# Measuring Time-Dependent Induced Quantum Coherences via Two-Dimensional
Coherence Spectroscopy

William Barford<EMAIL_ADDRESS>Department of Chemistry,
Physical and Theoretical Chemistry Laboratory, University of Oxford, Oxford,
OX1 3QZ, United Kingdom  Allison Nicole Arber  Department of Chemistry,
Physical and Theoretical Chemistry Laboratory, University of Oxford, Oxford,
OX1 3QZ, United Kingdom  Balliol College, University of Oxford, Oxford, OX1
3BJ, United Kingdom  Fynn McLennan  Department of Chemistry, Physical and
Theoretical Chemistry Laboratory, University of Oxford, Oxford, OX1 3QZ,
United Kingdom  Magdalen College, University of Oxford, Oxford, OX1 4AU,
United Kingdom  Max Marcus  Department of Chemistry, Physical and Theoretical
Chemistry Laboratory, University of Oxford, Oxford, OX1 3QZ, United Kingdom

###### Abstract

We propose a two-dimensional spectroscopic protocol for measuring the time-
dependent coherences between the stationary states of a system induced by a
time-dependent system-bath interaction. We also investigate the role of
temporally-correlated noise on coherence dephasing. This protocol enables
dynamical information about the system and its coupling to the environment to
be determined. Our results are based on the quantum-trajectory method, and are
obtained from both approximate, analytical and exact, numerical solutions of
the time-dependent Schrödinger equation. As an example, we show how this
protocol can be used to investigate exciton dynamics in conjugated polymers
induced by the coupling of their torsional modes with the environment.

##  I Introduction

The question of whether electronic and vibrational coherences exist in
macromolecular systems, e.g., $\pi$-conjugated polymers and light harvesting
complexes, has been an outstanding one for a number of years[1, 2, 3, 4, 5, 6,
7, 8]. This question is motivated by the attempt to understand how coherences
— if they exist — can survive under ambient conditions. It is also motivated
by the expectation that coherences might enhance the efficiency of energy and
charge transport (but see Ref. [9] for a counter argument), thus improving the
efficiency and providing design principles for synthetic photovoltaic devices.

Recently, borrowing concepts from quantum information theory, quantum process
tomography has been proposed as a method to ‘witness’ coherences[10, 11, 12].
In this paper we propose a conceptually straightforward protocol based on the
well-established technique of two-dimensional coherence spectroscopy to
determine dynamical coherences. For reviews of two-dimensional coherence
spectroscopy, see Refs[13, 14, 3, 15, 6].

Before discussing how coherences are established and measured, let us first
define what we mean by them. In this paper we define coherences as the off-
diagonal matrix elements of the system’s density matrix when expressed in the
energy eigenbasis of the system. Such coherences in a system can be induced
via coupling to the environment in a variety of ways. For example, a coherent
light source might excite a number of energy eigenstates, thus creating a non-
stationary state. In this case, assuming no dephasing or dissipation, the
populations are constant, while coherences have constant magnitudes but
oscillate with the transition angular frequencies. A more interesting scenario
is when the excited system interacts with an environment that couples these
eigenstates and hence causes interstate transitions. This will cause
population transfers and the magnitude of the coherences will change in time.
Moreover, if the energy eigenstates are spatially separated, it will cause
energy transfer. It is this latter scenario that is the principal
investigation of this paper.

Here, we set up a simple two-level system subject to a time-dependent periodic
interaction. We show how the induced coherences can be observed by using
2D-spectroscopy with particular choices of the electric field polarization.
This results in a characteristic fingerprint of dynamical coherences, from
which dynamical information about the system and environment can be
determined. We then demonstrate how a noisy (dephasing) environment destroys
the coherences. Our results are obtained from both approximate, analytical and
exact, numerical solutions of the time-dependent Schrödinger equation.

Having established this model system, we next discuss how it might be realized
in practice. In particular, we envisage a conjugated polymer subject to
torsional fluctuations via the Brownian impulses of its environment. We derive
realistic parameters for both intra and inter chromophore energy transfer, and
investigate whether the coherence signals described in Section II.4 might be
observed.

##  II Two-Level System

###  II.1 Model

To introduce the key ideas behind the spectroscopic technique proposed here,
we begin by considering a closed two-level system described by the
Hamiltonian,

$\hat{H}_{\text{S}}=\hbar\omega_{a}\ket{a}\bra{a}+\hbar\omega_{b}\ket{b}\bra{b}.$
(1)

Note that the true ground state, $\ket{\text{GS}}$, is not part of this model,
as the kets $\ket{a}$ and $\ket{b}$ span the excited state Hilbert space.
Consequently, the energies $\hbar\omega_{j}$ are excitation energies, rather
than absolute energies. Defining $\hat{a}_{j}^{\dagger}$ as the creation
operator for ket $\ket{j}$, then
$\ket{j}=\hat{a}_{j}^{\dagger}\ket{\text{GS}}$. For convenience we also define
the Bohr angular frequency $\omega_{0}=(\omega_{b}-\omega_{a})>0$.

The system is coupled to a bath via the interaction

$\hat{H}_{SB}=V(t)\left(\ket{a}\bra{b}+\ket{b}\bra{a}\right),$ (2)

where we take the system-bath interaction, $V(t)$, to be of the general form

$V(t)=2\hbar\varpi\cos(\omega t+\phi(t)+\chi).$ (3)

Here, $2\hbar\varpi$ is the interaction strength, $\omega$ is the driving
frequency and $\phi(t)$ acts as a temporally-correlated phase that causes
dephasing of the coherences. $\chi$ is a constant phase whose origin will be
explained later.

An electric field pulse acting on the ground state will excite the system into
these two states. We denote the electric field as

${\bf E}(t)=E(t){\bf e},$ (4)

where $E(t)$ is the temporal envelope of the pulse and ${\bf e}$ is its
polarisation. If the pulse (centered at time $t=0$) is sufficiently narrow in
time it will act instantaneously on the system and thus the excited state is
given as,[16, 17]

$\ket{\Psi(t=0)}=\frac{\text{i}}{\hbar}\sum_{j=a,b}({\bm{\mu}}_{j}\cdot{\bf
e})\tilde{E}(\omega_{j})\ket{j}=\sum_{j=a,b}\psi_{j}(0)\ket{j},$ (5)

where $\bm{\mu}_{j}=\bra{j}\bm{\mu}\ket{\text{GS}}$ is the transition dipole
moment and $\tilde{E}(\omega_{j})$ is the Fourier transform of $E(t)$ at
frequency $\omega_{j}$. At a subsequent time the system is subject to
evolution as determined by the total Hamiltonian, $\hat{H}_{S}+\hat{H}_{SB}$.
In general,

$\ket{\Psi(t)}=\sum_{j=a,b}\psi_{j}(t)\text{e}^{-\text{i}\omega_{j}t}\ket{j},$
(6)

where the amplitudes $\psi_{j}$ are time-dependent as the time-dependent
interaction on the system induces transitions between the eigenstates.

###  II.2 Response Functions

In this work we determine the third-order non-linear response functions using
the method of quantum trajectories, as described by Marcus _et al._[18]. This
wavefunction method is particularly convenient for determining numerical
solutions in Hilbert space, as it avoids dealing with the much larger
Liouville space that is necessary in a denisty matrix formalism.

The first order trajectory, $\ket{\Psi_{\beta}(t_{\delta})}$, is defined as an
excitation from the ground state by a single pulse at $t=t_{\beta}$ and
subsequent evolution to a time $t_{\delta}\geq t_{\beta}$. Thus,

$\ket{\Psi_{\beta}(t_{\delta})}=\hat{U}(t_{\delta},t_{\beta})\hat{O}_{\beta}\ket{\text{GS}},$
(7)

where $\hat{O}_{\beta}$ is an excitation operator defined for a given
spectroscopic protocol and $\hat{U}$ is the evolution operator. Similarly, the
third order trajectory, $\ket{\Psi_{\alpha\gamma\delta}(t_{\delta})}$, is
created by three pulses at times $t_{\alpha}\leq t_{\gamma}\leq t_{\delta}$.
Thus,

$\ket{\Psi_{\alpha\gamma\delta}(t_{\delta})}=\hat{O}_{\delta}\hat{U}(t_{\delta},t_{\gamma})\hat{O}_{\gamma}\hat{U}(t_{\gamma},t_{\alpha})\hat{O}_{\alpha}\ket{\text{GS}},$
(8)

again where the (de)excitation operators will be explained later.

The ordering of $t_{\beta}$ relative to $t_{\alpha}$, $t_{\gamma}$ and
$t_{\delta}$ determines the type of diagram in 2D-spectroscopy, i.e.,
rephasing or non-rephasing[15, 18]. To measure dynamically-induced coherences
a rephasing (or photon echo) diagram is required. We choose the ‘stimulated
emission’ diagram, illustrated in Fig. 1, where $t_{\alpha}\leq t_{\beta}\leq
t_{\gamma}$. 111This diagram is labeled ‘B’ in Ref.[18] and $R_{1}$ in
Ref.[15].

