis but also could be used to derive other exited states in
the Schrödinger picture. One can expand two last terms in the equation (38)
and compare $\lambda_{k}^{n}$ terms with with their counterpart in (36) to
find all exited states. Before ending this section, it is worth to mention a
very important point that all information about kind of a state is hidden in
mode functions. For example, by looking at (37) one cannot observe any
entanglement in this state. Therefore, to see this entanglement between
excited states, we have to study mode functions of one observer and compare it
with another observer.

##  IV ABH State

In 2014, Albrecht, Bolis and Holman (ABH) Holman-entanglement  introduced an
interesting state which describes entanglement between two non-interacting
scalar fields in FLRW background. They applied it in Schrödinger picture and
use it as an ansats to solve Schrödinger equation. In this section, we have a
short review of their derivation for more details see (Holman-entanglement, ).
Suppose two fields $\phi$ and $\chi$ coupled to gravity with no any other
interactions,

$\displaystyle S=\frac{1}{2}\int d^{4}x\
a^{4}(\eta)\left[\frac{1}{a^{2}(\eta)}\left(\Phi^{\prime}(\eta,\vec{x})^{2}-\left(\nabla\Phi(\eta,\vec{x})\right)^{2}\right)-m_{\Phi}^{2}\
\Phi(\eta,\vec{x})^{2}\right.+$
$\displaystyle\left.\frac{1}{a^{2}(\eta)}\left(\chi^{\prime}(\eta,\vec{x})^{2}-\left(\nabla\chi(\eta,\vec{x})\right)^{2}\right)-m_{\chi}^{2}\
\chi(\eta,\vec{x})^{2}\right],$ (39)

To be able to make a functional Schrödinger equation for a such action one
needs the Hamiltonian in terms of conjugate momenta;

$\displaystyle H=\int d^{3}x\
\left[\frac{\pi_{\Phi}^{2}}{2a^{2}(\eta)}+\frac{1}{2}a^{2}(\eta)\left(\left(\nabla\Phi(\eta,\vec{x})\right)^{2}+a^{2}(\eta)m_{\Phi}^{2}\
\Phi(\eta,\vec{x})^{2}\right)+\right.$
$\displaystyle\left.\frac{\pi_{\chi}^{2}}{2a^{2}(\eta)}+\frac{1}{2}a^{2}(\eta)\left(\left(\nabla\chi(\eta,\vec{x})\right)^{2}+a^{2}(\eta)m_{\Phi}^{2}\
\chi(\eta,\vec{x})^{2}\right)\right],$ (40)

where $\pi_{\phi}$ and $\pi_{\chi}$ are the canonically conjugate momenta for
$\phi$ and $\chi$. Like before section, it would be more appropriate to go to
Fourier space as following for $\phi$

$\Phi(\eta,\vec{x})=\sum_{\vec{k}}\frac{\phi_{\vec{k}}}{\sqrt{V}}e^{-i\vec{k}\cdot\vec{x}},\quad\pi_{\Phi}(\eta,\vec{x})=\sum_{\vec{k}}\frac{\pi_{\Phi,{\vec{k}}}}{\sqrt{V}}e^{-i\vec{k}\cdot\vec{x}},$
(41)

and with a similar transformation for $\chi$. It is easy to see with this
decomposition, the Hamiltonian has the following form

$\displaystyle H=H_{\Phi}+H_{\chi}$ $\displaystyle
H_{\Phi}=\sum_{\vec{k}}H_{\Phi,\vec{k}},\quad
H_{\Phi,\vec{k}}=\frac{\pi_{\Phi,\vec{k}}\pi_{\Phi,-\vec{k}}}{2a^{2}(\eta)}+\frac{1}{2}a^{2}(\eta)\left(k^{2}+m_{\Phi}^{2}a^{2}(\eta)\right)\phi_{\vec{k}}\phi_{-\vec{k}}$
$\displaystyle H_{\chi}=\sum_{\vec{k}}H_{\chi,\vec{k}},\quad
H_{\chi,\vec{k}}=\frac{\pi_{\chi,\vec{k}}\pi_{\chi,-\vec{k}}}{2a^{2}(\eta)}+\frac{1}{2}a^{2}(\eta)\left(k^{2}+m_{\chi}^{2}a^{2}(\eta)\right)\chi_{\vec{k}}\chi_{-\vec{k}}.$
(42)

