The article information
 Yuxiang Weng
 翁羽翔
 Detection of Electronic Coherence via TwoDimensional Electronic Spectroscopy in Condensed Phase
 凝聚相中电子相干态的二维电子光谱测量
 Chinese Journal of Chemical Physics, 2018, 31(2): 135151
 化学物理学报, 2018, 31(2): 135151
 http://dx.doi.org/10.1063/16740068/31/cjcp1803055

Article history
 Received on: March 30, 2018
 Accepted on: April 21, 2018
Electronic coherence arising from the superposition of the excitedstate wave functions of the atoms or molecules has been gradually shaped from a theoretical concept of quantum to the experimentally detectable reality, thanks to the advent of the ultrafast pulsed lasers. Among the various timeresolved spectroscopic methods [14], twodimensional electronic nonlinear spectroscopy is one of the powerful tools in detection of electronic coherence, which has been widely used to study the coherent electronic energy transfer in a variety of photosynthetic lightharvesting systems [510], organic coherent intrachain energy migration in a conjugated polymer [11], exciton valley coherence in monolayer WSe
In measuring quantum coherence with ultrafast laser pulses, the information expected from the measurement at least should include (ⅰ) coherence time which can be as short as tens of femtosecond, and (ⅱ) the multiple energy levels comprising the coherence states. However, it seems that, in principles, there are two physical barriers that have to be overcome. One is the uncertainty principle; the other is the collapse of the wavepacket when subjected to any external perturbation like measurement. For the uncertainty principle, roughly speaking, it states that one cannot simultaneously determine the exact values of a pair of conjugate observables such as momentum and position of a physical system. Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals. The duality relations lead naturally to an uncertainty relation between them, in physics called the Heisenberge uncertainty principle [24]. While in matrix mechanics, this corresponds to that pair of observables whose operators do not commute [25]. The fact that short pulses probe dynamics and long pulses probe energy levels is indicative of the uncertainty principle, that time and energy resolution are related to each other through the Fourier transform [26].
A. Uncertainty Principle forThe Heisenberg uncertainty principle is a relationship between certain types of physical variables like position and momentum. The most wellknown expression takes the position and momentum to be the conjugate variables. Another uncertainty relation which is often referenced in discussion of quantum mechanics is the energytime uncertainty principle, which is directly related to the coherence detection. It is tempting to interpret this energytime relation as the statement that a system may fluctuate in energy by an arbitrarily large amount over a sufficiently short time scale. This explanation is often given as a description for particleantiparticle production and annihilation, where a particle and its antiparticle appear spontaneously from the vacuum briefly via "borrowed" energy before colliding and returning to vacuum. However, this explanation is not very precise and the given inequality is not so welldefined in quantum mechanics despite the nice physical interpretation. The reason that is not welldefined is because there is no operator in quantum mechanics corresponding to the measurement of time, although the Hamiltonian is the operator corresponding to energy. Nevertheless, there are some ways to make sense of an energytime uncertainty principle by considering how the measurement of an arbitrary operator changes in time.
Since time is not an operator, it is unclear how time enters quantum mechanics at all. The answer is that time is incorporated into the Schrödinger equation, where it describes the time rate of change of a wave function. Physically, the passage of time is recorded by noting that certain physical observables are changing over time: for instance, perhaps the position of a particle is changing, which one interprets as motion over time, or the momentum of a particle is changing, which one interprets as accelerating or decelerating over time.
