# Investigations of Amplitude and Phase Excitation Profiles in Femtosecond Coherence Spectroscopy

Abstract

We present an effective linear response approach to pump-probe femtosecond coherence spectroscopy in the well separated pulse limit. The treatment presented here is based on a displaced and squeezed state representation for the non-stationary states induced by an ultrashort pump laser pulse or a chemical reaction. The subsequent response of the system to a delayed probe pulse is modeled using closed form non-stationary linear response functions, valid for a multimode vibronically coupled system at arbitrary temperature. When pump-probe signals are simulated using the linear response functions, with the mean nuclear positions and momenta obtained from a rigorous moment analysis of the pump induced (doorway) state, the signals are found to be in excellent agreement with the conventional third order response approach. The key advantages offered by the moment analysis based linear response approach include a clear physical interpretation of the amplitude and phase of oscillatory pump-probe signals, a dramatic improvement in computation times, a direct connection between pump-probe signals and equilibrium absorption and dispersion lineshapes, and the ability to incorporate coherence such as those created by rapid non-radiative surface crossing. We demonstrate these aspects using numerical simulations, and also apply the present approach to the interpretation of experimental amplitude and phase measurements on reactive and non-reactive samples of the heme protein Myoglobin. The role played by inhomogeneous broadening in the observed amplitude and phase profiles is discussed in detail. We also investigate overtone signals in the context of reaction driven coherent motion.

## I Introduction

Femtosecond coherence spectroscopy (FCS) is an ultrafast pump-probe technique that allows the experimentalist to create and probe coherent vibrational motions and ultrafast chemical reactions in real time [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. In a typical pump-probe experiment, an ultrashort pump laser pulse is used to excite the sample of interest. The subsequent non-stationary response of the medium is monitored by an optically delayed probe pulse. Owing to the large spectral bandwidth available in a short laser pulse, it can generate non-stationary vibrational states in a molecular system as shown in Fig. 1(a). The subsequent nuclear dynamics modulates the optical response as detected by the probe pulse. Coherent vibrational motion in the ground state has been observed in crystalline and liquid phase systems [2, 12], and in biological specimens having short excited state lifetimes such as bacteriorhodopsin[8, 9], and myoglobin[14]. For molecules that have long-lived excited states, however, the excited state coherence is dominant and has been identified in several dye molecules in solution[1, 6, 20], in small molecules in the gas phase[5, 7, 10], and in photosynthetic reaction centers[11, 17].

Apart from “field driven” coherence directly prepared by the laser fields, vibrational coherence can also be driven by rapid non-radiative processes. For example, if we consider a third electronic state that is coupled non-radiatively to the photo-excited state as in Fig. 1(b), the wave-packet created in the excited state by the pump can cross over to , leaving a vibrationally coherent product [15]. Fig. 1(b) suggests the importance of taking a multidimensional view of the problem, whereby the surface crossing between the reactant excited state and the product state, along the reaction coordinate is accompanied by the creation of a vibrational coherence along the coordinate that is coupled to the non-radiative transition. In earlier work, we have presented expressions for the time dependent population and first moment evolution of vibrational dynamics following a Landau-Zener surface crossing[21]. These expressions are rigorously valid, provided the quantum yield for the reaction is unity (a condition that holds for NO and CO photolysis from heme proteins[22, 23, 24]).

A common theoretical formulation for pump-probe spectroscopy is based on the third order susceptibility formalism, which provides a unified view of four wave mixing spectroscopies [25, 26, 27, 28] with different combinations of fields, irrespective of whether they are continuous wave or pulsed. However, the separation of the pump and probe events is not clear in this formalism, since the pump induced density matrix is implicitly contained in the third order response functions. Thus, it is also attractive to treat the pump and probe processes separately in the well separated pulse (WSP) limit. For example, the “doorway/window” picture has been developed[28] which can be used to represent the pump and probe events as Wigner phase space wave packets. This readily enables a semi-classical interpretation of pump-probe experiments[29, 30, 31]. Another view of the WSP limit is based on the effective linear response approach. In this approach, the pump induced medium is modeled using a time-dependent linear susceptibility[32, 33, 34, 8, 27]. This has the appealing aspect that a pump-probe experiment is viewed as the non-stationary extension of steady state absorption spectroscopy.

While the above theoretical treatments are primarily concerned with field driven processes, the need to incorporate additional non-radiative mechanisms into the theory is evident from recent experimental observations [14, 21, 35]. Recent studies have addressed population transfer during non-adiabatic transitions in electron transfer systems[36, 37]. Numerical wave packet propagation techniques have also been presented to model coherent dynamics due to non-adiabatic excited state processes[38, 39].