Now, in this level one can use the separation of variables method to solve
Schrödinger equation for each $k$-mode

$i\partial_{\eta}\psi_{\vec{k}}\left[\phi_{\vec{k}},\chi_{\vec{k}};\eta\right]=\left(H_{\Phi,\vec{k}}+H_{\chi,\vec{k}}\right)\psi_{\vec{k}}\left[\phi_{\vec{k}},\chi_{\vec{k}};\eta\right],$
(43)

but such solution is only able to give an entangled state between exited
states of one kind of fields not entanglement between different fields. To
make a entangled state between different fields, we use the following ansats

$\psi_{\vec{k}}\left[\phi_{\vec{k}},\chi_{\vec{k}};\eta\right]=N_{k}(\eta)\exp\left[-\frac{1}{2}\left(A_{k}(\eta)\phi_{\vec{k}}\phi_{-\vec{k}}+B_{k}(\eta)\chi_{\vec{k}}\chi_{-\vec{k}}+C_{k}(\eta)\left(\phi_{\vec{k}}\chi_{-\vec{k}}+\chi_{\vec{k}}\phi_{-\vec{k}}\right)\right)\right].$
(44)

where $C_{k}$ plays the role of entanglement between two fields. Inserting
(43) into (IV) and then matching the powers of the field modes gives us the
following equations

$\displaystyle
i\frac{N_{k}^{\prime}}{N_{k}}=\frac{\left(A_{k}+B_{k}\right)}{2a^{2}(\eta)}$
$\displaystyle
iA_{k}^{\prime}=\frac{A_{k}^{2}+C_{k}^{2}}{a^{2}(\eta)}-\Omega_{\Phi,k}^{2}a^{2}(\eta),\quad\Omega_{\Phi,k}^{2}\equiv
k^{2}+m_{\Phi}^{2}a^{2}(\eta)$ $\displaystyle
iB_{k}^{\prime}=\frac{B_{k}^{2}+C_{k}^{2}}{a^{2}(\eta)}-\Omega_{\chi,k}^{2}a^{2}(\eta),\quad\Omega_{\chi,k}^{2}\equiv
k^{2}+m_{\chi}^{2}a^{2}(\eta)$ $\displaystyle
i\frac{C_{k}^{\prime}}{C_{k}}=\frac{\left(A_{k}+B_{k}\right)}{a^{2}(\eta)}.$
(45)

One could reach a more suitable form of these equations using the following
definitions

$iA_{k}(\eta)\equiv
a^{2}(\eta)\left(\frac{f_{k}^{\prime}(\eta)}{f_{k}(\eta)}-\frac{a^{\prime}(\eta)}{a(\eta)}\right),\quad
iB_{k}(\eta)\equiv
a^{2}(\eta)\left(\frac{g_{k}^{\prime}(\eta)}{g_{k}(\eta)}-\frac{a^{\prime}(\eta)}{a(\eta)}\right).$
(46)

Then the resulting differential equation under these definitions would be

$\displaystyle
f_{k}^{\prime\prime}+\left(\Omega_{\Phi,k}^{2}-\frac{a^{\prime\prime}(\eta)}{a(\eta)}\right)f_{k}=\frac{C_{k}(\eta)^{2}}{a^{4}(\eta)}f_{k}$
$\displaystyle
g_{k}^{\prime\prime}+\left(\Omega_{\chi,k}^{2}-\frac{a^{\prime\prime}(\eta)}{a(\eta)}\right)g_{k}=\frac{C_{k}(\eta)^{2}}{a^{4}(\eta)}g_{k}$
(47)

and furthermore $C_{k}(\eta)$ would be determined by

$\frac{C_{k}(\eta)}{a^{2}(\eta)}=\frac{c_{k}}{f_{k}(\eta)g_{k}(\eta)},$ (48)