Figure 1: Our proposed spectroscopic protocol for measuring coherences induced
by an external interaction, as described in Section II.4. The upper panel
shows the evolution of the first-order trajectory, $\ket{\Psi_{\beta}}$, after
an excitation into $\ket{b}$ and subsequent oscillation between $\ket{b}$ and
$\ket{a}$. The lower panel shows the third-order trajectory,
$\ket{\Psi_{\alpha\gamma\delta}}$, after an excitation into $\ket{b}$,
subsequent oscillation between $\ket{b}$ and $\ket{a}$, then de-excitation and
re-excitation of the amplitudes of both $\ket{a}$ and $\ket{b}$. The dashed
horizontal lines indicate oscillations between $\ket{b}$ and $\ket{a}$.

The third-order response is then given by the overlap of these two
trajectories, namely, 222Setting $\hat{O}_{\delta}\equiv\hat{\mu}$ gives
$\left\langle\Psi_{\alpha\gamma\delta}|\Psi_{\beta}\right\rangle\equiv\textrm{Tr}\left\\{\hat{\mu}\hat{\rho}^{(3)}\right\\}$,
where $\hat{\rho}^{(3)}$ is the third-order density operator[15] for a pure
state. Similarly, an ensemble average over $S(t_{1},t_{2},t_{3})$ implies that
$\hat{\rho}^{(3)}$ is the third-order density operator for a mixed state.
Indeed, Eq. (9) is the equivalent in third-order non-linear spectroscopy to
the Loschmidt echo,
$S(t)=\left\langle\Psi_{\alpha}|\Psi_{\beta}\right\rangle$, in linear
spectroscopy[16].

$S(t_{1},t_{2},t_{3})=\left\langle\Psi_{\alpha\gamma\delta}|\Psi_{\beta}\right\rangle.$
(9)

Defining the time variables as $t_{1}=t_{\beta}-t_{\alpha}$ (coherence time),
$t_{2}=t_{\gamma}-t_{\beta}$ (waiting time) and $t_{3}=t_{\delta}-t_{\gamma}$
(echo time), and Fourier transforming with respect to $t_{1}$ and $t_{3}$, we
obtain the two-dimensional in frequency-space and one-dimensional in time-
space third-order response,

$\tilde{S}(\omega_{1},t_{2},\omega_{3})=\int
S(t_{1},t_{2},t_{3})\exp(\textrm{i}(\omega_{1}t_{1}+\omega_{3}t_{3}))\textrm{d}t_{1}\textrm{d}t_{3}.$
(10)

Details of how the third-order signal is measured may be found in text books;
see Hamm and Zani[15], for example.

###  II.3 Closed System, $\hat{H}_{SB}=0$

To motivate our experimental protocol for measuring dynamically-induced
coherences, as described in Section II.4, we first consider two simple
examples of determining a 2D-spectrum in a closed system, i.e.,
$\hat{H}=\hat{H}_{S}$.

####  II.3.1 Stationary State

Let us first consider the case of a targeted excitation within the two-level
system. We assume that the pulse at $t_{\beta}$ only excites the system into
the $\ket{b}$ state. In this case $\hat{O}_{\beta}=\hat{a}_{b}^{\dagger}$ and
the first-order trajectory is,

$\ket{\Psi_{\beta}}=\hat{U}(t_{\delta},t_{\beta})\hat{a}_{b}^{\dagger}\ket{\text{GS}}=\text{e}^{-\text{i}\omega_{b}(t_{\delta}-t_{\beta})}\ket{b}.$
(11)

Similarly, for the third-order trajectory with
$\hat{O}_{\alpha}=\hat{O}_{\gamma}^{\dagger}=\hat{O}_{\delta}=\hat{a}_{b}^{\dagger}$,
we obtain,

$\ket{\Psi_{\alpha\gamma\delta}}=\text{e}^{-\text{i}\omega_{b}(t_{\gamma}-t_{\alpha})}\ket{b},$
(12)

as in the ground state the evolution operator is just the identity.

Then,

$S(t_{1},t_{2},t_{3})=\text{e}^{\text{i}\omega_{b}(t_{\beta}-t_{\alpha}+t_{\gamma}-t_{\delta})}=\text{e}^{\text{i}\omega_{b}(t_{1}-t_{3})}$
(13)

and

$\begin{split}\tilde{S}(\omega_{1},t_{2},\omega_{3})=\delta(\omega_{3}-\omega_{b})\delta(\omega_{1}+\omega_{b}).\end{split}$
(14)

This shows that the resulting 2D spectrum consists of a single peak at
$(-\omega_{b},\omega_{b})$ in the $(\omega_{1},\omega_{3})$ plane, which is
independent of $t_{2}$. As, once excited the population of $\ket{b}$ does not
change, this peak is a direct signal for the population of the investigated
state. A schematic spectrum is shown in Fig. 2(a).

Figure 2: Schematics of the analytical spectra of a two-level system. (a)
shows a single excitation of a closed system (i.e., a stationary state) and
(b) a collective excitation into both states (i.e., a non-stationary state).
(c) and (d) shows the analytical spectra, determined using the RWA (and given
in Appendix A), of the system under the influence of a time-dependent
interaction with the excitation sequence shown in Fig. 1. (c) on-resonant
interaction while (d) is off-resonant. The red and blue signals correspond to
the population signals of states $\ket{a}$ and $\ket{b}$, while the purple and
orange signals are signatures of coherence between the states. The interaction
shifts the signals at $\omega_{3}=\omega_{a}$ to the right by the driving
frequency, $\omega$, and splits each signal into four by the Rabi frequency,
$\Omega$. At off-resonance the detuning leads to a shift of the four groups of
four with respect to the positions of the on-resonance peaks (shown faintly in
(d)). The waiting time $t_{2}=0$.

####  II.3.2 Non-stationary State

Let us now consider the 2D-spectrum obtained via the coherent simultaneous
excitation of $\ket{a}$ and $\ket{b}$ and the non-stationary state’s
subsequent evolution under the action of $\hat{H}_{S}$. In this case,

$\hat{O}_{\alpha}=\hat{O}_{\beta}=\hat{O}_{\gamma}^{\dagger}=\hat{O}_{\delta}=\frac{1}{\sqrt{2}}\left(\hat{a}_{a}^{\dagger}+\hat{a}_{b}^{\dagger}\right).$
(15)

The first-order trajectory is then

$\ket{\Psi_{\beta}}=\frac{1}{\sqrt{2}}\left(\text{e}^{-\text{i}\omega_{a}(t_{\delta}-t_{\beta})}\ket{a}+\text{e}^{-\text{i}\omega_{b}(t_{\delta}-t_{\beta})}\ket{b}\right),$
(16)

while the third-order trajectory is

$\ket{\Psi_{\alpha\gamma\delta}}=\frac{1}{2\sqrt{2}}\text{e}^{-\text{i}\omega_{a}(t_{\gamma}-t_{\alpha})}\left(1+\text{e}^{-\text{i}\omega_{0}(t_{\gamma}-t_{\alpha})}\right)\left(\ket{a}+\ket{b}\right).$
(17)

The time-resolved overlap is

$S(t_{1},t_{2},t_{3})=\frac{1}{4}\left[\text{e}^{-\text{i}\omega_{a}(t_{3}-t_{1})}+\text{e}^{-\text{i}\omega_{b}(t_{3}-t_{1})}+\text{e}^{-\text{i}\omega_{a}(t_{3}-t_{1})}\left(\text{e}^{-\text{i}\omega_{0}(t_{3}+t_{2})}+\text{e}^{\text{i}\omega_{0}(t_{2}+t_{1})}\right)\right],$
(18)

while the frequency-resolved spectrum is,

$\begin{split}\tilde{S}(\omega_{1},t_{2},\omega_{3})&=\frac{1}{4}\left[\delta(\omega_{1}+\omega_{a})\delta(\omega_{3}-\omega_{a})+\delta(\omega_{1}+\omega_{b})\delta(\omega_{3}-\omega_{b})\right.\\\
&+\left.\left(\delta(\omega_{1}+\omega_{a})\delta(\omega_{3}-\omega_{b})\text{e}^{-\text{i}\omega_{0}t_{2}}+\delta(\omega_{1}+\omega_{b})\delta(\omega_{3}-\omega_{a})\text{e}^{\text{i}\omega_{0}t_{2}}\right)\right].\end{split}$
(19)

Here we see that the resulting spectrum has four peaks: two are on the
diagonal of the $(-\omega_{1},\omega_{3})$ axes, representing the populations
of the two eigenstates, while the other two are off-diagonal and oscillate
with a frequency $\omega_{0}$ in $t_{2}$. These off-diagonal peaks are the
coherences, $\psi_{a}^{*}\psi_{b}$, between the two eigenstates. A schematic
spectrum is shown in Fig. 2(b).