The present work is motivated by the need to interpret FCS experiments of reactive and non-reactive samples of myoglobin[35]. Here, we make the WSP approximation, and demonstrate the usefulness and accuracy of the effective linear response approach, with the non-stationary states represented using unitary displacement and squeezing operators[59]. This representation readily leads to rigorous analytic expressions for the effective linear response functions. The analytic expressions are physically intuitive and allow for the efficient calculation of oscillatory amplitude and phase profiles over a wide range of carrier frequencies. The resulting information is the FCS analogue of a resonance Raman excitation profile. The computational ease of this approach eliminates the need for impulsive or semi-classical approximations[39, 40, 41, 42, 43, 44, 45]. Furthermore, when the linear response functions are used in combination with a moment analysis of the pump induced (doorway) state[46] (presented in Appendix A), excellent agreement is found with the full third order response approach[27, 28]. In addition, coherence driven by rapid non-radiative processes as in Fig. 1(b) can be readily incorporated into this formalism. This is illustrated using the first moments (presented in Appendix B) of a Landau-Zener driven coherence[21].

The general outline of the paper is as follows. In Sec. II, we briefly review the third order response approach and its connection with the effective linear response approach in the well separated pulse limit. In Sec. III, we obtain analytic expressions for the effective linear response function using a displaced thermal state representation for the doorway state. We further consider undamped vibrational motion and derive arbitrary temperature expressions for the dispersed and open band pump-probe signals. In Sec. IV, we present simulations to demonstrate the accuracy and physically intuitive aspects of the first moment based effective linear response approach. Next, we include the effects of inhomogeneous broadening of the electronic lineshape. Finally, we consider the application of the effective linear response approach to reaction driven coherence, and discuss the magnitude of overtone signals using displaced and squeezed initial states.

## Ii Third order and effective linear response

Here, we briefly review the connection between the third order and effective linear response functions in the well separated pulse limit. In the electric dipole approximation, we ignore the spatial dependence of the electric fields and write the total electric field of the incident pulses as . The subscripts ‘’ and ‘’ refer to the pump and the probe, and is the delay between the two pulses. The detected signal in the open band detection scheme, is the differential probe energy transmitted through the medium

(1) |

where is the material polarization induced by the pump and probe fields. If we assume that the delay is larger than the pulse durations, the lowest order in the polarization that is detected in pump-probe spectroscopy is given by

(2) |

Here, is the tensorial third order susceptibility. We have written the sequential contribution[29] ignoring the coherent coupling/tunneling terms[18, 19] that involve the overlap of the pump and probe fields. The state of the system in between the pump and probe pulses is not exposed in the the above expression for the polarization. If we group the and integrations in Eq. (2) that involve only the pump interactions, then we are left with the integral over the probe field and we may rewrite Eq. (2) as

(3) |

where is the effective linear susceptibility describing the non-stationary medium created by the pump interaction[27, 32]. Recall for a stationary medium that , which describes the equilibrium response of the system.

Eqs. (2) and (3) are formally equivalent expressions for the polarization induced by the well separated pump and probe pulses. However, Eq. (3) is more important than a mere rephrasing of the third order polarization expression in the well separated limit. The non-stationary response function can equally well describe coherence induced by mechanisms other than (and including) the pump field interaction; e.g. a rapid non-radiative surface crossing. To see what is involved, we evaluate the induced polarization quantum mechanically (in the interaction picture[47]) as

(4) |

is density operator describing the quantum-statistical state of the system, and is the electric dipole moment operator. We write the total Hamiltonian for the molecule and its interaction with the laser fields as where the free Hamiltonian and the interaction Hamiltonian have the following form for a molecular system with two electronic levels:

(5) |

where and , are, respectively, the Born-Oppenheimer Hamiltonians for the ground and excited electronic states. is defined as the difference potential that specifies the electron nuclear coupling and is the vertical electronic energy gap at the equilibrium position of the ground state. With the above definitions, the third order susceptibility takes the standard form

(6) |

Here, is the initial density operator of the system before the pump and the probe pulses and denotes the trace operation. Expansion of the above triple commutator leads to four non-linear response functions and their complex conjugates[8, 28]. On the other hand, if we substitute Eq. (6) in Eq. (2), and group the pump field interactions separately, a comparison of the resulting expression with Eq. (3) leads to

(7) |

Here, is the perturbation in the density matrix due to the pump interaction (also referred to as the density matrix jump[48] and the doorway function[29, 31]) and is given (in the interaction picture) by,

(8) |

In arriving at the second equality in Eq. (7), we have used the invariance of the trace under permutation. In the WSP limit, where the overlap between the pump and probe pulse is negligible, we can extend the upper limit in Eq. (8) to infinity. The interaction picture density matrix is then time independent.