To quantify this statement, consider the Ehrenfest theorem governing the dynamics of the expectation value of an operator in terms of the commutator with the Hamiltonian [27]:
$ \begin{eqnarray} \frac{\textrm{d}}{{\textrm{d}t}}\left\langle A \right\rangle = \frac{i}{\hbar }\left\langle {\left[{\hat H, \hat A} \right]} \right\rangle + \left\langle {\frac{{\partial \hat A}}{{\partial t}}} \right\rangle \end{eqnarray} $  (1) 
Assuming that
$ \begin{eqnarray} \sigma _H^2 \sigma _A^2 \ge \left {\frac{1}{{2i}}\left\langle {\left[{\hat H, \hat A} \right]} \right\rangle } \right^2 \hspace{0.15cm}& =&\hspace{0.15cm}\left {\frac{1}{{2i}}\frac{\hbar }{i}\frac{{\textrm{d}\left\langle {\hat A} \right\rangle }}{{\textrm{d}t}}} \right^2 \nonumber\\ &=&\hspace{0.15cm} \frac{{\hbar ^2 }}{4}\left {\frac{{\textrm{d}\left\langle {\hat A} \right\rangle }}{{\textrm{d}t}}} \right^2 \end{eqnarray} $  (2) 
Notably, since the Hamiltonian is the energy operator,
Taking square roots now gives the relation:
$ \begin{eqnarray} \sigma _H \sigma _A \ge \frac{\hbar }{2}\left {\frac{{\textrm{d}\left\langle {\hat A} \right\rangle }}{{\textrm{d}t}}} \right \end{eqnarray} $  (3) 
Define
$ \begin{eqnarray} \begin{array}{l} \displaystyle{\sigma _A = \left {\frac{{\textrm{d}\left\langle {\hat A} \right\rangle }}{{\textrm{d}t}}} \right\sigma _t } \\ \displaystyle{\frac{{\sigma _A }}{{\textrm{d}\left\langle {\hat A} \right\rangle /\textrm{d}t}} \approx \frac{{\sigma _A }}{{\sigma _A /\sigma _t }} = \sigma _t} \\ \end{array} \end{eqnarray} $  (4) 
In other words,
$ \begin{eqnarray} \sigma _H \sigma _t \ge \frac{\hbar }{2} \end{eqnarray} $  (5) 
Ultrashort pulse excitation creates a linear superposition of the eigenstates within the spectrum of the laser rather than a single eigenstate of the system excited by continuous working laser. The linear superposition is such as to create wave packet. The superposition principle states that a system is in all possible states at the same time, until it is measured [27]. In fact, such states are very fragile in the presence of dissipation, and would rapidly collapse to eigenstates after measurement, thus destroying the original configuration leaving no unusual interference features. This is also known as the collapse of the wave packet: when the system is excited with a laser pulse expressed as a pulse electric field
$ \begin{eqnarray} \psi = \sum\limits_{n = 1} {a_n \varphi _n } \end{eqnarray} $  (6) 
where
$ \begin{eqnarray} a_n = A\left\langle {\varphi _n } \right\left. {\chi _0 } \right\rangle \int\limits_{  \infty }\hspace{0.15cm}&&\hspace{0.15cm} \exp [i(\omega _n\omega _0 )t] \times\nonumber\\&& \exp (  t^2 /\alpha \tau ^2 )\cos (\omega t)\textrm{d}t \end{eqnarray} $  (7) 
$ \begin{eqnarray}a_n \hspace{0.15cm}&=&\hspace{0.15cm} A\left\langle {\varphi _n } \right\left. {\chi _0 } \right\rangle \exp [(\omega _n\omega _0 )^2 \alpha \tau ^2 /4] \end{eqnarray} $  (8) 
$ \begin{eqnarray} \psi = A\sum\limits_{n = 1}^{} {\left\langle {\varphi _n } \right\left. {\chi _0 } \right\rangle \varphi _n } = \sum\limits_{n = 1} {c_n \varphi _n } \end{eqnarray} $  (9) 
where
$ \begin{eqnarray} c_n = A\left\langle {\varphi _n } \right\left. {\chi _0 } \right\rangle \end{eqnarray} $  (10) 
When considering the timeevolution, one has
$ \begin{eqnarray} \psi (t) = \sum\limits_{n = 1}^{} {c_n \exp \left( {  iE_n t/\hbar } \right)\varphi _n } \end{eqnarray} $  (11) 
This wave packet is a coherent superposition of excited eigenstates. Over time, these states dephase with respect to one another.