We see that if both states are populated the spectrum becomes richer. As the
system here is closed, the states have to be populated directly by the
electric field pulses, as otherwise no coherences occur. However, if the
system is subject to an interaction and population transfer can happen during
its evolution, then the initial excitation can be into a single state, thereby
probing the possibility of population (and hence energy) transfer. We discuss
this situation in the next section.

###  II.4 Open System with a Time-Dependent Interaction, $\hat{H}_{SB}$

We now turn to the key result of this work: a protocol for measuring the
coherences of a system induced by a time-dependent interaction. In this case
we assume that both the first and third order trajectories are initially
prepared in one of the eigenstates of $\hat{H}_{S}$. We choose this to be the
upper state, $\ket{b}$. Then, under the action of $\hat{H}_{SB}$ the system
oscillates coherently between $\ket{b}$ and $\ket{a}$. Specifically, the
amplitudes in Eq. (6) are given by the time-dependent Schrödinger equations,

$\frac{\textrm{d}\psi_{a}(t)}{\textrm{d}t}=-\text{i}2\varpi\cos(\omega
t+\phi(t)+\chi)\text{e}^{-\text{i}\omega_{0}t}\psi_{b}(t)$ (20)

and

$\frac{\textrm{d}\psi_{b}(t)}{\textrm{d}t}=-\text{i}2\varpi\cos(\omega
t+\phi(t)+\chi)\text{e}^{\text{i}\omega_{0}t}\psi_{a}(t).$ (21)

These equations can be solved within the rotating-wave or secular
approximation (RWA), which assumes that
$(\omega+\omega_{0})\gg|\omega-\omega_{0}|$. Using the method of Laplace
transforms, with the initial conditions of $\psi_{a}(0)=0$ and
$\psi_{b}(0)=1$, and setting the temporal phase $\phi(t)=0$, we obtain

$\psi_{a}(t)=\left(\frac{\varpi}{\Omega}\right)\exp(\textrm{i}\chi)\exp(\textrm{i}\Delta\omega
t/2)\left(\exp(\textrm{i}\Omega t/2)-\exp(-\textrm{i}\Omega t/2)\right)$ (22)

and

$\psi_{b}(t)=\frac{1}{2}\exp(-\textrm{i}\Delta\omega
t/2)\left(\left(1+\left(\frac{\Delta\omega}{\Omega}\right)\right)\exp(\textrm{i}\Omega
t/2)+\left(1-\left(\frac{\Delta\omega}{\Omega}\right)\right)\exp(-\textrm{i}\Omega
t/2)\right).$ (23)

The detuning parameter is

$\Delta\omega=(\omega-\omega_{0})$ (24)

and we define the Rabi angular frequency as

$\Omega=(4\varpi^{2}+(\Delta\omega)^{2})^{1/2}.$ (25)

The time-dependent system-bath interaction causes population to be exchanged
between $\ket{b}$ and $\ket{a}$ with a Rabi time period
$T_{\Omega}=2\pi/\Omega$. For the pure quantum state defined by Eq. (6) the
populations of the eigenkets are $|\psi_{a}(t)|^{2}$ and $|\psi_{b}(t)|^{2}$,
while their coherences are $\psi_{a}^{*}(t)\psi_{b}(t)$. Off-resonance (i.e.,
$\Delta\omega\neq 0$) there is an incomplete transfer of population with the
maximum population of $\ket{a}$ being $4({\varpi}/{\Omega})^{2}$.

We notice the phase factor of $\exp(\textrm{i}\chi)$ in the expression for
$\psi_{a}(t)$, which arises from the $t=0$ phase in $V(t)$. As explained
shortly, this phase factor has important implications for the definition of
$\ket{\Psi_{\beta}}$, which is created at a time
$t_{1}=(t_{\beta}-t_{\alpha})$ after $\ket{\Psi_{\alpha}}$ at which point
$\chi=\omega t_{1}$.

We now propose a 2D-spectroscopic protocol for observing these induced
coherences. The first-order trajectory $|\Psi_{\beta}\rangle$ is given by Eq.
(7), where $\hat{O}_{\beta}=\hat{a}^{\dagger}_{b}$ and
$\hat{U}(t_{\delta},t_{\beta})$ is now determined by $\hat{H}_{S}$ and
$\hat{H}_{SB}$. Thus, $|\Psi_{\beta}\rangle$ is an excitation from the
groundstate to $\ket{b}$ at time $t_{\beta}$, followed by evolution in the
excited state manifold (ESM) under the action of $(\hat{H}_{S}+\hat{H}_{SB})$
to time $t_{\delta}$, by which time it has acquired amplitude in both
$\ket{a}$ and $\ket{b}$. This trajectory is illustrated schematically in red
in Fig. 1.

Similarly, the third-order trajectory $\ket{\Psi_{\alpha\gamma\delta}}$ is
given by Eq. (8), where $\hat{O}_{\alpha}=\hat{a}^{\dagger}_{b}$, but now
$\hat{O}_{\gamma}=(\hat{a}_{a}+\hat{a}_{b})/\sqrt{2}$ and
$\hat{O}_{\delta}=(\hat{a}^{\dagger}_{a}+\hat{a}^{\dagger}_{b})/\sqrt{2}$.
Thus, $\ket{\Psi_{\alpha\gamma\delta}}$ is (i) an excitation from the ground
state to $\ket{b}$ at time $t_{\alpha}$, (ii) evolution in the ESM under the
action of $(\hat{H}_{S}+\hat{H}_{SB})$ to time $t_{\gamma}$, (iii) de-
excitation via a linear combination of $\ket{a}$ and $\ket{b}$ to the ground
state at time $t_{\gamma}$, (iv) evolution in the ground state to time
$t_{\delta}$, and finally (v) re-excitation via a linear combination of
$\ket{a}$ and $\ket{b}$ at time $t_{\delta}$. This trajectory is illustrated
schematically in blue in Fig. 1. The selective (de)excitation of $\ket{a}$ and
$\ket{b}$ is achieved by different polarizations of the laser light, as
illustrated in Section III.

By a time $t_{\gamma}$ the state
$\ket{\Psi_{\alpha}}=\hat{U}(t_{\gamma},t_{\alpha})\hat{O}_{\alpha}|\textrm{GS}\rangle$
has acquired amplitude in both $\ket{a}$ and $\ket{b}$. These amplitudes are
transferred to the ground state via $\hat{O}_{\gamma}$ and re-excited at
$t_{\delta}$ via $\hat{O}_{\delta}$. The interference of these amplitudes with
those of $|\Psi_{\beta}\rangle$, via the overlap
$\langle\Psi_{\alpha\gamma\delta}|\Psi_{\beta}\rangle$, allow the dynamics —
and in particular, the coherences — of the system to be determined.

Since the wavefunction overlap,
$\langle\Psi_{\alpha\gamma\delta}|\Psi_{\beta}\rangle$, obtained via the RWA
(i.e., Eq. (22) and (23)) is a sum of a product of complex exponentials, its
Fourier transform with respect to $t_{1}$ and $t_{3}$ results in a sum of a
product of delta-functions. In particular,
$\tilde{S}(\omega_{1},t_{2},\omega_{3})$ corresponds to four groups of four-
peaks whose analytical expressions are given in Appendix A. We now describe
the key features of the resulting 2D-spectrum for cases of on-resonance and
off-resonance interactions.

####  II.4.1 On-Resonance, $\omega=\omega_{0}$

At resonance there is complete population transfer between $\ket{b}$ and
$\ket{a}$. The resulting spectrum is given in Appendix A and illustrated by
Fig. 2(c). From this we observe:

  1. 1.

As explained in Section II.3.1, in the absence of induced coherences, i.e.,
when $V(t)=0$, the system would remain in the stationary state $\ket{b}$ and
there would be a single population peak at $(-\omega_{b},\omega_{b})$ (as in
Fig. 2(a)).

  2. 2.

With induced coherences, however, there are four groups of four peaks. These
groups of peaks correspond to the populations of $\ket{b}$ (i.e., at BB) and
$\ket{a}$ (i.e., at AA), and their coherences (i.e., AB and BA).

  3. 3.

Notice, however, that the two groups of peaks at
$\omega_{3}=\omega_{a}-\Delta\omega/2$ (i.e., AA and BA) are displaced along
the $-\omega_{1}$ axis by $\omega$. This ‘boost’ along $\omega_{1}$ at
$\omega_{3}=\omega_{a}-\Delta\omega/2$ is a consequence of the phase factor
$\chi=\omega t_{1}$ that the amplitude of $\ket{a}$ acquires when
$\ket{\Psi_{\beta}}$ is created, as shown by Eq. (22).

  4. 4.

Each group of peaks is split into four peaks whose splitting is determined by
$\Omega=2\pi/T_{\Omega}$, where $T_{\Omega}$ is the (Rabi) population transfer
period.

  5. 5.