Eq. (7) expresses the effective linear response as the average value of the standard linear response Green’s function[49] with respect to the pump induced (non-stationary) density matrix . Note that does not necessarily have to be induced by the pump fields as in Eq. (8). It could, for example, describe the perturbation in the state of the system due to a non-radiative interaction that follows the pump excitation to a dissociative state. It might even be a pump induced state with higher order pump interactions included as would be required for very strong pump fields[50]. The only assumptions are that the probe field interrogates the non-stationary medium after the coherence has been created, and the probe interaction with the medium is linear.

We express the full perturbed density matrix in terms of the ground and excited electronic state nuclear sub-matrices. The point is that the second order density matrix in Eq. (8) has no electronic coherence due to the even number of dipole interactions. Hence

(9) |

Correspondingly, the effective linear susceptibility Eq. (7) can be decomposed into a ground and excited state linear response function

(10) |

where

(11) |

In general, the pulse induced nuclear density matrices and contain vibrational coherence, i.e. off-diagonal elements in the phonon number state representation. This coherence translates into time dependent wave-packets in a semi-classical phase space Wigner representation of the density matrix [31, 51, 52, 28]. On the other hand, the highly localized (in and ) nature of the impulsively excited non-stationary states suggests that we calculate their moments using

(12) |

With as the creation (destruction) operator for the phonon mode, represents the dimensionless quadrature operators , or their higher powers. The time is larger than the pulse duration. Let the initial () values of the first moments of the position and momentum , for (say) or be denoted by . These uniquely determine the subsequent first moment dynamics . Since and denote shifts from thermal equilibrium, we may represent the pump induced nuclear density matrix for the electronic state

(13) |

where is the equilibrium thermal density matrix corresponding to the electronic level .

(14) |

is the quantum mechanical displacement operator[53] defined as

(15) |

Here is the initial displacement (in phase space) of the coherent state induced on the potential surface . When Eq. (13) is substituted into Eq. (11), the response functions are readily calculated as we show below. Furthermore, Eq. (13) offers a general scheme to represent the non-stationary density matrix on any given electronic level with only a knowledge of the first moment . The effective linear response function is thus not restricted to coherences driven by a second order pump interaction. This is the point of departure of the present work from earlier treatments[32, 33, 8, 27] which generally utilize the second order pump induced density matrix Eq. (8), and are thus formally identical to the approach. This development can be extended to include the higher moments of and , but for harmonic potentials we expect the higher moments to play a less significant role in the overall dynamics. Second moment changes are incorporated in a manner analogous to Eq. (13) in Appendix D.

### ii.1 Detection schemes

The two common experimental detection schemes are the open band and the dispersed probe configurations. For simplicity, we assume that the medium is isotropic and take the pulse fields to be scalar quantities. We then write the susceptibilities without tensor subscripts. The measured open band signal is given by Eq. (1). For an almost monochromatic laser pulse, we can write the electric field and the induced polarization as:

(16) |

where is the carrier frequency, and and are slowly varying envelope functions. Typically, the envelope function for the probe field is a real Gaussian centered at time ; , where is the field strength of the probe pulse. The envelope function for the polarization can be obtained by using the definitions in Eq. (16) and Eq. (3). After making a change of variables , we find

(17) |

Employing Eq. (16), Eq. (1), and the rotating wave approximation (RWA) in which highly oscillating non-resonant terms are ignored, we get

(18) |

Since for the present case is real, the measured (dichroic) signal is directly related to . If were made imaginary as in heterodyne techniques[54, 41], then the resulting (birefringent) signal is related .

In the dispersed probe detection scheme, the measured quantity is spectral density of the transmitted energy through the sample defined by

(19) |

In the RWA,

(20) |

where and denote the Fourier transforms of the envelope functions, and .

## Iii Evaluation of the response functions

In this section, we evaluate both the third order and the effective linear response functions. Consider the ground state response function given by Eq. (11) without the tensor subscripts. We make the Condon approximation and ignore the coordinate dependence of the dipole moment operator. If we then let and use the interaction picture time evolution , we can write,

(21) |

where we have defined the two-time correlation function for the ground state response:

(22) |

Here, the subscript denotes time ordering and evolves in time via . The density matrix is obtained from Eq. (8) as , assuming :

(23) | |||||

where denotes Hermitian conjugate.

For the excited state response function, we similarly write

(24) |

where is the two-time correlation function for the excited state response:

(25) |

with the subscript denoting anti-time ordering. The excited state density matrix is obtained as :

(26) | |||||

The above expressions are valid for a two level system with arbitrary difference potentials. In what follows, we take , with dimensionless and relative displacement . The electron-nuclear coupling force is expressed as , where and are, respectively, the reduced mass and frequency of the mode.

