Strong and Long Makes Short: Strong-Pump Strong-Probe

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Strong and Long Makes Short: Strong-Pump Strong-Probe Spectroscopy Maxim F. Gelin,*,† Dassia Egorova,‡ and Wolfgang Domcke† †

Department of Chemistry, Technische Universit€ at M€ unchen, D-85747 Garching, Germany, and Institute of Physical Chemistry, Christian-Albrechts-Universit€ at zu Kiel, D-24098 Kiel, Germany

‡

ABSTRACT We propose a new time-domain spectroscopic technique that is based on strong pump and probe pulses. The strong-pump strong-probe (SPSP) technique provides temporal resolution that is not limited by the durations of the pump and probe pulses. By numerically exact simulations of SPSP signals for a multilevel vibronic model, we show that the SPSP signals exhibit electronic and vibrational beatings on time scales which are significantly shorter than the pulse durations. This suggests the possible application of SPSP spectroscopy for the realtime investigation of molecular processes that cannot be temporally resolved by pump-probe spectroscopy with weak pump and probe pulses. SECTION Kinetics, Spectroscopy

I

n the past, studies have made remarkable progress with the detection of various physical and chemical processes in real time, from time scales of daily life down to femtoseconds and attoseconds. An overview of the progress up to the end of the last millennium can be found in the transcript of the Nobel Lecture of A. H. Zewail1 (for recent achievements in attoscience, see, e.g., the reviews2,3). It seems intuitively evident that, for the probing of system dynamics with a characteristic time scale τS, the duration of the probe must be shorter than τS. The aim of the present work is to show that a probe that is longer than τS can still provide good temporal resolution on the time scale τS. However, such a probe must interact with the system strongly, that is, in a nonperturbative manner. Specifically, consider a molecule possessing two electronic states (the ground state and an excited state), each of which accommodates a number of vibrational levels. Consider the standard optical pump-probe (PP) experiment with transform-limited pulses, which consists of exciting the molecule by a pump pulse that is resonant with the electronic transition and subsequently probing it by a probe pulse.4 A weak pump pulse that is short on the time scale of molecular vibrations creates vibrational wavepackets in the excited electronic state and in the ground electronic state. These wavepackets are interrogated by the weak and short probe pulse. If the pump pulse is weak, but its duration is much longer than τS, only one or at most a few vibrational levels that are in resonance with the carrier frequency of the pulse are excited, provided that the transitions have nonvanishing FranckCondon factors. No wavepackets are created. Subsequent probing of the system by a long and weak probe pulse yields a detectable signal if the probe pulse is tuned into resonance with the excited vibrational levels. Such a PP experiment provides virtually no temporal resolution.

r 2011 American Chemical Society

The situation changes if the pump and probe pulses are longer than τS, but are rather strong. The precise meaning of “strong” will be clarified and quantified later. For the present qualitative discussion, the pulses are assumed to be strong in the sense that the system-field interaction can no longer be described in the lowest-order perturbation theory. In this case, the molecule experiences several electronic Rabi cyclings during the action of the pump and probe pulses. The modus operandi of such a PP experiment is determined by the following three conditions. (i) In contrast to a weak and long pump pulse, the outcome of the excitation by a strong and long pump pulse does not crucially depend on its carrier frequency. Such a pulse establishes a coupling between the ground and excited electronic states during its action and creates vibrational wavepackets in these states (see, e.g., refs 5-7). (ii) If the (strong) pump and (strong) probe pulses are temporally well separated, the system evolution between the pulses is determined by its intrinsic (field-free) dynamics. (iii). Analogously to condition (i), the strong and comparatively long probe pulse interrogates wavepacket dynamics in the ground and excited electronic states. The probing is not limited to the few vibrational levels that are in resonance with the carrier frequency of the probe pulse. In the present work, these qualitative expectations are confirmed by simulations of strong-pump strong-probe (SPSP) signals for a model system that represents characteristic photophysical dynamics in large molecules with strong nonadiabatic and electron-vibrational couplings (Figure 1). We write the system Hamiltonian H as the sum of an electronic Received Date: November 10, 2010 Accepted Date: December 21, 2010 Published on Web Date: January 04, 2011