Items (3) and (4) illustrate the characteristic 2D-spectrum obtained via this
2D-protocol that provides a characteristic fingerprint of dynamically-induced
coherences.

  6. 6.

The peaks exhibit complex dynamics as a function of $t_{2}$. For example, the
off-diagonal components of the population group of peaks oscillate with a
period of $T_{\Omega}$, while the diagonal components of the coherence group
of peaks oscillate with a period of $T=2\pi/\omega$.

####  II.4.2 Off-Resonance, $\omega\neq\omega_{0}$

Figure 3: The 2D-spectrum for $\Delta\omega/\omega_{0}=0.9$,
$\varpi/\omega_{0}=0.25$ and $t_{2}=0$. These results are obtained by
computing the overlap $\langle\Psi_{\alpha\gamma\delta}|\Psi_{\beta}\rangle$
numerically using the TNT Library[21] and evaluating Eq. (10) using a discrete
Fourier transform (DFT). This should be compared to the analytical solution
obtained via the RWA, shown in Fig. 2(d). Note, however, that additional low-
intensity peaks appear here because this an exact numerical calculation of the
wavefunction overlap. The peak features (i.e., elongations along the axes) are
artefacts of the DFT of a finite length signal.

Off-resonance there is incomplete population transfer between $\ket{b}$ and
$\ket{a}$. The resulting spectrum is given in Appendix A and illustrated by
Fig. 2(d). From this we observe:

  1. 1.

The center of the peaks is shifted by $|\Delta\omega/2|$ in each direction,
depending on the peak group.

  2. 2.

The splitting of peaks within a group increases from $2\varpi$ to
$\sqrt{(2\varpi)^{2}+(\Delta\omega)^{2}}$.

  3. 3.

The intensity of the coherence peaks (i.e., AB and BA) relative to the
dominant population peak (i.e., BB) is proportional to $2\varpi/\Omega$, and
is a measure of the strength of the coherences. Similarly, the intensity of
the population peak AA relative to BB is proportional to
$4(\varpi/\Omega)^{2}$, and is a measure of the maximum population transfer
between $\ket{b}$ and $\ket{a}$.

  4. 4.

All but one of the peaks in group BB vanish for off-resonance if
$2\varpi\ll\Omega$, as in this case no coherences between $\ket{b}$ and
$\ket{a}$ are established.

Fig. 3 shows a calculated spectrum of the two-level system. The wavefunction
overlap, $S(t_{1},t_{2},t_{3})$, was performed numerically exactly via a
Trotter decomposition of the evolution operator using the TNT Library[21],
while its Fourier transform, $\tilde{S}(\omega_{1},t_{2},\omega_{3})$, was
performed via a discrete Fourier transform. Clearly visible are the four
groups of four peaks in accordance with the schematic picture shown in Fig.
2(d). Fig. 2(d) and Fig. 3 illustrate the central result of our work, namely
that our proposed protocol gives a very different 2D spectrum from the
standard spectrum of coherence illustrated in Fig. 2(b).

###  II.5 Noisy Environment

Figure 4: (a) The population of $\ket{b}$, $\rho_{bb}$, as a function of time
at resonance, $\omega=\omega_{0}$ (where $\rho_{aa}=1-\rho_{bb}$). (b) The
coherences between $\ket{a}$ and $\ket{b}$, $\rho_{ab}$, as a function of
time. The autocorrelation function of the system-bath interaction satisfies
Eq. (26). Purple (no dephasing), green (dephasing time
$\gamma^{-1}=2T_{\Omega}$, where $T_{\Omega}$ is the Rabi period). We note
that in the case of no-dephasing $\rho_{bb}(t)$ deviates from
$|\psi_{b}(t)|^{2}$ with $\psi_{b}(t)$ given by Eq. (23) because this is a
numerically exact solution of the two-level system and not the RWA. Figure 5:
Dephasing acting on the system and the effect on the spectra. The upper panel
shows fast dephasing with $\gamma^{-1}=T_{\Omega}/4$, while the lower panel
shows slower dephasing with $\gamma^{-1}=T_{\Omega}$, where $T_{\Omega}$ is
the Rabi period. In both cases the fine structure of the peak groups is
destroyed while the boost in $-\omega_{1}$ at
$\omega_{3}=\omega_{a}-\Delta\omega/2$ is maintained.
$\Delta\omega/\omega=0.9$, $\varpi/\omega_{0}=0.25$ and $t_{2}=0$.

Noise destroys coherences, so we now consider its role in changing our
predicted spectra. We can incorporate noise into the interaction by
randomising the phase. In particular, we choose $\phi(t)$ so that the
autocorrelation function of the system-bath interaction satisfies

$\langle V(t)V(0)\rangle=|V(0)|^{2}\cos(\omega t)\exp(-\gamma t),$ (26)

where $\gamma$ is the dephasing rate. The form of Eq. (26) is chosen so as to
represent damped, harmonic system-bath interactions at ambient temperatures,
e.g., torsional oscillations of monomers subject to Brownian fluctuations. Eq.
(26) is achieved if $\phi(t)$ in Eq. (3) satisfies

$\phi(t)=\int_{0}^{t}\delta\omega(t^{\prime})\text{d}t^{\prime},$ (27)

where $\delta\omega(t)$ is a temporally-correlated noise function satisfying

$\left\langle\delta\omega(t)\delta\omega(0)\right\rangle=(\delta\omega)^{2}\text{e}^{-t/\tau_{c}}$
(28)

and $\delta\omega\tau_{c}\ll 1$.[22, 15]

The numerical simulations of the model with the stochastic interaction given
by Eqs. (27) and (28) were performed via a Trotter decomposition of the
evolution operator using the TNT Library[21]. By taking the ensemble average
over 100 trajectories with different temporal disorder we construct the
system’s reduced density matrix.

The effect of a system-bath dephasing interaction is illustrated in Fig. 4.
Fig. 4(a) shows that the populations of $\ket{a}$ and $\ket{b}$ equilibrate to
equal values, while Fig. 4(b) shows that the coherences between $\ket{a}$ and
$\ket{b}$ decay.

The effect of dephasing on the 2D-spectrum is shown in Fig. 5. For fast
dephasing ($\gamma^{-1}=T_{\Omega}/4$, where $T_{\Omega}$ is the Rabi period),
coherences between $\ket{a}$ and $\ket{b}$ do not have time to become fully
established, so a single population peak BB (at
$\sim(-\omega_{b},\omega_{b})$) dominates the spectrum. For intermediate
dephasing ($\gamma^{-1}=T_{\Omega}$), however, coherences are established, but
noise destroys the resolution of each of the four sub-peaks within a group of
peaks. Nevertheless, one of the characteristic fingerprints of induced
dynamical coherences, namely the boost along $\omega_{1}$ at
$\omega_{3}=\omega_{a}-\Delta\omega/2$, is still evident.

##  III Realistic Model

So far we have considered a theoretical two-level system and have shown how
the dynamics of that system can be observed via 2D-spectrum. In this section
we describe practical realizations of that system. In particular, we consider
two examples of exciton dynamics in conjugated polymer systems, namely (A),
energy transfer between two chromophores and (B), energy relaxation on the
same chromophore. In both cases we assume that the ‘external’ interaction
driving the dynamics is the damped torsional motion of the monomers. We also
describe the electric field polarizations required to selectively excite and
de-excite the eigenstates.

Here we are concerned with the lowest energy excited states of a chromophore.
A convenient theory to describe these states is the Frenkel exciton model. We
show in Appendix B how this model maps onto our two-level system with a time-
dependent coupling of the eigenstates determined by the monomer rotations.

###  III.1 Energy Transfer

Figure 6: A schematic of the exciton center-of-mass wavefunctions $c_{a}$ (in
red) and $c_{b}$ (in blue) corresponding to the stationary states $\ket{a}$
and $\ket{b}$. (a) neighboring chromophores of a polymer segment (represented
by the dashed black curve), where $N_{b}<N_{a}$; (b) the same curved
chromophore (represented by the dashed black curve). $\bm{\mu}_{a}$ and
$\bm{\mu}_{b}$ represent the transition dipole moments of these states.
$N_{j}$ is the number of monomers in chromophore $j$.

We first consider energy transfer between two almost orthogonal chromophores
on a bent portion of a conjugated polymer, as shown in Fig. 6(a). In practice,
chromophores are defined by the spatial extent of local exciton ground states
(LEGSs), whose spatial extent is determined by Anderson localization of the
exciton center-of-mass wavefunction[23]. The boundary between chromophores is
determined by the spatial distribution of disorder and possible ‘conjugation
breaks’. In all cases, the LEGSs overlap at the boundaries.