Ⅲ. HOMOGENEOUS AND INHOMOGENEOUS SPECTRAL BROADENING CAUSED BY ELECTRONIC DEPHASINGThe following describes the concepts of line broadening and optical dephasing pertaining to an ensemble comprising of twolevel systems [28]. Considering an ensemble of identical oscillators that do not interact with the surrounding environment or each other, the system is free to evolve over time and will be correlated for long times, limited only by the natural lifetime
$ \begin{eqnarray} \frac{1}{{T_2 }} = \frac{1}{{2T_1 }} + \frac{1}{{T_2^* }} \end{eqnarray} $  (12) 
In the second hypothetical scenario each oscillator has a slightly different local environment, which does not change significantly over the course of time. In this case, the interaction between the inhomogeneous environments and the oscillators results in a shift in the energy separation away from the unperturbed energy
An absorption or fluorescence spectrum of a sample in a real solution may have both homogeneous and inhomogeneous characteristics. One could imagine a hypothetical twolevel system between the two limiting cases in which a distribution of inhomogeneous environments arises from extremely slow solvent motions while dynamical interactions from faster solvent motions occur within each environment. Unfortunately, the linear response of system does not distinguish between dephasing processes of different time scales. The advantage of some nonlinear techniques is their ability to selectively eliminate inhomogeneous contributions and extract only the part for which dynamical (homogeneous) interactions dominate.
Ⅳ. PRINCIPLE OF 3PPEThe molecular system for nonlinear optical measurement is not a molecule of eigenstates, but a statistical ensemble of the inhomogeneous system, the temporal evolution of the ensemble can be described by the density function theory. Due to the ensemble inhomogeneity, any optical information retrieval occurs in the form of a photon echo [30], where photon echo is one of the first examples of an optical analogue of NMR.
For an ensemble of twolevel systems characterized by a ground state
$ \begin{eqnarray}\left {\psi (t)} \right\rangle = c_g e^{  i\omega _g t} \left g \right\rangle + c_e e^{  i\omega _e t} \left e \right\rangle \end{eqnarray} $  (13) 
It should be noted that the observables corresponding to operators that do not commute with the Hamiltonian, such as the dipole operator, will oscillate over time with frequency
$ \begin{eqnarray} \rho &=& \sum\limits_{ij} {\left\langle {c_i (t)c_j^* (t)} \right\rangle } \left {\varphi _i } \right\rangle \left\langle {\varphi _j } \right \nonumber\\ &= &\sum\limits_{ij} {\rho _{ij} \left {\varphi _i } \right\rangle \left\langle {\varphi _j } \right} \end{eqnarray} $  (14) 
In the matrix representation of the operator, each diagonal element