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DOI: 10.1021/jz1015247 |J. Phys. Chem. Lett. 2011, 2, 114–119

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(a = 1,2) and vertical excitation energies εi f εi - εe (i = B, D) are used throughout the paper. The Liouville-von Neumann equation for the system density matrix F(t) reads (p = 1) ∂t FðtÞ ¼ - i½H þ Hint ðtÞ, FðtÞ

ð8Þ

The system's response to the applied fields is determined by the total complex nonlinear polarization (angular brackets indicate the trace) PðtÞ ÆX† FðtÞæ

Figure 1. Schematic view of the potential-energy surfaces; G: electronic ground state; B: bright excited state; D: dark excited state.

which contains contributions corresponding to all possible values of the wave vector k = l1k1 þ l2k2, with la being arbitrary integers. The PP polarization PPP(t) corresponds to the phase-matching condition l1 = 0, l2 = 1. PPP(t) can be retrieved from the total polarization P(t) by the formula8 Z 1 2π PPP ðtÞ ¼ dj2 expf - ij2 gPðj2 ; tÞ ð10Þ 2π 0

ground-state Hamiltonian, HG ¼ jGæhG ÆGj

ð1Þ

and an excited-state Hamiltonian, HE ¼ jBæðhB þ εB ÞÆBj þ jDæðhD þ εD ÞÆDj þ UBD ðjBæÆDj þ jDæÆBjÞ

ð2Þ

Here, P(j2;t) is computed via eqs 8 and 9, but for specific values of the phase angles (k1r = 0 and k2r = j2) in the system-field interaction Hamiltonian 4. Note that eq 10 is valid for arbitrary pulse strengths. Once PPP(t) has been determined, the transient absorption integral PP signal is obtained as follows:4 Z ¥ SPP ðTÞ ¼ Im dtE 2 ðtÞðPPP ðtÞ - Poff ð11Þ PP ðtÞÞ

The bracket notation is used to denote the electronic ground state |Gæ and the diabatic excited states |Bæ and |Dæ. The state |Bæ is assumed to be optically bright (that is, it has a nonvanishing transition dipole moment with the electronic ground state), while the state |Dæ is optically dark. UBD is the intramolecular electronic coupling of the excited states, and εi is the vertical excitation energy of the excited state |iæ. We consider a single vibrational mode that is described by the dimensionless coordinate Q and momentum P in the harmonic approximation. The vibrational Hamiltonians read ð3Þ hi ¼ ΩfðP2 þ Q2 Þ=2 - Δi Qg, i ¼ G, B, D

-¥

Poff PP(t)

Here, pulse, and

E a ðtÞ ¼ λa Ea ðt - τa Þ expfiðka r - ωa tÞg

According to ref 8, the choice n = 4 yields the exact PPP(t) for weak (and possibly overlapping) pump and probe pulses. This choice is also exact for temporally well-separated pump and probe pulses of any intensity. For all calculations discussed in the present paper, the choice n = 8 was sufficient for obtaining converged results. The dynamics of the system introduced above has been studied in detail elsewhere.6,7 For specific calculations, we assume the following values of the system parameters: vibrational frequency Ω = 0.05 eV, dimensionless horizontal displacements ΔB = -2 and ΔD = -1, intramolecular electronic coupling UBD = Ω/5, and εB - εD = 3.5Ω. The system possesses three characteristic frequencies that manifest themselves in the time domain through the corresponding beatings. The most pronounced beating is the vibrational beating with the period τΩ = 2π/Ω ≈ 82.7 fs.