To achieve the protocol described in Section II.4, exciton $\ket{b}$ is first
excited, which means that the electric field is initially polarized along the
transition dipole moment of $\ket{b}$. Thus, ${\bf e}\propto\bm{\mu}_{b}$, and
$\hat{O}_{\alpha}\propto\hat{\bf{\mu}}_{b}$ and
$\hat{O}_{\beta}\propto\hat{\bf{\mu}}_{b}$. The transfer of amplitude to
exciton $\ket{a}$ is then determined by setting the electric field
polarization along components of the transition dipole moments of both
$\ket{a}$ and $\ket{b}$. Thus, ${\bf e}\propto(\bm{\mu}_{a}+\bm{\mu}_{b})$,
and $\hat{O}_{\gamma}\propto(\hat{\bf{\mu}}_{a}+\hat{\bf{\mu}}_{b})/\sqrt{2}$
and $\hat{O}_{\delta}\propto(\hat{\bf{\mu}}_{a}+\hat{\bf{\mu}}_{b})/\sqrt{2}$.

For a typical conjugated polymer at room temperature, e.g., poly(para-
phenylene), $\bar{\delta\theta}\approx 0.1$ rads, $J\approx 2$ eV and
$J_{SE}\sin 2\theta\approx 1$ eV (see Appendix B). The wavefunction overlap is
determined by disorder, but is typically $\sim 0.1$. Thus,
$\hbar\varpi\lesssim 0.005$ eV, implying at resonance a population transfer
period $T_{\Omega}\sim 1$ ps. The torsion oscillation energy is
$\hbar\omega=0.02$ eV and thus resonance is achieved if the difference in
chromophore sizes satisfies333Because $E(N)=a-b/N^{2}$ for $N\gg 1$, where $a$
and $b$ are constants. $\Delta N/N^{3}\approx 5\times 10^{-4}$, e.g.,
$N_{b}=18$ and $N_{a}=22$. In this case $\varpi/\omega_{0}=0.25$, which
corresponds to the parameters used for Fig. 3. In practice, noise will smear
the spectrum, so a result more like Fig. 5 is expected.

###  III.2 Energy Relaxation

As a second example we consider energy relaxation from a higher to a lower
energy exciton, as shown in Fig. 6(b). Here, $c_{a}$ is the LEGS while $c_{b}$
is a locally excited exciton state (LEES)[23]. Importantly, $c_{b}$ has one
node so that on a curved chromophore its transition dipole moment is almost
orthogonal to that of $c_{a}$[25]. To achieve the protocol described in
Section II.4, $\hat{O}_{\alpha}\propto\hat{\bf{\mu}}_{b}$,
$\hat{O}_{\delta}\propto\hat{\bf{\mu}}_{b}$,
$\hat{O}_{\beta}\propto(\hat{\bf{\mu}}_{a}+\hat{\bf{\mu}}_{b})/\sqrt{2}$ and
$\hat{O}_{\gamma}\propto(\hat{\bf{\mu}}_{a}+\hat{\bf{\mu}}_{b})/\sqrt{2}$. In
this case, resonance is achieved if the chromophore size $N\approx 50$, when
again $\varpi/\omega_{0}=0.25$.

Before concluding this section, we note that the monomer rotations that cause
interstate coupling also introduces a time-dependent diagonal term into the
two-level model. However, because the angular frequency of this motion, i.e.,
$\omega$, is so much smaller than $\omega_{a}$ and $\omega_{b}$, this term has
a negligible effect on the 2D-spectra.

##  IV Concluding Remarks

We have proposed a protocol for performing 2D-spectroscopy experiments that
measure induced time-dependent coherences between the stationary states of a
two-level system. This protocol gives a rich 2D-spectrum characteristic of
dynamical coherences that differs from the observed coherences of the non-
stationary state $\ket{\Psi(t=0)}=(\ket{a}+\ket{b})/\sqrt{2}$. The 2D-spectrum
for the non-stationary state consists of populations at
$(-\omega_{a},\omega_{a})$ and $(-\omega_{b},\omega_{b})$, and coherences at
$(-\omega_{a},\omega_{b})$ and $(-\omega_{b},\omega_{a})$. In contrast, the
2D-spectrum for measuring induced dynamical coherences proposed here differs
in two distinct ways. First, each peak splits into a group of four peaks, with
the splitting of the peaks determined by the population transfer (Rabi)
period, $T_{\Omega}$. Second, at $\omega_{3}\sim\omega_{a}$ there is a boost
along the $\omega_{1}$ axis by $\omega$, the angular frequency of the system-
bath interaction that induces these dynamics. These two features imply that
this proposed protocol provides a unique fingerprint for induced dynamical
coherences and the system-bath interactions. However, for weak system-bath
interactions coherences are only established at or close to resonance, i.e.,
$\omega\simeq\omega_{0}$. Furthermore, coherences are destroyed by strong
dephasing, meaning that only the population at $(-\omega_{b},\omega_{b})$
would be observed.

We illustrated this approach using one of the ‘rephasing’ (or photon-echo)
diagrams in 2D-spectroscopy (i.e., the stimulated emission diagram, where
$t_{\alpha}\leq t_{\beta}\leq t_{\gamma}$). However, the ground state bleach
diagram (where $t_{\gamma}\leq t_{\beta}\leq t_{\delta}$)444This diagram is
labeled ‘C’ in Ref.[18] and $R_{2}$ in Ref.[15]. is also equally appropriate,
as in both of these diagrams the first and third order trajectories evolve in
the excited state manifold during either $t_{1}$ or $t_{3}$.

We next discussed exemplars of this model, namely a conjugated polymer whose
torsional fluctuations act as the time-dependent interaction. The two
stationary states are either excitons localized on neighboring chromophores,
or the two lowest excited states on the same chromophore. Since both the
angular frequency of the torsional oscillations and the strength of the
interaction that drives the dynamics is small (i.e., $\sim 0.01-0.02$ eV) it
is important that the system is on or close to resonance for coherences to be
induced and observed via 2D-spectroscopy. In addition, since these energy
scales are much smaller than other possible electronic sources of homogeneous
and inhomogenous broadening, e.g., by fast density fluctuations, we might
expect that our example is only relevant for a polymer in an inert environment
not subject to sources of large dynamical and static disorder.

We hope that this proposed technique will have applications in other fields of
molecular and condensed matter physics in the study of quantum coherences.

###### Acknowledgements.

This work was performed using the Tensor Network Theory Library, Beta Version
1.2.1 (2016), S. Al-Assam, S. R. Clark, D. Jaksch, and the TNT Development
team, www.tensornetworktheory.org.

##  Appendix A Analytical Results

Since the wavefunction overlap,
$\langle\Psi_{\alpha\beta\gamma}|\Psi_{\delta}\rangle$, obtained via the RWA
(i.e., Eq. (22) and (23)) is a sum of a product of complex exponentials, its
Fourier transform with respect to $t_{1}$ and $t_{3}$ results in a sum of a
product of delta-functions. In particular,
$\tilde{S}(\omega_{1},t_{2},\omega_{3})$ corresponds to four groups of four-
peaks. Referring to Fig. (2), we identify the four groups as follows.

  * •

The group AA are associated with the population of $\ket{a}$:

$\displaystyle\tilde{S}_{AA}$
$\displaystyle=\left(\frac{\varpi}{\Omega}\right)^{2}[\delta(\omega_{1}+(\omega_{a}+\omega-(\Delta\omega\pm\Omega)/2))\delta(\omega_{3}-(\omega_{a}-(\Delta\omega\pm\Omega)/2))$
$\displaystyle-\delta(\omega_{1}+(\omega_{a}+\omega-(\Delta\omega\pm\Omega)/2))\delta(\omega_{3}-(\omega_{a}-(\Delta\omega\mp\Omega)/2))\exp(\pm\textrm{i}\Omega
t_{2})].$

  * •

The group BB are associated with the population of $\ket{b}$:

$\displaystyle\tilde{S}_{BB}$
$\displaystyle=\frac{1}{4}\left(1\mp\frac{\Delta\omega}{\Omega}\right)^{2}\delta(\omega_{1}+(\omega_{b}+(\Delta\omega\pm\Omega)/2))\delta(\omega_{3}-(\omega_{b}+(\Delta\omega\pm\Omega)/2))$
$\displaystyle+\frac{1}{4}\left(1-\left(\frac{\Delta\omega}{\Omega}\right)^{2}\right)\delta(\omega_{1}+(\omega_{b}+(\Delta\omega\pm\Omega)/2))\delta(\omega_{3}-(\omega_{b}+(\Delta\omega\mp\Omega)/2))\exp(\pm\textrm{i}\Omega
t_{2}),$

where $\Omega$ is given by Eq. (25).