which we call $c_{k}$ the entanglement parameter and if this parameter
vanishes then we regain (25). Note that this parameter is constant of the
integral and only depends on $k$. Here it is enough to solve coupled
differential equations in (46) with given initial conditions. ABH showed that
these equations have a perturbative solution of even orders of entanglement
parameter $c_{k}^{2n}$ for $f_{k}$ and odd orders $c_{k}^{2n+1}$ for $C_{k}$.
In fact if $v_{k}$ and $u_{k}$ be solutions of eq. (IV) for $f_{k}$ and
$g_{k}$ respectively when $c_{k}=0$, then they found

$f_{k}(\eta)=v_{k}(\eta)\left(1+c_{k}^{2}\mathcal{F}_{k}(\eta)+\cdots\right),$
(49)
$g_{k}(\eta)=u_{k}(\eta)\left(1+c_{k}^{2}\mathcal{G}_{k}(\eta)+\cdots\right)$
(50)

and

$C_{k}=\frac{c_{k}}{u_{k}(\eta)v_{k}(\eta)}-\frac{c_{k}^{3}}{u_{k}(\eta)v_{k}(\eta)}\left(\mathcal{F}_{k}(\eta)+\mathcal{G}_{k}(\eta)\right)+\cdots,$
(51)

where

$\mathcal{F}_{k}(\eta)=\left(\int_{\eta_{0}}^{\eta}d\eta_{1}\frac{1}{v_{k}^{2}(\eta_{1})}\int_{\eta_{0}}^{\eta_{1}}d\eta_{2}\frac{1}{u_{k}^{2}(\eta_{2})}\right),\
\ \ \ \ \ \ \ \ \
\mathcal{G}_{k}(\eta)=\left(\int_{\eta_{0}}^{\eta}d\eta_{1}\frac{1}{u_{k}^{2}(\eta_{1})}\int_{\eta_{0}}^{\eta_{1}}d\eta_{2}\frac{1}{v_{k}^{2}(\eta_{2})}\right).$
(52)

There some questions might arise about the entity of this state. We know that
this state can not be a vacuum state and should be written in terms of a
superposition of excited states of $\phi$ and $\chi$. As a result, to gain
such view one has to go to Heisenberg language of the quantum field. In next
section, we shall try to obtain this goal and find a solution for (46). Note
that the eq. (44) is not the general entanglement solution for the system
under study because it can not include all entangled states such as a state of
form $|0_{\phi},1_{\chi}\rangle+|1_{\phi},0_{\chi}\rangle$.

##  V Entanglement Transformation

In this section, we introduce a transformation like the Bogoliubov
transformation which help us to reconstruct the previous entangled state in
the Heisenberg picture. For simplicity, consider annihilation operator (27)
for $\phi$ in the new form

$a_{k}=\mu_{k}\phi_{k}(\eta)+i\sigma_{k}\frac{\delta}{\delta\phi_{-k}}$ (53)

where

$\mu_{k}\equiv\frac{1}{i\sqrt{2}}\left[(av_{k}^{{}^{\prime}}(\eta)-a^{{}^{\prime}}v_{k}(\eta))\right]$
(54)

and

$\sigma_{k}\equiv\frac{1}{i\sqrt{2}}\frac{v_{k}}{a}(\eta).$ (55)

The similar annihilation operator for $\chi$

$b_{k}=\tilde{\mu}_{k}\chi_{k}(\eta)+i\tilde{\sigma}_{k}\frac{\delta}{\delta\chi_{-k}},$
(56)

where $\tilde{\mu}$ and $\tilde{\sigma}$ are similar to ${\mu}$ and ${\sigma}$
just with this different that $v_{k}\rightarrow u_{k}$ and $u_{k}\rightarrow
v_{k}$. Note that we have absorbed $(2\pi)^{3}$ factor just by redefinition of
commutators (20) and (22). Now, we transform ${a}_{k}$ and ${b}_{k}$ to the
new annihilation and creation operators as following