$ \begin{eqnarray} \frac{{\partial \rho _{nm} }}{{\partial t}} \hspace{0.15cm}&=& \hspace{0.15cm}\frac{i}{\hbar }\left({\sum\limits_k {\rho _{nk} H_{km}  H_{nk} \rho _{km} } } \right) \nonumber\\ &=&\hspace{0.15cm} \frac{i}{\hbar }\left[{\rho _n H_mH_n \rho _m } \right] \nonumber\\ &=&\hspace{0.15cm}\frac{i}{\hbar }\left[{\hat \rho, \hat H} \right] \end{eqnarray} $  (15) 
where
$ \begin{eqnarray}\rho _{nm} = c_n c_m^* \end{eqnarray} $  (16) 
Considering an optical transition in a twolevel system, e.g., the ground state and an excited state of a dye molecule having two different energies
Using
$ \begin{eqnarray} V \hspace{0.15cm}&= &\hspace{0.15cm}  \mu \cdot {\bf{E(r, t)}} \end{eqnarray} $  (17) 
$ \begin{eqnarray} \mu _{gg} \hspace{0.15cm} &=&\hspace{0.15cm} \mu _{ee} = 0 \end{eqnarray} $  (18) 
$ \begin{eqnarray} \mu _{ge} \hspace{0.15cm}&=&\hspace{0.15cm}\ \mu _{eg} = \mu \end{eqnarray} $  (19) 
When molecules interact with electrical field
$ \begin{eqnarray} \left\langle \mu \right\rangle &=& \textrm{tr}(\rho \mu ) \nonumber\\ &=& \rho _{ge} \mu _{eg} + \rho _{eg} \mu _{ge} + \rho _{gg} \mu _{gg} + \rho _{ee} \mu _{ee} \nonumber\\ &=& \mu (\rho _{ge} + \rho _{eg} ) \end{eqnarray} $  (20) 
Then consider the radiation field whose frequency fulfills
Introducing Rabi frequency
$ \begin{eqnarray} \Omega \hspace{0.15cm}&=& \hspace{0.15cm}\frac{V}{\hbar }\nonumber\\ \hspace{0.15cm}&=& \hspace{0.15cm}\frac{{  {\bf{ \pmb{\mathit{ μ}} }} \cdot {\bf{E}}(r, t)}}{\hbar } \end{eqnarray} $  (21) 
$ \begin{eqnarray} H(t) \hspace{0.15cm}&=& \hspace{0.15cm}H_0 + V(t) \end{eqnarray} $  (22) 
$ \begin{eqnarray} \frac{{\textrm{d}\rho _{gg} }}{{\textrm{d}t}}\hspace{0.15cm}& = &\hspace{0.15cm}  \frac{i}{\hbar }(V_{ge} \rho _{eg}  V_{eg} \rho _{ge} ) \end{eqnarray} $  (23) 
$ \begin{eqnarray} \frac{{\textrm{d}\rho _{ee} }}{{\textrm{d}t}}\hspace{0.15cm}& =&\hspace{0.15cm}  \frac{i}{\hbar }(V_{eg} \rho _{ge}  V_{ge} \rho _{eg} ) \end{eqnarray} $  (24) 
$ \begin{eqnarray} \frac{{\textrm{d}\rho _{ge} }}{{\textrm{d}t}} \hspace{0.15cm}&=&\hspace{0.15cm}  \frac{i}{\hbar }(E_g  E_e )\rho _{ge}  \frac{i}{\hbar }V_{ge} (\rho _{ee}  \rho _{gg} ) \nonumber\\ &=& i\omega _{eg} \rho _{ge}  i\Omega (\rho _{ee}  \rho _{gg} ) \end{eqnarray} $  (25) 
$ \begin{eqnarray} \frac{{\textrm{d}\rho _{eg} }}{{\textrm{d}t}} &=&  \frac{i}{\hbar }(E_e  E_g )\rho _{eg}  \frac{i}{\hbar }V_{eg} (\rho _{gg}  \rho _{ee} ) \nonumber\\ &=&  i\omega _{eg} \rho _e g + i\Omega (\rho _{ee}  \rho _{gg} ) \end{eqnarray} $  (26) 
Apparently the solutions of
$ \begin{eqnarray} \hat \rho _{gg} &=& \rho _{gg} \end{eqnarray} $  (27) 
$ \begin{eqnarray} \hat \rho _{ee} &= &\rho _{ee} \end{eqnarray} $  (28) 
$ \begin{eqnarray} \tilde \rho _{ge}&\equiv&\textrm{e}^{  i\omega t} \cdot \rho _{ge} (t) \end{eqnarray} $  (29) 
$ {\tilde \rho _{eg}}{\rm{ }} \equiv {\rm{ }}{{\rm{e}}^{i\omega t}} \cdot {\rho _{{e_g}}}(t) $  (30) 
Then we have
$ \begin{eqnarray} \frac{{\textrm{d}\tilde \rho _{ge} }}{{\textrm{d}t}} =  i(\omega  \omega _{eg} ) \cdot \tilde \rho _{ge}  i\textrm{e}^{  i\omega t} \cdot \Omega \cdot (\rho _{ee}  \rho _{gg} ) \end{eqnarray} $  (31) 