  * •

The group AB are associated with coherences between $\ket{a}$ and $\ket{b}$:

$\displaystyle\tilde{S}_{AB}$
$\displaystyle=\mp\frac{\varpi}{2\Omega}\left(1\mp\frac{\Delta\omega}{\Omega}\right)\delta(\omega_{1}+(\omega_{a}-(\Delta\omega\mp\Omega)/2))\delta(\omega_{3}-(\omega_{b}+(\Delta\omega\pm\Omega)/2))\exp(-\textrm{i}\omega
t_{2})$
$\displaystyle\mp\frac{\varpi}{2\Omega}\left(1+\frac{\Delta\omega}{\Omega}\right)\delta(\omega_{1}+(\omega_{a}-(\Delta\omega\pm\Omega)/2))\delta(\omega_{3}-(\omega_{b}+(\Delta\omega\mp\Omega)/2))\exp(-\textrm{i}(\omega\mp\Omega)t_{2}).$

  * •

Similarly, the group BA are associated with coherences between $\ket{a}$ and
$\ket{b}$:

$\displaystyle\tilde{S}_{BA}$
$\displaystyle=\mp\frac{\varpi}{2\Omega}\left(1\mp\frac{\Delta\omega}{\Omega}\right)\delta(\omega_{1}+(\omega_{b}+\omega+(\Delta\omega\pm\Omega)/2))\delta(\omega_{3}-(\omega_{a}-(\Delta\omega\pm\Omega)/2))\exp(\textrm{i}\omega
t_{2})$
$\displaystyle\mp\frac{\varpi}{2\Omega}\left(1+\frac{\Delta\omega}{\Omega}\right)\delta(\omega_{1}+(\omega_{b}+\omega+(\Delta\omega\pm\Omega)/2))\delta(\omega_{3}-(\omega_{a}-(\Delta\omega\mp\Omega)/2))\exp(\textrm{i}(\omega\mp\Omega)t_{2}).$

##  Appendix B Derivation of the Two-State Model for Conjugated Polymers

The dynamics of Frenkel excitons in conjugated polymers is described by the
Frenkel exciton model[23], defined by

$\hat{H}=\sum_{n=1}^{N}\epsilon_{n}\ket{n}\bra{n}+\sum_{n=1}^{N-1}J_{n}(t)\left(\ket{n}\bra{n+1}+\ket{n+1}\bra{n}\right),$
(33)

where $n$ labels a monomer and $\epsilon_{n}$ is the excitation energy of a
monomer. $J_{n}$ is the exciton transfer integral,

$J_{n}(t)=J_{DD}+J_{SE}\cos^{2}\theta_{n}(t),$ (34)

where $J_{DD}$ is the through-space, dipole-dipole contribution,
$J_{SE}\cos^{2}\theta_{n}(t)$ is the through-bond, superexchange contribution,
and $\theta_{n}$ is the dihedral angle between monomers. In general,
$\theta_{n}(t)=\theta+\delta\theta_{n}(t)$, where the dynamical component
$\delta\theta_{n}(t)$ arises from thermal fluctuations.

Assuming that $\delta\theta_{n}(t)\ll\theta$ we can linearize Eq. (34) to
obtain

$J_{n}(t)=J-J_{SE}\delta\theta_{n}(t)\sin 2\theta,$ (35)

where

$J=J_{DD}+J_{SE}\cos^{2}\theta.$ (36)

This linearization allows us to partition the Hamiltonian into a time-
independent (system) part,

$\hat{H}_{S}=\sum_{n=1}^{N}\epsilon_{n}\ket{n}\bra{n}+\sum_{n=1}^{N-1}J\left(\ket{n}\bra{n+1}+\ket{n+1}\bra{n}\right),$
(37)

and a time-dependent (system-bath) part,

$\hat{H}_{SB}=-J_{SE}\sin
2\theta\sum_{n=1}^{N-1}\delta\theta_{n}(t)\left(\ket{n}\bra{n+1}+\ket{n+1}\bra{n}\right).$
(38)

Transforming into the eigenkets of $\hat{H}_{S}$, i.e.,

$\ket{j}=\sum_{n=1}^{N}c_{nj}\ket{n}$ (39)

(where $c_{nj}$ is the exciton wavefunction, i.e., $j=a,b$, shown in Fig. 6)
we have the system Hamiltonian

$\hat{H}_{S}=\sum_{j=1}^{N}\hbar\omega_{j}\ket{j}\bra{j},$ (40)
$\hbar\omega_{j}=2J\sum_{n}c_{n,j}c_{n+1,j}.$ (41)

Similarly, the system-bath Hamiltonian is

$\hat{H}_{SB}=\sum_{ij}V_{ij}(t)\ket{i}\bra{j},$ (42)

where

$V_{ij}(t)=-J_{SE}\sin
2\theta\sum_{n=1}^{N-1}\delta\theta_{n}(t)\left(c_{n,i}c_{n+1,j}+c_{n,j}c_{n+1,i}\right).$
(43)

For a polymer subject to stochastic thermal fluctuations, the dynamical
component of the torsional angle satisfies

$\delta\theta_{n}(t)=\bar{\delta\theta}\cos(\omega t+\phi_{n}(t)),$ (44)

where $\bar{\delta\theta}=\left(k_{B}T/K_{\textrm{rot}}\right)^{1/2}$,
$\omega$ is the rotational angular frequency and $K_{\textrm{rot}}$ is the
rotational force constant.

Finally, by only retaining two states, $\ket{a}$ and $\ket{b}$, this model
maps onto the two-level model introduced Section II.1 and we find that the
interaction strength is,

$2\hbar\varpi=-J_{SE}\bar{\delta\theta}\sin
2\theta\sum_{n=1}^{N-1}\left(c_{n,a}c_{n+1,b}+c_{n,b}c_{n+1,a}\right).$ (45)

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11institutetext:  Auckland University of Technology, Auckland, New Zealand,
22institutetext: University of Tasmania, Tasmania, Australia
33institutetext: Nagoya Institute of Technology, Japan
33email<EMAIL_ADDRESS>weihua.li<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Privacy Information Classification: A Hybrid Approach

Jiaqi Wu 11 Weihua Li 11 Quan Bai 22 Takayuki Ito 33 Ahmed Moustafa 33

###### Abstract

A large amount of information has been published to online social networks
every day. Individual privacy-related information is also possibly disclosed
unconsciously by the end users. Identifying privacy-related data and
protecting the online social network users from privacy leakage turn out to be
significant. Under such a motivation, this study aims to propose and develop a
hybrid privacy classification approach to detect and classify privacy
information from OSNs. The proposed hybrid approach employs both deep learning
models and ontology-based models for privacy-related information extraction.
Extensive experiments are conducted to validate the proposed hybrid approach,
and the empirical results demonstrate its superiority in assisting online
social network users against privacy leakage.

###### Keywords:

Privacy detection Online Social Networks Deep Learning Privacy Information.

##  1 Introduction

With the proliferation and popularisation of the World Wide Web, Online Social
Networks (OSNs) become of one of the essential channels for social
interactions and communications [1, 2]. OSNs provide great convenience to the
users, but these online social platforms also raise potential risks, such as
privacy leakage. A vast amount of private information can be accessed publicly
through OSNs, such as preferences, email address, marital status, hometown,
activities attended, etc., which may lead to severe security issues.

Therefore, it is significant to explore a useful and practical approach to
protect OSN users from privacy information disclosure. Users should be
reminded before posting any privacy-related messages to the public. In the
context of OSNs, “user privacy” refers to a sequence of words, stating or
implying any individual’ s personal information, preferences, events that he
or she involved; privacy leakage describes a situation when an individual
shares stories including private information with their contacts or even those
they are not familiar with. Thus, it is necessary to develop a tool to detect
and identify all the possible privacy-related information contained in any
posting messages [20, 7]. More importantly, the justifications of privacy-
related information classification would be helpful for OSN users to get rid
of posting similar messages again.

In our previous research work [11], we conducted preliminary studies and
developed a generic framework of privacy leakage detection for OSN users based
on deep learning models. The proposed framework is capable of capturing
privacy-related entities after giving sufficient training. However, two
significant limitations are to be covered. Firstly, it can only remind users
regarding the possibility of privacy leakage. As a result, detailed leaking
information in terms of what kind of leakage is missing. Secondly, the privacy
model is not conceptually modelled or presented.

Ontology models conceptually reflect the domain-specific knowledge in the form
of terms and demonstrate two apparent advantages, i.e., shareability and
reusability [22]. Therefore, ontology-based privacy models can be easily
extended and applied to various OSNs [16]. Moreover, as ontology organises the
concepts in the form of taxonomy or hierarchy based on a pre-defined natural
relationship, it is suitable to introduce ontology into the research of
privacy-related information extraction and classification.

As an extension of our previous work [11], in this paper, we leverage a hybrid
privacy classification approach, incorporating both deep learning and ontology
models, for individual users of OSNs. The extended framework is capable of
addressing the privacy leakage problem for individuals by effectively
identifying privacy information and classifying into a detailed category. More
specifically, the proposed hybrid approach is composed of two major
components, i.e., a deep learning based approach to detect the privacy leakage
on online social data and an ontology privacy model classifying the privacy-
related information into fine-grained privacy categories. Deep learning models
are utilised to conduct the Name Entity Recognition (NER) and detect the pre-
defined privacy-related entities. An ontology model is developed based on the
analysis of massive data collected from real-world OSNs. Given the predictive
results carried out by the deep learning model, the privacy ontology model
further classifies the recognised privacy-related entities into sub-classes.