$\tilde{a}_{k}={\alpha}_{k}(\eta)a_{k}+{\beta}^{*}(\eta)_{k}b_{-k}^{\dagger}$
(57)

$\tilde{b}_{k}={\alpha}_{k}(\eta)b_{k}+{\beta}^{*}_{k}(\eta)a_{-k}^{\dagger}.$
(58)

Here $\alpha_{k}$ and $\beta_{k}$ are like what we saw in Bogoliubov
transformation with this difference that a Bogoliubov transformation is time-
independent and so doesn’t change the space of mode functions. This
transformation just transforms a set of two independent mode functions into
another set, but above entangled transformation mixes modes of different
fields and also can have time-dependency. Unfortunately, Kanno (Kanno:2015ewa,
) supposed that this kind of transformation as a time-independent
transformation without any reason. Here we consider the general case and
suppose this transformation can be time-dependent and finally describe why
this can not be independent of time. If we request that these new operators
satisfy their own commutation relations like (22) then we get

$|{\alpha}_{k}(\eta)|^{2}-|{\beta}_{k}(\eta)|^{2}=1.$ (59)

Consequently, one can find out that the new vacuum state $|\tilde{0}\rangle$
(which annihilated by $\tilde{a}_{k}$ and $\tilde{b}_{k}$) is

$|\tilde{0}\rangle=N(\eta)\exp\left(-\sum_{k}\frac{\lambda_{k}(\eta)}{2}a_{k}^{\dagger}a^{\dagger}_{-k}\right)\
|0\rangle,$ (60)

where $\lambda_{k}=\frac{\beta^{*}_{k}}{\alpha_{k}}$. Now, let us find these
states in Schrödinger picture. To this end, we consider the following
relations

$\langle\phi_{k},\chi_{k}|\tilde{a}_{k}|\tilde{0}\rangle=\langle\phi_{k},\chi_{k}|\tilde{b}_{k}|\tilde{0}\rangle=0$
(61)

and use (53) and (56) to obtain

${\alpha}_{k}\left[\mu_{k}\phi_{k}(\eta)\Psi[\phi,\chi]+i\sigma_{k}\frac{\delta\Psi[\phi,\chi]}{\delta\phi_{-k}}\right]+{\beta}^{*}_{k}\left[\tilde{\mu}^{*}_{k}\chi_{k}(\eta)\Psi[\phi,\chi]+i\tilde{\sigma}^{*}_{k}\frac{\delta\Psi[\phi,\chi]}{\delta\chi_{-k}}\right]=0$
(62)

and

${\alpha}_{k}\left[\tilde{\mu}_{k}\chi_{k}(\eta)\Psi[\phi,\chi]+i\tilde{\sigma}_{k}\frac{\delta\Psi[\phi,\chi]}{\delta\chi_{-k}}\right]+{\beta}^{*}_{k}\left[{\mu}^{*}_{k}\phi_{k}(\eta)\Psi[\phi,\chi]+i{\sigma}^{*}_{k}\frac{\delta\Psi[\phi,\chi]}{\delta\phi_{-k}}\right]=0.$
(63)

These are just two simple functional differential equations which can be
solved directly. The interesting point is that the solution has the form of
(44) with the following kernels

$A_{k}=-ia^{2}\left(\frac{v_{k}^{{}^{\prime}}}{v_{k}}-\frac{a^{{}^{\prime}}}{a}\right)+\lambda_{k}C_{k}\frac{u^{*}_{k}}{v_{k}}$
(64)

and

$A_{k}=-ia^{2}\left((\frac{v_{k}^{{}^{\prime}}}{v_{k}})^{*}-\frac{a^{{}^{\prime}}}{a}\right)+{\lambda}_{k}^{-1}C_{k}\frac{u_{k}}{v_{k}^{*}}$
(65)

and the same for $B_{k}$ would be obtain only by $v_{k}\rightarrow u_{k}$ and
$u_{k}\rightarrow v_{k}$. These are two algebraic equations which have the
following solutions