Rewrite the cosine form of the electrical field in the Rabi frequency into a complex form:
$ \begin{eqnarray} \Omega (t) \hspace{0.15cm}&=&\hspace{0.15cm} \frac{1}{2} \cdot \left \Omega \right \cdot (\textrm{e}^{i\omega t} +\textrm{ e}^{  i\omega t} ) \end{eqnarray} $  (32) 
$ \begin{eqnarray} \tilde \Omega (t) \hspace{0.15cm}&=&\hspace{0.15cm} \Omega (t) \cdot \textrm{e}^{  i\omega t} = \frac{1}{2} \cdot \left \Omega \right \cdot (1 + \textrm{e}^{  2i\omega t} ) \end{eqnarray} $  (33) 
$ \begin{eqnarray} \frac{{\textrm{d}\tilde \rho _{ge} }}{{\textrm{d}t}} =  i\Delta \cdot \tilde \rho _{ge}  \frac{i}{2}\Omega \cdot (\rho _{ee}  \rho _{gg} ) \end{eqnarray} $  (34) 
$ \begin{array}{l} \frac{{\rm{d}}}{{{\rm{d}}t}}\left[{\begin{array}{*{20}{c}} {{{\tilde \rho }_{ge}}}\\ {{{\tilde \rho }_{eg}}}\\ {{{\tilde \rho }_{gg}}}\\ {{{\tilde \rho }_{ee}}} \end{array}} \right] =  i\left[{\begin{array}{*{20}{c}} \Delta &0&{\tilde \Omega (t)}&{\tilde \Omega (t)}\\ 0&{\Delta }&{{{\tilde \Omega }^*}(t)}&{{{\tilde \Omega }^*}(t)}\\ {  {{\tilde \Omega }^*}(t)}&{\tilde \Omega (t)}&0&0\\ {{{\tilde \Omega }^*}(t)}&{  \tilde \Omega (t)}&0&0 \end{array}} \right]\left[{\begin{array}{*{20}{c}} {{{\tilde \rho }_{ge}}}\\ {{{\tilde \rho }_{eg}}}\\ {{{\tilde \rho }_{gg}}}\\ {{{\tilde \rho }_{ee}}} \end{array}} \right]\\ =  i\left[{\begin{array}{*{20}{c}} \Delta &0&{\frac{1}{2}\Omega }&{\frac{1}{2}\Omega }\\ 0&{\Delta }&{\frac{1}{2}\Omega }&{\frac{1}{2}\Omega }\\ {  \frac{1}{2}\Omega }&{\frac{1}{2}\Omega }&0&0\\ {\frac{1}{2}\Omega }&{  \frac{1}{2}\Omega }&0&0 \end{array}} \right]\left[{\begin{array}{*{20}{c}} {{{\tilde \rho }_{ge}}}\\ {{{\tilde \rho }_{eg}}}\\ {{{\tilde \rho }_{gg}}}\\ {{{\tilde \rho }_{ee}}} \end{array}} \right] \end{array} $  (35) 
The resulting Liouvillevon Neumann equation can now be recast in the form
$ \frac{{{\rm{d}}\tilde \rho }}{{{\rm{d}}t}}{\rm{ }} = \frac{i}{\hbar }\left[{{{\hat H}_{eff, }}\hat \rho } \right] $  (36) 
$ {\hat H_{eff}} = \left[{\begin{array}{*{20}{c}} {\hbar \Delta }&{\frac{\hbar }{2}\Omega }\\ {\frac{\hbar }{2}\Omega }&0 \end{array}} \right] $  (37) 
In this model, no relaxation is introduced so far. Now we introduce relaxation by two phenomenological constants. The population of the excited state decays to the ground state at a decay rate written as 1/
$ \begin{eqnarray} \begin{array}{l} \displaystyle{\frac{{\textrm{d}\tilde \rho _{ee} }}{{\textrm{d}t}} =  \frac{i}{2}\Omega \cdot (\tilde \rho _{eg}  \tilde \rho _{ge} )  \tilde \rho _{ee} /T_1 } \\[2ex] \displaystyle{\frac{{\textrm{d}\tilde \rho _{gg} }}{{\textrm{d}t}} =  \frac{i}{2}\Omega \cdot (\tilde \rho _{ge}  \tilde \rho _{eg} ) + \tilde \rho _{ee} /T_1} \\[2ex] \displaystyle{\frac{{\textrm{d}\tilde \rho _{eg} }}{{\textrm{d}t}} = i\Delta \cdot \tilde \rho _{eg} + \frac{i}{2}\Omega \cdot (\tilde \rho _{ee}  \tilde \rho _{gg} )  \tilde \rho _{eg} /T_2 } \\[2ex] \displaystyle{\frac{{\textrm{d}\tilde \rho _{ge} }}{{\textrm{d}t}} =  i\Delta \cdot \tilde \rho _{ge}  \frac{i}{2}\Omega \cdot (\tilde \rho _{ee}  \tilde \rho _{gg} )  \tilde \rho _{ge} /T_2 } \\ \end{array} \end{eqnarray} $  (38) 