The rest of the paper is organised as follows. Section 2 reviews the existing
literature of privacy information classification approaches on OSNs. Section 3
introduces the entire privacy information detection and fine-grained
classification approach. In Section 4, experiments are conducted to evaluate
the ontology-based approach on the dataset crawled from twitter. Section 5
concludes the findings of this paper and points out the limitations and future
direction.

##  2 Related Work

Most contemporary privacy information classification approaches aim to detect
some specific categories of privacy information on OSNs or perform a binary
classification, i.e., sensitive or non-sensitive, rather than identifying and
classifying privacy information for the end users. [8] indicates that it is
easy for users to leak the matters or activities that one involves anytime
anywhere unconsciously. For example, a tweet saying that user’s family will go
out for a holiday implies no one stays at home, which may cause robbery
accidents. Therefore, the privacy information of OSNs users should be prompted
before sharing with the public.

[6] proposes a machine learning classification approach by adopting Named
Entity Recognition technology, which can classify privacy information on OSNs
into different categories, e.g., electronic devices and brands. Similarly,
[13] present a privacy classifier for three kinds of sensitive tweets, i.e.,
drunk, holiday and disease, and classifies tweets into binary results, i.e.,
sensitive tweets or nonsensitive tweets.

From the research above, we can see even there are several papers about
classifying privacy revealing information on OSNs, most of them pay attention
to specific categories by machine learning approaches. Very few studies
classify them according to a domain privacy ontology and protect individual
OSNs users from online privacy leaks. Consequently, an ontology-based
classification approach is comparatively a new research field of privacy
information classification on OSNs. Ontology-based classification approaches
show outstanding performance in many areas, e.g., online job offers [3] and
trust requirements on semantic web services [4]. Whereas, it is necessary to
build an ontology into the OSNs [10].

[5] survey the privacy ontology and point out that although some security
ontologies for fulfilling security requirements have been presented, these
studies focus on security rather than privacy [19]. They present a novel
privacy ontology to identify the key concepts and relations to satisfy the
privacy requirement. However, it aims to deal with the privacy requirements
for software engineers. Actually, there is little research about privacy
information ontology which can apply for OSNs.

Therefore, in this paper, we aim to build an ontology about privacy
information on online social media. Because almost all the privacy-related
entities are words or phrases, the semantic similarity degree calculating
plays a significant role in the proposed ontology-based privacy approach.

##  3 Automated Hybrid Privacy Detection Framework

The proposed automated hybrid privacy information detection and classification
approach are demonstrated in Fig 1. There are two key parts in the proposed
framework, i.e., the privacy-related entities recognition and further
classification based on ontology models. In the former, a deep learning model
is trained to detect four types of entities that potentially cause privacy
leakage. While, in the latter, a privacy ontology model is developed based on
the analysis of messages posted by the OSN users.

Figure 1: Automated Hybrid Privacy Detection Framework

The rationale of utilising a hybrid model of both deep learning and ontology-
based classification is clarified as follows: ontology-only approach
classifies privacy information merely based on the domain-specific vocabulary
of terms or concepts, which have to be systematically defined [17]. Deep
learning based NER does not necessarily require a lexicon or domain-specific
words, but it is difficult to control the classification of specific terms as
deep learning models turn out to be a “black-box”. By considering the factors
mentioned above, a combination of deep learning and ontology model is adopted.

The details of these two key modules are introduced in the following
subsections.

###  3.1 Deep-learning Based Privacy Information Detection

Users’ data collection turns out to be the preliminary of deep-learning based
privacy information detection. OSN users’ public data can be collected from
OSNs through web crawlers or available APIs. The collected raw data are
supposed to be filtered and pre-processed, e.g., removing messages which are
advertisements or spams, removing meaningless words and characters and parsing
word sequences to tokens.

Next, the processed data are enriched by running through the pre-annotation
process if a privacy-detection NER model is already available. Given the pre-
annotated dataset, privacy-related entities are required to be annotated
manually, and the data size of the training and testing data for the deep
learning model are nearly 20k. As mentioned previously, in the context of
OSNs, “privacy” is associated with an individual’ s personal information,
preferences, events that he or she involved. Thus, messages containing four
types of entities, i.e., “PERSON”, “TRAIT”, “PREF”, and “EVENT”, potentially
cause privacy leakage. After annotation, the annotated dataset is then fed
into the deep learning model for training.

###  3.2 Ontology-based Privacy Information Classification

As the second part of our hybrid privacy detection approach, an ontology-based
classification approach can remind individual users regarding what is to be
disclosed instead of simply giving general information.

####  3.2.1 The Domain and Scope of The Privacy Domain Ontology

The defining domain and scope of “privacy” is the first step to build a
private-related information ontology model [17]. Naturally, the privacy
information concepts describing different subclasses (class corresponds to the
entity in this thesis) of the four privacy-related entities will be fed into
our ontology. Specifically, the privacy domain ontology on OSNs includes:

  1. 1.

A hierarchical classification of privacy concepts from general classes to
specific subclasses.

  2. 2.

A set of relations between privacy classes to link concepts in a more
complicated way that implied by an underlying hierarchy.

####  3.2.2 Privacy-related Keywords Extraction

Initially, it is significant to obtain a comprehensive list of privacy-related
terms and concepts in order to form the hierarchy of privacy ontology. To
construct an ontology, we extracted all the values of privacy-related entities
recognized by the deep learning based model. Next, based on the word
frequencies, representative keywords are selected as the major indicators of
the subcategories of the entities. For example, we want to find some keywords
regarding private events under the “Event” main class. Some verbs
representative of private events, e.g., eating, shopping, etc., as well as
some nouns, e.g., concert, meeting, journey, etc., are frequently mentioned in
event-related entities. Additionally, some words are significant indicators of
privacy-related entities, which can imply the user is leaking his/her ’TRAIT’
sensitive information.

Because the creation process of an ontology is an interactive process [12], we
searched the selected keywords in the dataset to find out the occurrence of
these words and how important of them according to the term frequency [17].
Through the interactive procedure, the terms and concepts of the privacy
information ontology are finally determined. For example, because the keyword
of “interview” frequently appears in the extracted entities, it turns out to
be a keyword, representing the subclass of “Corporate Event”.

Table 1 shows the representative keywords extracted from the collected data,
which are used for building the privacy domain ontology.

Table 1: Corresponding Keywords with Classes and Subclasses Class | Subclass |  Keywords
---|---|---
Person | Individual |  I
| Third Party |  you, we, they, he, she, classmate, uncle
Preference | Item |  book, chocolate, keyboard, tea
| Hobby |  cosplay, paint, fishing, dancing, reading
| Specific Person |  girlfriend, teacher
Event | Private Event |  eat, shopping, concert, movie, exercise, spa
| Corporate Event |  wedding, interview, meeting, conference, festival, party, parade, salon
| Journey |  fly, holiday, travel, island, hotel, airport
Trait | Individual Identity |  years-old, Auckland
| Linked Information |  lawyer, female, gay, Christian, married, white, disable

####  3.2.3 Privacy Ontology

Among the possible approaches in developing a class hierarchy, a top-down
process has been selected by considering the relationships among the privacy
concepts in this paper [17]. The ontology hierarchy presents a tree structure,
having most general classes on top and specific associative classes connected
with the general ones. For example, given “PERSON” as a superclass,
“Individual” and “Third Party” can be the subclasses based on the recognised
entity-value pairs. Three other superclasses, i.e., “TRAIT”, “EVENT”, and
“PREFERENCE”, are also included.

TRAIT describes personally identifiable information(PII) or sensitive personal
information, which is defined and classified into two types from the usage of
the PII in United Stated legal fields, i.e., distinguish identity and relating
information [14]. Similarly, TRAITS can be classified into two subclasses as
below:

  1. 1.

Any information can be used to distinguish an individuals identity, such as
birth date and hometown.

  2. 2.

Any information can be linked to an individual, such as medical, educational,
marital status and employment information.

Through this kind of classification approach, two subclasses of Trait are
identified: Individual identity and Linked information. For example, the date
of birth can be recognised as an individuals identity. Whereas, race, gender,
sexual orientation, marital status, religion, belief, and education background
are categorized as linked information. Similarly, EVENT can be classified as a
private event, corporate event, and journey in terms of the event is social or
non-social. Among the subclasses of “Event”, tweets about the journey is
individually classified because we think users who reveal their journey plans
will make them very vulnerable to theft crimes. Preference can be classified
as a specific person, item and hobby according to the characteristic of the
leaking hobby information. Therefore, the privacy domain ontology for OSNs is
presented in Fig. 2:

Figure 2: Privacy Ontology

####  3.2.4 Semantic Phrase Similarity Degree Based On GloVe

To classify the privacy-related entities into fine-grained subclasses, it is
essential to find an approach in finding the subclass where the entities have
the highest belonging degree. Moreover, the highest belonging degree is
decided by the highest similarity degree between the extracted entities and
the representative terms. Comparing the similarity of word embedding of each
word can decide their similarity degree [9]. Word embedding is defined as
semantic vector space models that use vectors to represent each word.