If the trace in Eqs. (22) and (25) were evaluated with respect to the equilibrium thermal density matrices rather than , then the correlation functions, would be the equilibrium absorption and emission correlators[56]; i.e.

(27) |

where we have introduced homogeneous dephasing through the electronic damping constant . For linearly displaced and undamped oscillators, is given by

(28) |

where . Exact expressions for the damped harmonic oscillator[57] can be easily incorporated into the present development. The half-Fourier transforms of determine complex lineshape functions,

(29) |

whose imaginary parts are directly related to the absorption and emission cross-section: and .

### iii.1 Full third-order response

Analytic expressions have been derived for the third order susceptibility for a two level system coupled to a multimode set of linearly displaced harmonic oscillators, and expressed in terms of non-linear response functions () [26, 28]. For completeness, these are listed in Appendix C. When the second order pump induced density matrices in Eqs. (23) and (26) are substituted in Eqs. (22) and (25), the ground and excited state correlation functions are related to the non-linear response functions as expressed in Eqs. (60) and (63).

The expressions of Appendix C cast the effective linear response functions in terms of the conventional non-linear response functions. The two approaches are entirely equivalent at this stage. The key point from Eqs. (60) and (63) is that the ground and excited state correlation functions involve a double integration over the pump electric fields. In the next section, we will show that the displaced state representation for the non-stationary density matrix Eq. (13) directly leads to analytic expressions for .

### iii.2 Effective linear response

#### iii.2.1 Displaced thermal state

Following Eq. (13), we represent the pump induced non-stationary state as

(30) |

This representation can be used to calculate the effective linear response in a straightforward manner[58]. Substituting the ground state part of Eq. (30) into the correlation function in Eq. (22) and using the permutation invariance of the trace and the unitarity of the displacement operator, we obtain

(31) |

The effect of the displacement operator is to shift the position operator by a time dependent classical function (c-number), namely the mean pump induced displacement[59]:

(32) |

where is the time dependent mean position of the oscillator in the ground state. If we substitute the above expression into Eq. (31), then the c-number can be removed outside the time ordering and the thermal average simply reduces to the equilibrium absorption correlator in Eq. (27). Using Eq. (51) of Appendix A for the first moment dynamics, we find

(33) |

where and are the amplitude and phase for the coherent wave-packet motion in the ground electronic state. The above expression has an intuitively appealing form: the ground state correlation function for the pump induced non-stationary medium is expressed as a modulation of the standard equilibrium (linear) absorption correlator[56] by the first moment dynamics of the ground state wave-packet motion. It is clear that the corresponding non-stationary ground state response function , Eq. (21), would translate in the frequency domain into dynamic absorption and dispersion lineshapes[32, 8, 27], which determine the final probe response to the non-stationary medium. After performing the integral, Eq. (33) reads

(34) | |||||

It is interesting to note that the strength of the first moment modulation appears through the product of the initial displacement and the optical coupling, .

For the excited state correlation function, we similarly find

(35) | |||||

Thus, the excited state non-equilibrium correlation function is expressed as the modulation of the equilibrium (fluorescence) correlation function by the excited state wave-packet.

The analytic expressions in Eqs. (34) and (35) together with the first moments presented in Eqs. (49), (50) and (52), effectively replace the non-linear response expressions derived in Eqs. (60) and (63). The generalization of the above single mode results to the multimode case is straightforward. It easy to show that the multimode expressions for and , factor into a product of single mode correlation functions. The results derived using the first moments are approximate, since the higher moments of the wave-packet motion (which are induced by the pump pulse[46]) are neglected. Nevertheless, they provide a fully quantum mechanical description of the probe response due to modulation by coherent nuclear dynamics. Moreover, we expect that for a harmonic system, the first moment dynamics will constitute a major part of the wave-packet motion detected by the probe. This expectation is verified when we present simulations comparing the first moment based linear response approach with the full third order approach.