The term
The Bloch vector formalism consists in replacing the density matrix elements by three real components
$\begin{eqnarray} \begin{array}{l} \displaystyle{ \tilde \rho _{gg} = \frac{{1  w}}{2}} \\[2ex] \displaystyle{ \tilde \rho _{ee} = \frac{{1 + w}}{2}} \\[2ex] \displaystyle{\tilde \rho _{eg} = \frac{{u  iv}}{2}} \\[2ex] \displaystyle{ \tilde \rho _{ge} = \frac{{u + iv}}{2}} \\[2ex] \end{array} \end{eqnarray} $  (39) 
or
$ \begin{eqnarray}\begin{array}{l} w = \tilde \rho _{ee}  \tilde \rho _{gg} \\ u = \tilde \rho _{eg} + \tilde \rho _{ge} \\ v = i(\tilde \rho _{eg}  \tilde \rho _{ge} ) \\ \end{array} \end{eqnarray} $  (40) 
Notice that
The Bloch vector equations of motion thus can be written as
$ \begin{eqnarray} \begin{array}{l} \dot u =  u/T_2 + \Delta v + \Omega w \\ \dot v =  \Delta u  v/T_2  \Omega w \\ \dot w = \Omega (v  u)  (1 + w)/T_1 \\ \end{array} \end{eqnarray} $  (41) 
$ \begin{eqnarray} \left[{\begin{array}{*{20}c} {\dot u} \\ {\dot v} \\ {\dot w} \\ \end{array}} \right] = \left[{\begin{array}{*{20}c} 0&\Delta &\Omega \\ {\Delta }&0&{\Omega } \\ {\Omega }&\Omega &0 \\ \end{array}} \right]\left[{\begin{array}{*{20}c} u \\ v \\ w \\ \end{array}} \right]  \left[{\begin{array}{*{20}c} {u/T_2 } \\ {v/T_2 } \\ {(1 + w)/T_1 } \\ \end{array}} \right] \end{eqnarray} $  (42) 
If the relaxation process is disregarded, the optical Bloch equation has the same form as that on NMR by setting
$ \begin{eqnarray} \vec S(t) = \left[{\begin{array}{*{20}c} u \\ v \\ w \\ \end{array}} \right] \end{eqnarray} $  (43) 
$ \begin{eqnarray} \vec \Omega (t) = \left[{\begin{array}{*{20}c} \Omega \\ \Omega \\ {\Delta } \\ \end{array}} \right] \end{eqnarray} $  (44) 
And the coherent evolution part can be represented by
$ \begin{eqnarray} \frac{{\textrm{d}\vec S(t)}}{{\textrm{d}t}} = \vec \Omega (t) \times \vec S(t) \end{eqnarray} $  (45) 
Neglecting the relaxation processes, the length of the Bloch vector is thus
The whole Bloch diagram is rotating around the
The detuning factor
An electric field turns the Bloch vector around the
In photo echo experiments, the
$ \begin{eqnarray} \vec k_e = 2\vec k_2  \vec k_1 \end{eqnarray} $  (46) 
If the temporal order of the two writing beams is changed, the echo is emitted in the direction implied as a dashed arrow above, which provides a convenient way to acquire the rephasing and nonrephaisng signals in the 2DES.
Although in the time sequence, there appear two excitation pulses called twopulse photo echo. However, in terms of nonlinear wave mixing, it consists of three beams, i.e., one beam 1 and two degenerated beam 2 having the same propagating direction and a zero respective delay time. Therefore, by nature, 2PPE is a simplified version of threephoto echo emission.