Consequently, in this subsection, we will propose a semantic phrase similarity
degree approach based on Glove. It is divided into two steps. Firstly, a word
semantics vector space model is decided. Secondly, the construction of the
classification model will be described by taking “Event” entity as an example.

$\bullet$ Word Semantics Vector Space Model

In the first step, the main step is to choose a word semantics vector space
model. A pre-trained statistical model (called “en_core_web_lg” in spacy) is
used in this paper, which is trained on blogs, news, and comments with GloVe.
Global Vectors for Word Representation (GloVe) is a state-of-art tool using
word embedding techniques. Most word embedding approaches like Word2Vec exist
a disadvantage, which is the lack of co-occurrence between words. Luckily, the
GloVe approach trains global word-word co-occurrence counts which fills this
gap, outperforming than other current word embedding approaches in common word
similarity tasks[18] [15]. That is the reason we choose this word semantics
vector space model for us to use in order to compare the similarity degree
between words.

$\bullet$ The Construction of The Classification Model

To explain the construction of the classification model, we make use of
“Event” entities as an example. In Fig. 2, the entity consists of three
subclasses, “Private Event”, “Corporate Event” and “Journey”. Moreover, in
Table 1, there are 20 representative terms representing the subclasses
associated with them. Each representative term can be represented as
$term_{i}$, where $i$ belongs to {0,1,2,…,19}. Among them, ’Private
Event’:{0,1,2,…,5}, ’Corporate Event’:{6,7…,13}, and ’Journey’:{14,15…,19}.
Similarly, each word in the extracted entities can be represented as
$entity_{j}$.

Consequently, $S_{ij}$ can represent the semantic similarity degree between
each word in the extracted entities and each representative term:

$S_{ij}=similarity(entity_{j},term_{i})$ (1)

So the similarity of event-related entities and the event representative terms
in the ontology can be shown as follows:

$S_{i(sum)}=\sum_{j=1}^{n}S_{ij}$ (2)

After the calculation of all the similarity degree, then the subclass of the
extracted event entities can be decided according to the maximum degree of
$S_{i(sum)}$, which means the subclass which obtains the maximum $S_{i(sum)}$
is the corresponding subclass of the event-related entity:

$\lambda=\max\left(S_{0}{(sum)},S_{1}{(sum)},...,S_{19}{(sum)}\right)$ (3)

Then the $i$ which obtains the maximum degree of $S_{i(sum)}$ can be decided
and the corresponding subclass can be decided as below:

$subclass=\left\\{\begin{array}[]{ll}PrivateEvent&\textrm{$i\in{0,1,...,5}$}\\\
CorporateEvent&\textrm{$i\in{6,7,...,13}$}\\\
Journey&\textrm{$i\in{14,15,...,19}$}\end{array}\right.$

Hence, by using the semantic information which the word embedding technique
capture [21], we can classify privacy-related information into the
corresponding subclass in the privacy ontology like the “Event” procedure we
list.

##  4 Experiments

Experiments have been conducted to evaluate the proposed hybrid privacy
detection approach. The experiment uses a real-world testing dataset that aims
to classify fine-grained privacy-related entities. Moreover, we also focus on
one interesting problem around private tweets: What type of personal private
information is leaked most on OSNs?

###  4.1 Data Description

Twitter 111https://twitter.com/ is one of the largest Online Social Media
platforms as a micro-blogging service, where a large amount of information is
broadcast publicly by individual users. In Twitter, the information posted by
end users is named as tweets. Twitter provides APIs, allowing developers to
search and store tweets based on certain criteria. Therefore, we search and
collect 18k tweets from Twitter through API. Most tweets are selected by
searching for keywords related to sensitive activities and plans.

Consequently, we use Twitter API to search for some terms which contain
sensitive keywords, e.g., sensitive activities and plans, which may result in
privacy information leakage and cause negative consequences. Then we collect
around 18k tweets as a testing dataset in this experiment. As we demonstrated
before, the tweets will be conducted with the deep learning based NER approach
and the ontology-based classification approach in this chapter, then the
corresponding subclasses will be decided.

###  4.2 Evaluation

To evaluate the performance of the hybrid privacy detection approach, we
manually annotate the subclasses of privacy leaking information. Moreover, the
”ground truth” is prepared by allowing the users themselves to provide
opinions on whether they leak the privacy and what types of private
information they are leaking on tweets. For example, a tweet “I watch a
movie.”, has “I” annotated as “Individual” and “watch a movie” annotated as
“Private Event”.

Four traditional measures to utilised to evaluate the performance of the
proposed hybrid approach:

  1. 1.

Accuracy: the fraction of correct classification in the testing data set;

  2. 2.

Precision: the fraction of correct classification among all results are
classified in this subclass in the testing data set;

  3. 3.

Recall: the fraction of correct classification among all actual results belong
to this subclass in the testing data set;

  4. 4.

F1-value: the harmonic value of precision and recall, which is a balance
measurement.

###  4.3 Experimental Results

Our hybrid privacy classification approach utilizes a deep learning based NER
approach and an ontology-based approach to perform classification of specific
privacy information. After the NER, we use the ontology vocabulary for
performing semantic phrase similarity degree calculation with the extracted
privacy-related entities.

We evaluate our hybrid approach on the testing dataset and get a considerable
performance, as shown in Table 2, with high accuracy in each type. We believe
this accuracy is high enough to demonstrate the effectiveness of our automated
detection and classification approach. However, we observe the accuracy of
categories under “TRAIT” entity is much lower than other types of entities.
The extraction result of “TRAIT” entity is lower than other types of entity
and a trait-related privacy entity will be classified to categories under
other types of entities. That is why the accuracy value of classification of
“TRAIT” entity is lower than other entities.

Table 2: Performance of Hybrid Privacy Information Classification Class |  Subclass |  Accuracy
---|---|---
Person |  Individual |  0.94
|  Third Party |  0.85
Preference |  Item |  0.82
|  Hobby |  0.81
|  Specific Person |  0.76
Event |  Private Event |  0.74
|  Corporate Event |  0.76
|  Journey |  0.77
Trait |  Individual Identity |  0.62
|  Linked Information |  0.64

###  4.4 Privacy Leaking Information Categorization

After the whole automated hybrid privacy information detection approach, all
the types of private information leaks on the 18k testing dataset can be
extracted. Then we plot the distribution of the results of the privacy
information types of the testing dataset in Fig. 3, which includes all the
subclasses under the “TRAIT”, “PREF”, and “EVENT” entities. According to the
privacy rule in this detection approach, each tweet with privacy-related
entities which is identified as the private tweet contains the “Person”
entity, so the distribution of the “Person” entity is not necessary to be
analyzed its categorization.

Figure 3: Distribution of Different Types of Privacy Information Leaking

In this section, we also explore the question of what type of sensitive
information users leak most. We counted the percentage of the eight types of
privacy information in the 18k testing dataset. The results are shown in Fig.
3. From Fig. 3, we can see the most leaking information is the “Event” entity,
where “Private Event” counts the most in it. Additionally, “Journey” under the
“Event” and “Specific person” under the “Preference” are also leaked a lot in
tweets, which means the information about all subclasses under the entity of
“Event” both is leaked a lot on OSNs. From the results, we suggest that OSNs
users should exercise a little more restraint about posting relevant tweets
about these types. On the other hand, people are more conservative about
“Individual Identity” and “Hobby” information because the privacy leaks (as
the percentage) of the two types is much smaller than other types.

##  5 Conclusions and Future Works

In this paper, we propose a hybrid approach for classifying private
information generated by OSN users. Through characterising the nature of
privacy information leaking on OSNs, a deep learning model has been employed
for privacy-related entities recognition, and an ontology-based classification
approach is conducted to automatically classify fine-grained privacy
information, i.e., nine subtypes of private leaking. The ontology-based
approach calculated the semantic similarity between entities extracted from
the deep learning model and the representative terms. We evaluated the result
with the accuracy value, which demonstrates it gains a considerable
performance. Moreover, what specific types of personal private information
users are leaking on OSNs can be understood.

This research can be extended by investigating the following directions.
Firstly, we can recognise privacy-related entities on tweets other than on all
tweets by one user. In the future, all privacy information revealing by a user
can be collected by the model and protect the user. Secondly, different tweets
are associated with different degrees of privacy leakage. In this paper, we
demonstrate what types of privacy-related entities OSNs users reveal. However,
we do not analyse the detailed privacy leakage degree of users. In the future,
we plan to evaluate the privacy leakage degree and explore the insights based
on predictive results.

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