$C_{k}(\eta)=\frac{2a^{2}(\eta)}{-\lambda_{k}(\eta)u_{k}^{*}(\eta)v_{k}^{*}(\eta)+{\lambda}_{k}^{-1}(\eta)u_{k}(\eta)v_{k}(\eta)}$
(66)

and

$A_{k}(\eta)=-ia^{2}\left(\frac{v_{k}^{{}^{\prime}}}{v_{k}}(\eta)-\frac{a^{{}^{\prime}}}{a}(\eta)\right)+\lambda_{k}(\eta)\frac{2a^{2}(\eta)}{-\lambda_{k}(\eta)u_{k}^{*}(\eta)v_{k}^{*}(\eta)+{\lambda}_{k}^{-1}(\eta)u_{k}(\eta)v_{k}(\eta)}\frac{u^{*}_{k}}{v_{k}}(\eta).$
(67)

If the reader looks at the Kanno’s paper, he or she will find out that the
Kanno’s results for $A_{k},B_{k}$ and $C_{k}$ is different from what we have
derived here. This happens because Kanno did’t do the field redefinition and
also wrote a wrong condition for the wronskian of mode functions. Now, we
request that the new vacuum to be a solution of Schrödinger equation and with
this consideration that we should equate these kernels with those we have
found in (IV) and (46). This helps us to find expressions for $f_{k}$ and
$g_{k}$ which defined in (IV) as following

${f_{k}}={\rho_{k}}{v_{k}}(\eta)\left[{\exp\left({2i\int_{{\eta_{0}}}^{\eta}{\frac{{{u_{k}}^{*}}{\lambda_{k}}}{{{-\lambda_{k}}{u^{*}}_{k}|{v_{k}}{|^{2}}+\lambda_{k}^{-1}{u_{k}}v_{k}^{2}}}}}\right)}\right]$
(68)

and

${g_{k}}={\zeta_{k}}{u_{k}}(\eta)\left[{\exp\left({2i\int_{{\eta_{0}}}^{\eta}{\frac{{{v_{k}}^{*}}{\lambda_{k}}}{{{-\lambda_{k}}{v^{*}}_{k}|{u_{k}}{|^{2}}+\lambda_{k}^{-1}{v_{k}}u_{k}^{2}}}}}\right)}\right]$
(69)

where ${\rho_{k}}$ and ${\zeta_{k}}$ are constants. We can see the way which
we passed to finding an obvious description of the entangled state (by using
of entanglement transformation), leads us to the above relations for non-
linear equations in (IV). In the above relations to do the integrals in the
exponents we, have to know the time-dependent form of $\lambda_{k}$. But what
does happen if one supposes that $\lambda_{k}$ is time-independent? Let us to
look at it more precisely. Comparing eq. (66) with eq. (51) we find out that
$\lambda_{k}$ should be a function of entanglement parameter $c_{k}$. In fact
$c_{k}=0$ implies that $\lambda_{k}=0$. When $c_{k}\ll 1$ one can show that
$\lambda_{k}=c_{k}/2$. As a result, $\lambda_{k}$ is approximately constant
and eqs. (68, 69) are consistent with (49, 50). However when we go ahead and
consider the next powers of $c_{k}$ we find out that $\lambda_{k}$ can not be
time-independent. If we expand $\lambda_{k}$ in terms of the entanglement
parameter, one can show that

$\lambda_{k}(\eta)=c_{k}/2-c^{3}_{k}N_{k}(\eta)+\cdots,$ (70)

where

$N_{k}(\eta)=\frac{\mathcal{F}_{k}+\mathcal{G}_{k}}{2a^{2}}(\eta)+\frac{1}{8}\frac{u_{k}^{*}v_{k}^{*}}{u_{k}v_{k}}(\eta)$
(71)