ð5Þ ð6Þ

Here, λa, ka, ωa, and τa denote the amplitude, wave vector, frequency, and central time of the pulses. The dimensionless pulse envelope is assumed to be Gaussian, Ea ðtÞ ¼ expf - ðΓa tÞ2 g

ð7Þ

with τPa = 2(ln 2)1/2/Γa being the pulse duration (full width at half-maximum). Reduced carrier frequencies ωa f ωa - εe


ð12Þ

is the time delay between the pump and probe pulses. To calculate P(t), the Liouville-von Neumann eq 8 is converted into matrix form by an expansion in terms of the eigenstates of the system Hamiltonian. The field-matter interaction is treated exactly numerically, and the fourth-order Runge-Kutta scheme is employed for the propagation of F(t). The integral in eq 10 was evaluated as -1 1 nX 2π expf - imΘn gPðmΘn ; tÞ, Θn ð13Þ PPP ðtÞ ¼ n m¼0 n

a¼1

X† jBæÆGj

is the polarization induced solely by the probe T τ 2 - τ1

Here, Ω is the vibrational frequency, and ΔB and ΔD are the dimensionless horizontal displacements of the excited-state potential surfaces from the minimum of the electronic ground state (ΔG = 0). For optical transitions, the energy gap εe between the minima of the |Gæ and |Bæ potential energy surfaces is much larger than all other relevant energies. We thus invoke the rotating wave approximation (RWA) and write the interaction of the system with the pump (a = 1) and probe (a = 2) pulses as follows: 2 X ðX† E a ðtÞ þ XE a ðtÞÞ ð4Þ Hint ðtÞ ¼ X jGæÆBj,

ð9Þ

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Figure 2. PP signal SPP(T)excited and detected by short and weak (a), long and weak (b), and long and strong (c) pulses with carrier frequency ω1 = ω2 = 4Ω. (a) λ1= λ2 = 0.001 eV and Γ1 = Γ2 = Ω. (b) λ1 = λ2 = 0.001 eV and Γ1 = Γ2 = 0.3Ω. (c) λ1 = λ2 = 0.1 eV and Γ1 = Γ2 = 0.3Ω.

Figure 3. Fourier transforms of the PP signals shown in Figure 2a,b,c.

weak (upper panel), long and weak (middle panel), and long and strong (lower panel) pulses. The Fourier transforms of these signals are shown in Figure 3. Figure 2a corresponds to a PP experiment with weak pulses and good temporal resolution: the system is interrogated by weak (λ1 = λ2 = 0.001 eV) and short (Γ1 = Γ2 = Ω) pulses. Since the pulse duration τP1 = 18 fs is shorter than all three beating periods τΩ, τU, and τε, one expects that the PP transient SPP(T) will exhibit the corresponding oscillations. Indeed, SPP(T) in Figure 2a shows pronounced vibrational beatings with the period of τ Ω , which are superimposed on oscillations with lower and higher frequency and lower amplitudes. By Fourier transforming SPP(T), all three beating frequencies Ω, U BD, and 2Ω are uncovered (Figure 3a).

The slowest beating is the electronic beating with the period τU ∼ 420 fs, which is induced by the electronic coupling UBD = Ω/5. The fastest beating is related to the electronic energy gap, that is, the vertical separation of the potential energy surfaces |Bæ and |Dæ. In the present model, the period τε of this beating is 41.4 fs. It is shorter than the vibrational beating by a factor of 2. There exists another source of oscillations in the PP transients with a frequency of 2Ω, since the |Bæ-state wavepacket passes its mean position twice during the vibrational period. The goal of the PP experiment is to reveal these three types of beatings in the measured transients. Figure 2 depicts the calculated PP signal SPP(T) excited and detected by short and