#### iii.2.2 Analytic expressions for pump-probe signal: CARS and CSRS responses

An advantage of the analytic expressions in Eqs. (34) and (35), is that for the special case of undamped nuclear motion, they can be expanded in a Fourier-Bessel series, from which the polarization Eq. (17) can be evaluated in closed form. This yields an analytic expression for the dispersed probe signal Eq. (20) that provides insight into the origin of both resonant and non-resonant pump-probe signals. If the series expansion

(36) |

is placed into Eq. (34), we find for that

(37) |

We put this expression into Eqs. (21) and (17) to find the polarization within the RWA (which neglects the contribution of ). With Gaussian pulses, , where is a Gaussian spectral function. The ground state contribution to the frequency dispersed pump-probe signal is finally obtained:

(38a) | |||||

(38b) | |||||

where the amplitude and phase of the ’th overtone are |

(39a) | |||||

(39b) |

and where and are the quadrature amplitudes given by,

(40a) | |||||

(40b) |

In the above expressions, we have defined the constant for convenience, and introduced the product spectral function of the probe pulse:

(41) |

where is the detuning frequency and the absorptive and dispersive basis functions are defined as

(42a) | |||||

(42b) |

The basis functions depend only on , and the equilibrium lineshape functions and do not depend on the properties of the probe pulse. Note the symmetry conditions and . Eqs. (38-42) are rigorous expressions for the dispersed probe signal for a single undamped mode coupled to a two electronic level system. They are valid for arbitrary temperature, detection frequency and pulse width. The simple and direct connection between pump-probe signals and the equilibrium lineshape functions is exposed by these results. Equilibrium lineshapes thus enter the calculation at two separate stages: firstly in the expressions for the pump induced first moment amplitude and phase in Eqs. (49) and (50), and secondly in the final probe detected signal in Eqs. (42a-42b).

It is clear from Eqs. (38) that the detuning dependence of the dispersed probe signal is determined by the amplitudes and , which are in turn related to the product of the spectral function and the displaced lineshape functions . In studying the detuning dependence of , it is interesting to look at two opposite limits. When the electronic dephasing time is much longer than the pulse durations and the vibrational periods, the basis functions and consist of a well resolved Franck-Condon progression that acts like a filter in Eqs. (40a) and (40b) and determines the detuning dependence. In the opposite limit, the electronic dephasing time is much shorter than the pulse durations and vibrational periods. In this case, the lineshape functions are broad, exhibiting a much slower variation with respect to than the pulse envelope spectral function which then acts like a filter in Eqs. (40a) and (40b). The dispersed signal then consists of a superposition of red and blue shifted field envelope functions . The superposition weighting depends on the amplitude of the initial pump induced displacement and the displaced equilibrium lineshape functions. Thus, resonances occur at frequencies which correspond to the peaks of . These resonances can be identified with the well known coherent Stokes Raman scattering (CSRS) and coherent anti-Stokes Raman scattering (CARS) signals[60, 61].

The delay dependence of the dispersed signal is composed of oscillations at all harmonics of the fundamental frequency . The amplitude and the phase of the harmonic is given by and . It is clear from Eq. (39b) that the dispersed probe signal phase is not simply related to the initial phase of the coherent motion because of the additional frequency dependent functions and . However, it can be easily shown that when we integrate as specified in Eq. (19), the integral over is small due to approximately cancelling contributions from the Stokes and anti-Stokes shifted components in Eq. (40b). We then get for the ground state contribution to the open band signal

(43) |

where the amplitude and phase of the ’th overtone are given by

(44a) | |||||

(44b) |

Thus, the phase of the fundamental open band signal determines the initial phase of the nuclear motion to within an additive constant of . The additive constant phase arises from the fact that can have a positive or negative value depending on whether the initial transmission of the probe pulse is increased or decreased. It is easily shown that the open band signal will be of the form in Eq. (43) even when several vibrational modes are active. These results demonstrate that the phases of the open band signals are a direct reflection of the initial conditions of the non-stationary states.

It is also notable from Eqs. (43-44) that the open band (dichroic) signal vanishes as we tune off-resonance[62], where the imaginary lineshape function is negligible and is vanishingly small. In contrast, the off-resonant frequency dispersed signal is non-vanishing and depends on the real part of the lineshape through . We also have off-resonance using the results of Appendix A. We then get

(45) |

The dispersive functions are approximately constant in the off-resonant limit. It then follows from Eq. (40b) that the detuning dependence is mainly determined by the pulse spectral function . The dispersed probe signal consists of CARS and CSRS resonances that are centered near and oppositely phased[61, 55].

The expressions derived here make minimal assumptions and can be readily extended to incorporate optical heterodyne detection schemes[41, 42, 54, 55]. We have derived explicit expressions for the dichroic response . It can be shown for the birefringent response , that the roles of the quadrature amplitudes in Eq. (38a) will be reversed; i.e. that is the coefficient of the sine term and is the coefficient of the cosine term. Also, the CARS and CSRS contributions of will add constructively to give a non-vanishing birefringent open band signal off-resonance. This has been observed in transparent liquids[55]. For the resonant case, the results derived here are a multilevel, high temperature generalization of earlier treatments based on density matrix pathways[54] which considered a pair of vibrational levels in the ground and excited states. In contrast to prior treatments[41, 42], no assumption has been made with regard to the pulse durations in deriving Eqs. (38-42). Finally, we note that although we have given expressions for only the ground state response, analogous results are easily obtained for the excited state response. Here, the emission lineshape function in Eq. (29) plays a role similar to in the ground state expressions.