This implies that the entangled state defined by entanglement transformation
will coincide with the ABH state if $\lambda_{k}$ is time-dependent. It is
interesting to calculate the power spectrum in this picture. Kanno
Kanno:2015ewa  has shown that the power spectrum of $\phi$ in Schrödinger
picture is the same as the Heisenberg picture power spectrum only up to the
second order of $\beta_{k}$. Kanno expands the Albertch et.al’s result (for
power spectrum in the Schrödinger picture) with smallness assumption of
$\beta_{k}$. She didn’t get to her exact form for power spectrum which she
finds in the Heisenberg picture Kanno:2015ewa . In contrast with Kanno’s
analysis, we find out in our analysis the two power spectra do coincide
exactly without any approximation. In fact, with make using of (66) and (67)
and after some tedious calculations one can show that

$\langle\tilde{0}|\phi_{-k}\phi_{k}|\tilde{0}\rangle=\frac{1}{2a^{2}}|v_{k}(\eta)|^{2}\left(\frac{1+|\lambda_{k}|^{2}}{1-|\lambda_{k}|^{2}}\right)=\frac{1}{2}\frac{B_{kR}}{A_{kR}B_{kR}-C^{2}_{kR}},$
(72)

in which the sub-index $R$ stands for the real part of each variable. One
could see in this analysis, we are not only able to solve mode functions of
entangled space in term of entanglement transformation parameter but also re-
derive the power spectrum which people have found in Schrödinger analysis.
There just remains a point which we should clarify and that is by using
Heisenberg picture, one can access other entangled states in a simple way. It
is easy to see from (60) that the ABH state creates an entanglement between
states of two fields with the same number of particles i.e.
$|0_{\phi},0_{\chi}\rangle$ , $|1_{\phi},1_{\chi}\rangle$, …. To make an
entangled state in which states with different fields with a different number
of particles are entangled with each other, it is better (and usual) to go the
Heisenberg picture and by using of creation operators create an excited state
of arbitrary number of particles and then mix the resultant state with
momentum dependent coefficients. Consequently, without concern, the final
entangled state is a solution for Schrödinger equation.

##  VI Conclusion

An important aspect of the quantum physics is the entanglement which can have
the observational effect on cosmic structure formation which has originated
from quantum fluctuations in the early universe En-Observational . To study
the entanglement in the early universe, first Albrecht e.t Holman-entanglement
have shown that the initial entangled state can have some features on the
observable power spectrum in the early universe and then Kanno attempted to
construct these initial entangled state by making use of a transformation
which was similar to the Bogoliubov transformation Kanno:2015ewa . Besides
showing that the Kanno’s calculation and analysis were not correct in some
parts, in this article we have explored an entangled transformation (which is
introduced Heisenberg picture) role on the Schrödinger field theory of the
FLRW model. In accordance with this article analysis, we conclude with some
remarks:

  * •

The solution of the entangled coefficient $A,B,C$ in the Schrödinger picture
can be written in terms of the time-dependent entangled transformation
parameter in the Heisenberg formalism. The entangled coefficient in (57) is
not the Bogoliubov transformation since it mixes the mode functions of
different fields and can not be interpreted as a different observer physics
like in the Bogoliubov transformation.

  * •

We should mention that an entangled state (see Eq.(44)) defined by an
Entanglement transformation, is not in general a solution of Schrödinger
equation and one has to use Schrödinger equation to find
parameter$\lambda_{k}$ of the entanglement transformation while a Bogoliubov
transformation keeps us in space of solutions of Schrödinger equation.

  * •

We calculated the relation between two vacuum states which their mode
functions are transformed by the Bogoliubov transformation in the Schrödinger
picture.

  * •

We have derived the entangled parameter $C_{k}$ in the Schrödinger picture in
term of the entangled transformation parameter $\lambda_{k}$ in the Heisenberg
picture. This formalism verifies that the entanglement transformation depends
on the conformal time.

  * •

The power spectrum of the inflaton field can be presented both in terms of the
entangled parameter of Schrödinger picture and the entangled transformation
coefficient in the Heisenberg one which has the same value.

  * •

It was discussed and shown that the entanglement has a more deep meaning which
can not be presented in the form of the (44) and in term of the entanglement
transformation (57). We have presented an example that cannot describe the
entanglement (44) and (57). Moreover, the entanglement was discussed in the
equation (44) and (57) describe the entanglement between two fields and do not
quantify the entanglement between the excited state of each field.

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