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probe pulses are 3.3 times longer (τP1 = 61 fs) than those of Figure 2a, but otherwise are the same. In this case, the third vibrational level of the |Bæ state is predominantly excited and probed. Therefore, only the slowest (electronic) beating with the period τU ∼ 420 fs is resolved in the PP transient (Figure 2b) and its Fourier transform (Figure 3b). Figure 2c shows the PP signal obtained with strong and long pump and probe pulses. The pulse durations are the same as for Figure 2b (τP1 = 61 fs), but the pulse strengths are 100 times higher (λ1 = λ2 = 0.1 eV). In Figure 2c, the time domain |T| < 60 fs corresponds to overlapping pump and probe pulses. During that time, Rabi cycling occurs, which manifests itself in SPP(T) as the so-called coherent artifact. This regime has been analyzed in detail in refs 6 and 7. Here we concentrate on the time interval T > τP1, in which the pump and probe pulses are temporally well separated. SPP(T) for T > τP1 looks similar to its counterpart excited and detected by short and weak pulses (compare panels a and c in Figure 2) and its Fourier transform uncovers all three frequencies Ω, UBD, Ω/2 (see Figure 3c). We emphasize that these are the system frequencies that are not perturbed by the strong pulses. The frequency 2Ω corresponding to the vertical electronic energy gap is resolved, despite the fact that the pulse duration τP1 = 61 fs is longer than the corresponding beating period τε ≈ 41.4 fs. Interestingly, the long and strong PP pulses resolve this highest system frequency even better than short and weak pulses (compare panels a and c in Figure 3). Figure 4 shows a three-dimensional view of the PP signal, SPP(ω2,T), as a function of the interpulse delay T and the carrier frequency of the probe pulse, ω2. The spectra SPP(ω2,T) in Figure 4a,b,c are calculated for the same values of the pulse parameters as the transients SPP(T) in Figure 2a,b,c. If the pump and probe pulses are weak and short, SPP(ω2,T) mirrors the wavepacket motion (Figure 4a). More precisely, SPP(ω2,T) is the sum of a G-state contribution (bleaching), which is almost static, and a B-state contribution (stimulated emission), which reflects the wavepacket dynamics. SPP(ω2,T) for weak and long pulses (Figure 4b) exhibits only slow electronic beatings with the period of τU ∼ 420 fs. SPP(ω2,T) calculated for long and strong pulses exhibits pronounced oscillations (Figure 4c), but does not give a direct snapshot of the wavepacket dynamics. The spectrum in Figure 4c looks like a “reshuffled” version of the spectrum in Figure 4a: the cuts of SPP(ω2,T)at different ω2 in Figure 4a,c are shifted with respect to each other. However, every cut of SPP(ω2,T) in Figure 4c reflects the system dynamics and, by Fourier analysis, yields the system frequencies Ω, UBD, and 2Ω. In this sense, PPexperiments performed with either weak and short or long and strong pulses yield the same information. It is helpful to interpret the SPSP experiment in terms of the doorway-window picture.4 If the pump and probe pulses are temporally well separated, the pump pulse creates the doorway states D R(ω1), that is, the initial wavepackets in the ground state (R = G) and in the excited states (R = B,D). These doorway states evolve according to the field-free system dynamics (which is described by the evolution operators GR(T)) and is projected on the window state W R(ω2) state

Figure 4. PP spectrum SPP(ω2,T) for temporally well-separated short and weak (a), long and weak (b), and long and strong (c) pump and probe pulses. The spectra in panels a, b, and c are calculated for the same pulse parameters as the transients in Figure 2a,b,c. The time origin is set at T = 0.

Figure 2b shows the PP signal excited and detected by weak pulses with poor temporal resolution: both pump and


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created by the probe pulse, yielding the PP signal: X ÆW R ðω2 ÞGR ðTÞD R ðω1 Þæ SPP ðω2 , TÞ ¼

spacings and strong electronic couplings. However, the application of the SPSP technique is not limited to specific system Hamiltonians. It is only necessary that the system accommodates a number of discrete energy levels. In the application of SPSP spectroscopy to real systems, one has to consider the possibility of excited-state absorption, multiphoton processes, photoionization, and photodissociation. All these processes must be taken into account for the proper interpretation of SPSP signals. The excited-state absorption is, in fact, unavoidable in pump-probe experiments with strong pulses. Taking higher electronic states into account, the SPSP signal is still described by eq 14 for temporally well-separated pump and probe pulses. However, the summation over R in this formula now runs over all the electronic states involved. Therefore, the SPSP signal allows one to simultaneously interrogate the dynamics of several electronic states of the system under study. Of course, the higher the weight of a particular electronic state in the summation of eq 14, the better the system dynamics in this state can be resolved. If the transition dipole moments in excited electronic states have different orientations in the molecular reference frame (which usually is the case), a polarization-sensitive version of SPSP would allow the distinguishing of the contributions of different excited electronic states. Multiphoton processes will result in the modification of the doorway and window functions in the excited electronic states, while eq 14 is unaffected. As to photodissociation, its effect can be described through an effective “loss channel”. The results of the present study demonstrate that temporal resolution of molecular processes can be achieved by using strong pump and probe pulses whose durations are longer than the desired temporal resolution. We have shown that long and strong pulses may not only steer the wavepacket dynamics of the material system, but can also measure it. This suggests possible applications of SPSP spectroscopy for the study of processes that are faster than the time resolution of present-day lasers (such as the passage of wavepackets through conical intersections or monitoring molecular wavepackets in very short-lived excited states).