#### iii.2.3 Reaction driven coherence

As previously discussed, an advantage of the effective linear response approach is that it allows for a rigorous calculation of the probe response to non-radiatively driven coherence. As an example, we consider the multilevel system depicted in Fig. 1(b), and apply the effective linear response approach to the detection of reaction induced coherent nuclear motion along the degree of freedom in the product state .

The chemical reaction step and subsequent probe interaction minimally constitute a three electronic level problem comprising the electronic states , and . The states and are non-radiatively coupled. The probe interaction couples the ground state and excited state of the product, and is assumed to be well separated from the reaction step. The Hamiltonian for the reaction-probe stage of the problem may be written as

(46) |

Here, are the dissociative potentials along the classical reaction coordinate , for the states and . represent the quantum degree of freedom () coupled to the non-radiative transition. is the non-radiative coupling parameter which, for the case of MbNO photolysis discussed below, corresponds to the spin-orbit coupling operator needed to account for the spin change of the heme iron atom upon ligand photolysis. For simplicity, we also assume that the pump interaction that couples the states and is well separated from the chemical reaction. The sequence of interactions and the relevant coupling constants is then: pump, chemical reaction, and probe, . Each individual interaction is assumed to occur independently. The separation of these events allows us to consider the four electronic level problem in Fig. 1(b) as three sequential two electronic level problems.

We make the following two assumptions which are satisfied in the case of MbNO photolysis: (i) Before the curve crossing takes place, the system (in reactant state ) is assumed be in thermal equilibrium along the coordinate. (ii) The quantum yield for the dissociative reaction along is assumed to be unity; i.e. all the reactant molecules are transferred to the product state during the surface crossing. Assumption (i) holds if the coordinate is not optically coupled to the and electronic states. The pump pulse merely transports a fraction of the ground state electronic population to the excited state, leaving the vibrational state along unchanged. However, the degree of freedom is dissociative on the excited state potential surface and is left far from equilibrium after the pump excitation. For example, the Fe-His mode in MbNO is not optically coupled as revealed by its absence in the resonance Raman spectrum of MbNO. Its strong presence in the pump-probe signals demonstrates that this mode is triggered into oscillation following the highly efficient process[22, 23, 24] of ligand photo-dissociation in MbNO[14].

A multidimensional Landau-Zener theory was previously developed to describe vibrational coherence induced by non-radiative electronic surface crossing[21] based on assumptions (i) and (ii) above. For completeness of presentation, we have reproduced the key results of that work in Appendix B. We also present expressions for the amplitude and phase of the first moment of the nuclear motion induced on the product surface . Using these reaction driven initial conditions, we make the following representation for the non-stationary nuclear density matrix in the product state ;

(47) |

where is the complex displacement of the reaction induced coherence and is the equilibrium thermal density matrix for the nuclear Hamiltonian . Since only the ground state of the product is initially populated, the probe response due to the nuclear dynamics in the product state potential well can now be obtained along the same lines that led to Eq. (34). The correlation function for the non-stationary response is obtained by replacing and by and in Eq. (34). Here is the dimensionless coupling associated with the coupling of to the transition. The equilibrium optical absorption correlator is replaced by , which refers to the pair of electronic states and .

## Iv Simulations and Discussion

The numerical simulation of the open band signals involves multiple integrals of the time correlation functions which can be evaluated using standard algorithms. The full third order response approach involves quadruple integrations over time. The effective linear response approach replaces the correlation functions in Eq. (60) and (63) by the analytic expressions in Eq. (34) and (35). For field driven coherences, the analytic expressions require the evaluation of the first moments in Eqs. (49-50) and (52) of Appendix A. The moments can be evaluated in a single step before doing the time integrals. Hence, two of the time integrations are eliminated and the overall computation time is significantly reduced. The signal amplitude and phase in either of these approaches are obtained by performing a Fourier transform of the delay dependent pump-probe signal in Eq. (18). An alternative method that can be used when vibrational damping is neglected, is to directly calculate the amplitude and phase of the dispersed and open band signals using the analytic expressions derived in Sec. III.B.2. For calculations involving only a few modes, the analytic formula for the dispersed signal in Eq. (38b) can be easily evaluated, and a subsequent integration over frequencies as in Eq. (19) leads to the open band amplitude and phase.