The threepulse photon echo (3PPE) experiment involves three pulses with time sequence as
FIG. 5 shows the stimulated photon echo pulse sequence along with the 2
In summary, the conventional interpretation of a photon echo experiment represents the state of the system using a pseudo spin vector on the Bloch sphere. By transforming to the rotating frame, each member of the inhomogeneous ensemble precesses away from the origin at a rate proportional to the difference between the specific frequency
3PPE has been shown as potential technique for timedomain data storage [39] and word by word logic processing [40] possibly used in quantum computation. The idea of photonecho memory is to perform spectral holeburning memory [37] in the time domain by utilizing the temporal interference effect of two optical pulses. From the viewpoint of memory, the three excitation pulses are renamed as WRITE, DATA and READ pulses and, rather than being a deltafunction pulse, the DATA pulse carries some kind of temporal information as shown in FIG. 7. Calculation shows that this echo signal simply replicates the shape of the DATA pulse. In terms of memory, the WRITE pulse as a trigger and the DATA pulse with temporal information complete the storage of the data into the medium. The readout process starts by applying a trigger called a READ pulse, and the echo then appears as the OUTPUT DATA. The DATA can be understood in spectral region, since pulse 1 and pulse 2 would create spectral interference which provides spectral selectivity by tuning the relative delay
3PPE is one of implementations of 2DES [41]. In experiment, three ultrashort pulses are arranged in a unique temporal sequence to excite the sample, creating a 3rd order polarization
$ \begin{array}{l} {P^{(3)}}(\tau, T, t) \propto \int\limits_0^\infty {{\rm{d}}\tau \int\limits_0^\infty {{\rm{d}}T} } \int\limits_0^\infty {{\rm{d}}t} {R^{(3)}}(\tau, T, t){E_1}({{\vec k}_1}, \tau )\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{E_2}({{\vec k}_2}, T){E_3}({{\vec k}_2}, t) \end{array} $  (47) 
where the third order nonlinear response function is expressed as a function of the dipole moment and the density matrix of the unperturbed system
$ \begin{gathered} {R^{(3)}}({t_3}, {t_2}, {t_1}) = {\left( { \frac{i}{\hbar }} \right)^n}\left\langle {\mu ({t_3} + {t_2} + {t_1})[\mu ({t_2} + {t_1}), } \right. \hfill \\ \left. {\left[{\mu ({t_1}), [\mu ({t_0}), \rho (\infty )]]]} \right.} \right\rangle \hfill \\ \end{gathered} $  (48) 
Apparently,
The fourth homologous laser pulse serves as a local oscillator to detect the emitted signal by means of optical heterodyne detection as shown in FIG. 8 [42]. The detected signal by the spectrometer
$ \begin{eqnarray} \tilde S(\tau, T, \omega _t ) &=& i \cdot sig(\omega _t )\int_{  \infty }^\infty {P^{(3)} (\tau, T, t)\exp (i\omega _t t)\textrm{d}t} \nonumber\\ &=&i \cdot sig(\omega _t )\tilde P^{(3)} (\tau, T, \omega _t ) \end{eqnarray} $  (49) 
where the thirdorder polarization is related to the measured signal field by
$ \begin{eqnarray} \int_{  \infty }^\infty {P^{(3)} (\tau, T, \omega _t )\exp (i\omega _\tau t)\textrm{d}t} {\rm{ = }}\tilde P^{(3)} (\omega _\tau, T, \omega _t ) \end{eqnarray} $  (50) 
The 2D spectrum at a particular
The time resolution of dynamics over the waiting time, in both pumpprobe and 2D experiments, is dependent on the duration of the pump pulse, with shorter pulses giving higher time resolution. The commensurately larger bandwidth of a shorter pulse leads to a lower spectral resolution as restricted by the uncertainty principle. 2DFT spectroscopy use the Fourier transform methodology by converting time to the frequency, thus the restriction imposed by the uncertainty principle in 2D spectroscopy is circumvented [43], and the result has both high temporal and spectral resolution, limited only by the signaltonoise ratio. However, the wavepacket collapse to the eigenstates as reflected in both excitation and detection frequency axis at any fixed waiting time,
Doublesided Feynman diagrams describe the evolution of the density matrix in the presence of several coherent fields in time sequence [46, 47] by depicting interaction sequences between external perturbations such as electromagnetic fields and ensemble of molecules. One advantage of using Feynman diagrams is to physically capture the most important interaction paths without referring to the actual form of either the density matrix or the Hamiltonians describing the ensemble.