ð14Þ

R

Traditionally, time-resolved spectroscopy is performed with weak pulses, since the main goal is to probe the system and to learn about its intrinsic (unperturbed by external fields) dynamics. However, strong-pulse effects are ubiquitous in molecular spectroscopy. It is common practice to employ a strong (and short) pump pulse in conjunction with a weak and short probe pulse for the interrogation of high-lying states of the system under investigation.9 This gives rise, for example, to the dependence of the transient absorption10-12 and the fluorescence quantum yield13,14 of light-harvesting complexes II on the intensity of the pump pulse. A strong femtosecond infrared pump pulse and an attosecond extreme-ultraviolet probe pulse were employed to monitor spin wavepackets in Krþ ions.15 Strong-pulse phenomena were also detected in four-wave-mixing experiments: onset of nonlinear heterodyning in femtosecond rotational transients,16 pulsestrength-dependent three-pulse photon echo peak-shift of semiconducting single-walled carbon nanotubes,17,18 predicted pulse-intensity dependence of two-dimensional (2D) spectra of the photosynthetic antenna Fenna-Matthews-Olson (FMO) complex,19 manipulation of Liouville pathways in multiply resonant coherent multidimensional spectroscopy,20 and observation of triexciton quantum coherences in quantum wells.21 Various control schemes also rely on strong laser pulse(s) to steer the system to a prescribed target state.22 However, in all the experiments mentioned above, the strong pulses are used to manipulate the system dynamics. The role of the strong pulses in SPSP spectroscopy is fundamentally different. In the SPSP technique, strong, comparatively long, and temporally well-separated pulses are employed for the excitation of wavepackets and the probing of their field-free dynamics on a time scale which is shorter than the pulse durations. A related spectroscopic detection mechanism has recently been demonstrated experimentally.23 An XUV-pump XUV-probe experiment with strong (intensity ∼1013 1014W/cm2) laser pulses of ∼30 fs duration allowed not only the monitoring of vibrational beatings of the Dþ 2 ion wavepacket (with a period of ∼24 fs), but also the detection of sharp structures with a period of ∼7 fs. The SPSP experiment can be performed with a standard pump-probe setup; no phase locking of laser pulses is necessary. It requires moderately strong pulses. For example, the field amplitude of 0.1 eV used in our calculations corresponds to a field intensity of 5 1011 W/cm2 for a transition dipole moment of one atomic unit (2.54 D). SPSP spectroscopy works best if the system-field interaction energy is comparable with the average vibrational energies in the ground state and the excited electronic states (λ1,λ2 ∼ ÆhGæ,ÆhEæ). This condition guarantees multiple Rabi oscillations during the action of the pump and probe pulses. In the present work, we restricted ourselves to the consideration of a molecule in the gas phase. Our calculations show that SPSP spectroscopy can uncover the dynamics of a relatively complicated system with irregular energy level


AUTHOR INFORMATION Corresponding Author: *To whom correspondence should be addressed.

ACKNOWLEDGMENT This work has been supported by the

Deutsche Forschungsgemeinschaft (DFG) through a research grant and the DFG-Cluster of Excellence “Munich-Centre for Advanced Photonics” (www.munich-photonics.de). We thank Sergy Grebenshchikov and Enrico Benassi for fruitful discussions. We are grateful to Robert Moshammer and Joachim Ullrich for calling our attention to their recent paper (ref 23) and stimulating discussions.

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Strong and Long Makes Short: Strong-Pump Strong-Probe

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