In what follows, we demonstrate that amplitude and phase excitation profiles of field induced coherence are equally well predicted by the third order and the effective linear response approaches. We then apply the effective linear response approach to calculate coherent signals induced by rapid non-radiative reactions.

### iv.1 Temperature and carrier frequency dependence

One of the key parameters in a pump-probe experiment is the carrier frequency of the pump and probe laser pulses. Not only is this parameter the most easily accessible to the experimentalist, but measurements of the oscillatory amplitude and phase profiles through the resonant region provide crucial information regarding the origin of vibrational coherence[11, 15, 35]. The computational simplicity of the effective linear response approach, using the first moments, enables a direct calculation of phase and amplitude profiles over the entire absorption spectrum using realistic pulse widths.

In order to illustrate the accuracy and the physically intuitive aspects of the effective linear response approach, we treat a simple model system. Consider a single undamped mode with and coupled to a homogeneously broadened two level system with . We assume pump and probe pulses in a degenerate (same color for pump and probe) configuration. The short pulse width allows comparison with the third order response calculation which is quite formidable for long pulse durations .

Expressing the delay dependent oscillatory signal as

(48) |

we plot in Fig. 2 the degenerate open band amplitude and the phase profiles for the oscillations. The ground and excited state profiles are plotted for a range of carrier frequencies across the absorption maximum. The top panels of Fig. 2(a) and (b) show the ground and excited state amplitude and phase profiles for and . Both the third order response and the effective linear response outputs are plotted. The excellent agreement between the two approaches is evident over the entire range of carrier frequencies. We note that the ground and excited state amplitudes dip near the classical absorption and emission peaks, respectively, and . The phase of the ground state signal shows a variation of as the carrier frequency is detuned across . There is a sharp jump between at . In contrast to the ground state, the excited state phase remains constant apart from a phase jump at . At low temperature, the amplitude of the ground state signal drops by almost an order of magnitude. The approach of the phase toward on either side of the discontinuity at is steady and almost linear. For the high temperature case, the approach of the phase toward occurs much more sharply near . While the excited state phase is independent of temperature, the excited state amplitude becomes more asymmetric as the temperature is increased, as seen in Fig. 2(b).

All of the above aspects of the ground and excited state signals can be clearly understood using the first moments of the pump induced oscillations; see Appendix A. First, recall that the open band phase yields the initial phase of the pump induced oscillation to within an additive factor of as in Eq. (44b). This is shown for the ground state signal in Fig. 3. We plot the open band phase of the mode over an expanded range that includes the off-resonant limit, along with the initial phase of the wave-packet calculated using Eqs. (49) and (50).

Fig. 3(c) schematically depicts the effective initial conditions prepared by the pump pulse in the ground and excited states. Shown are the initial conditions for three different pump carrier frequencies, near resonance (indicated by downward pointing arrows in Fig. 3(a)) and off-resonance towards the red side of the absorption maximum. The arrows above the wave-packets indicate the direction of the momentum induced by the pump pulse. From Eqs. (49-50), we note that the mean nuclear position and momentum in the ground state depend on the derivative of the absorption and dispersion lineshapes respectively[46]. Thus, the wave-packet is undisplaced from equilibrium for excitation at . The momentum attains a maximum value at this frequency, and is signed opposite to the excited state equilibrium position shift[46, 44, 43].

As depicted in Fig. 3, when the pump pulse carrier frequency is tuned toward the red side of the absorption maximum , the ground state wave-packet is displaced toward decreased bond-lengths and receives a momentum kick in the same direction. Thus, a probe pulse of carrier frequency incident on the sample sees an initially depleted nuclear distribution (bleach) in the ground state. The probe difference transmission signal (pump-on minus pump-off) will at be positive and will simultaneously have a positive slope due to the sign of the wave-packet momentum. This is seen from the delay dependent signal plotted directly below the potentials in Fig. 3(c). The phase of the oscillatory signal in Eq. (48) will obey . On the other hand, for blue detuning from absorption maximum , the initial transmission signal will again be positive due to a depleted nuclear distribution in the probed region. However, the signal has a negative slope at , since now the momentum of the wave-packet points toward the probing region. In this case the signal phase will obey and is also precisely in phase with the wave-packet itself. When the carrier frequency is tuned to resonance at , the pump pulse merely imparts a momentum to the wave-packet so that the wave-packet phase is precisely . As described below, the CSRS and CARS resonances are oppositely phased for excitation at resonant maximum. The integrated signal therefore vanishes at and the phase becomes undefined. Incrementally to the red and blue sides of the absorption maximum, the open band phase shows a discontinuous jump of since the momentum impulse always points in the same direction in the resonance region. Note that Fig. 3(c) shows, for , the oscillatory signal incrementally to the red of the absorption maximum, before the phase jump.