The drawing of doublesided Feynman diagrams derives from taking separately each term of the commutator expansion (Eq.(47) and Eq.(48)) to create the corresponding diagram, using specific rules: (ⅰ) two vertical lines represent the time evolution of the ket (left line) and bra (right line) of the density matrix, with time running from the bottom to the top. (ⅱ) The interactions between the density matrix and the electric field are indicated by arrows acting either on the bra or ket side pointing towards or away from the system corresponding to absorption or emission. (ⅲ) The ket and bra sides of the diagram are the complex conjugate of each other and the last emission is always from the ket side, by convention. (iv) The last interaction must end in a population state. (v) An arrow pointing to the right represents an electric field with
Considering a molecular system with
Twodimensional spectroscopy has the capability to reveal the quantum coherence as the oscillatory behavior of the excitation dynamics of molecular systems. However, the situation is complicated regarding the exact origin of the observed coherent phenomena, i.e., what is actually being observed: excitonic or vibrational wavepacket motion or evidence of quantum transport. Especially, in some molecules and their aggregates, electronic transitions are coupled to various intraand intermolecular vibrational modes, with the magnitudes of the resonant couplings
Two generic model systems which exhibit distinct internal coherent dynamics have been compared [53]. One model system of pure electronic transition which shows similar spectroscopic properties but has completely different coherent internal dynamics without vibrations, is an excitonic dimer (ED) [54, 55]. It consists of two twolevel chromophores (sites) with identical transition energies
The other model is the vibronic system of an isolated molecular electronic excitation represented by two electronic states,
To reveal oscillatory contributions in the ED and DO systems, all contributions into either oscillatory or static can be compared, shown in FIG. 13(b) and 14(b). As a function of
Generally, the 2DES spectra are presented as the slices of pump and probe frequency projection at a fixed waiting time, and the coherence is given as oscillating amplitude at a certain point on the projection. To study the coherent oscillations, it is more straightforward to view the 2DES spectra along the beating frequency
Single vibrationalmode coupling to the vibronic transition is the simplest model. When multimode coupling is considered, the Hamiltonian as well as the corresponding sets of wave functions for a singlevibrational mode displaced oscillator coupled to the electronic states [5860] needs to be extended to the multimode case [61]. Considering a multimode coupled displaced oscillator in a chromophore molecule with only two electronic states,
$ \begin{eqnarray} H = H_g \left g \right\rangle \left\langle g \right + H_e \left e \right\rangle \left\langle e \right \end{eqnarray} $  (52) 
where
$ \begin{eqnarray} H_g \hspace{0.15cm}&=&\hspace{0.15cm} \sum\limits_i {\left( {\frac{{p_i^2 }}{{2\mu _{gi} }} + \frac{1}{2}\mu _{gi} \omega _{gi}^2 q_i^2 } \right)} \end{eqnarray} $  (53) 
$ \begin{eqnarray} H_e \hspace{0.15cm}& =&\hspace{0.15cm} h\omega _{eg} + \sum\limits_i {\left[{\frac{{p_i^2 }}{{2\mu _{gi} }} + \frac{1}{2}\mu _{ei} \omega _{ei}^2 (q_id_i )^2 } \right]} \end{eqnarray} $  (54) 
Here
The simplest model of multimode coupling in displaced oscillator is the two vibtrational modes system with their vibrational frequencies denoted as
Obviously, since the detected oscillation signals originate from all the possible Liouville pathways including the excitonic and singlemode or multimode vibrationally coupled coherences, new methods either of apparatus or data analysis are called for clear distinguishing the electronic coherence from those of vibronic. It has been shown that the polarization modulated [67, 68] 2DES can be an effective way toward this purpose as successfully demonstrated in a recent experiment, where the oscillation maps extracted from the doublecross polarized schemes (relative polarizations of
Quantum coherence manifests itself not only as a concept but also as a reality in many aspects. An emerging field of quantum computation involves coherence control and the quantum information storage, while 3PPE provides possible quantum information storage in time sequence. While in natural and artificial photosynthetic systems, it also has been suggested that coherent energy transport could be more efficient than classical transport. Development of advanced theories would provide more applicable feasibility while the new advances in the experimental methods and designs of novel quantum coherent systems would speed up the application of quantum coherence. Though quantum coherence was considered fragile, recent evidence of coherence in chemical and biological systems suggests that the phenomena are robust and can survive in the face of disorder and noise, suggesting that coherence can be used in complex chemical systems [71].
Ⅷ. ACKNOWLEDGMENTSThis work was supported by the National Natural Science Foundation of China (No.21227003, No.21433014, No.11721404). Mr. Xuan Leng and Ruidan Zhu are kindly acknowledged for proofreading and Figure preparation.
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