A similar analysis is possible for the excited state signal. The excited state response arises from the stimulated emission (by the probe) from the vibrationally coherent excited state. Since the excited state wave-packet oscillates about the shifted equilibrium position , the amplitude dip and phase flip of the pump-probe signal from the excited state occurs near the classical emission peak at . As depicted in Fig. 3(c), the excited state wave-packet does not receive a momentum impulse; see Appendix A. Furthermore, it is created on the same side of the excited state harmonic well irrespective of the pump carrier frequency[46]. Thus, the phase of the excited state wave-packet is independent of the pump carrier frequency. The corresponding signal phase is fixed at for blue detuning where there is increased stimulated emission at . The phase is fixed at for red detuning where the stimulated emission is minimum at . Note in Fig. 2(b) the nearly order of magnitude increase in the amplitude of the excited state signal compared to the ground state. The increase can be traced to the fact that for impulsive excitation using weak fields, the excited state wave-packet is situated very close to the vertical energy gap . This corresponds to the ground state hole that has a displacement much smaller than [50, 63, 46]. Thus, the amplitude of excited state oscillations about the equilibrium position is much larger than that of ground state oscillations about zero. The excited state coherence will thereby be the dominant contribution for systems with long-lived excited states[1, 5, 6, 7, 11, 20]. The asymmetry in the excited state amplitude profile can be seen in Fig. 3(c) to arise from the fact that the wave-packet amplitude in the excited state is larger for blue excitation than for red[46].

The temperature dependence of the oscillatory amplitude and phase in Fig. 2 can be understood via the relative temperature dependence of the initial position and momentum induced by the pump pulse. For high temperatures, the mean thermal phonon population and is enhanced according to Eq. (49), so that the complex displacement is dominated by the real part. Hence the initial phase of the wavepacket stays closer to zero or at higher temperatures, except near , where vanishes. Correspondingly, the open band phase also stays closer to zero (or ) except for the discontinuous transition near . When the temperature is lowered, drops sharply since and becomes comparable in magnitude to . Thus, at low temperatures, the amplitude of the ground state signal drops dramatically, and the phase varies almost linearly on either side of the discontinuity at as in Fig. 2(a). The temperature dependence of the excited state first moment is primarily determined by the relative magnitude of and the difference lineshape , which appears as the coefficient of in Eq. (52). The difference lineshape is responsible for enhancing the asymmetry in the amplitude profile at high temperatures as in Fig. 2(b).

An important observation to be made from Fig. 3(c) is that for the linearly displaced oscillator model, the amplitude and phase profiles of field driven coherence are independent of the sign of the electron nuclear coupling . If were negative, then the same arguments given above would apply. The amplitude and phase behavior would be the same as in Fig. 2. This can be seen directly from the fact that the non-linear response functions in Eqs. (61a-61b) and (64a-64b) depend only on through the function . However, the schematic depicted in Fig. 3 clearly indicates that the absolute sign of the potential displacements are not revealed in a pump-probe experiment.

The depiction of pump induced initial conditions in Fig. 3 contradicts prior predictions arising from time dependent wave-packet pictures of impulsive stimulated light scattering[64, 8, 65]. The prior work suggests that the ground state wave-packet is always created on the side of the ground state well that is closer to the excited potential minimum. This is not correct, as is shown by a careful analysis of the pump pulse interaction [46, 45, 44]. It is clear from Fig. 3(c) that the centroid of the ground state wave-packet induced by impulsive excitation is strongly sensitive to the carrier frequency of the laser pulse.

From Figs. 2 and 3 we conclude that the moment analysis based approach offers a physically intuitive and accurate interpretation of the observed amplitude and phase behavior of open band FCS signals. Another significant advantage offered by this approach is that the analytic form of the effective linear response functions reduces the computation times. For the comparisons made in Figs. 2 and 3 (top panels), we chose a very short pulse in order to make the calculations using the third order approach possible with reasonable computation times. However, in many experimental situations, including those reported here, the pulse width is often much longer (. For long pulse widths the two extra integrations involved in the third order approach makes the computation quite formidable. In Fig. 4, we compare the computation times for the linear response and third order response calculations as a function of pulse width . The advantage offered by the effective linear response approach is clear, especially when it is of interest to calculate the amplitude and phase profiles at many carrier frequencies. The computation time for calculating the open band amplitude and phase using the analytic formulae in Eqs. (38-42) for a single mode and a two mode case is also plotted in the figure. The integral of the analytic expressions is over the spectral profiles of the probe pulse so that a larger value of