On the mechanism of weak-field coherent control of retinal isomerization in bacteriorhodopsin

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Chemical Physics 341 (2007) 296–309 www.elsevier.com/locate/chemphys

On the mechanism of weak-field coherent control of retinal isomerization in bacteriorhodopsin Valentyn I. Prokhorenko a

a,*

, Andrea M. Nagy a, Leonid S. Brown b, R.J. Dwayne Miller

a,*

Institute for Optical Sciences, Departments of Chemistry and Physics, University of Toronto, 80 St. George Street, M5S3H6 Toronto, Ontario, Canada b Department of Physics, University of Guelph, N1G2W1, Guelph, Ontario, Canada Received 30 April 2007; accepted 23 July 2007 Available online 1 August 2007

Abstract Experimental studies of the short time reaction dynamics controlling the chemical branching ratio provide direct evidence for the mechanism of coherent control of the retinal photoisomerization in bacteriorhodopsin in the weak-field limit with respect to the previous report [V. Prokhorenko, A. Nagy, S. Waschuk, L. Brown, R. Birge, R. Miller, Science 313 (2006) 1257]. The phase sensitivity of the reaction dynamics is directly revealed using time- and frequency-resolved pump–probe measurements. The high degree of control of the reaction branching ratio is theoretically explained through a combination of spectral amplitude shaping and phase-dependent coupling to selectively excite vibrations most strongly coupled to the reaction coordinate. Coherent control in this context must involve reaction dynamics that occur on time scales comparable to electronic and vibrational decoherence time scales. Ó 2007 Elsevier B.V. All rights reserved. PACS: 33.80.b; 82.53.Kp; 32.80.Qk; 82.50.Nd Keywords: Bacteriorhodopsin; Coherence; Coherent control; Femtosecond; Isomerization; Pump–probe; Protein; Raman; Retinal; Theory

1. Introduction Coherent control experiments and theory explicitly exploit the wave nature of matter to create interference pathways to enhance or suppress an observable of interest. This approach provides additional insight into the phase relationships of the different degrees of freedom involved in defining the observable and competing processes of decoherence that average out such effects. We have recently used a coherent control approach to experimentally determine whether quantum coherence effects could persist long enough in biological systems to influence a biological function [1]. One of the most important observations made in this work dedicated to the coherent control of the photo* Corresponding authors. Tel.: +1 416 978 0354; fax: +1 416 978 0366 (V.I. Prokhorenko). E-mail addresses: [email protected] (V.I. Prokhorenko), [email protected] (R.J. Dwayne Miller).

0301-0104/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2007.07.031

isomerization of retinal in bacteriorhodopsin (bR) was the linear intensity dependence (within the accuracy of the measurements) of the photoisomerization yield on the shaped excitation laser pulses. The achieved control of the isomerization yield was 52% and 78% for the anti-optimal and optimal pulses, respectively (65% for transformlimited pulse). This experiment was the first to demonstrate both constructive and destructive interference effects in controlling a biological function, in this case the primary step in proton translocation. As we pointed out in [1], the possibility to control any photochemical reaction in the linear regime is known to be in contradiction to the existing theory of coherent control, developed for closed quantum systems [2]. For this reason, coherent control studies of molecular dynamics have been typically conducted in the high intensity regime where nonlinear interactions lead to multiple pathways to the same state and associated interference effects in the molecular response to the applied fields. This state preparation greatly limits the extension

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a Potential Energy

of coherent control to probe fundamental issues of biological systems, as high intensities necessarily involve multiphoton absorption processes and upper electronic levels that are typically not a part of the natural biological response. The question addressed by [1] was whether or not it was possible to extend coherent control protocols to the study of biological systems by increasing the sensitivity, so that it would be possible to study the quantum coherence effects at excitations which address levels similar to those under natural conditions. It was shown that it is possible to observe amplitude and phase dependent shaping effects at excitation levels at which less than 5% of the ground state was excited and could be tracked to levels as low as 0.5% (see Fig. 6 in [1]). The intensity dependence of the isomerization yield for different pulse shapes was specifically conducted to determine the order of the field interaction under these conditions; a control that is often ignored in coherent control experiments. The intensity dependence is observed to be approximately linear (quadratic in the applied field) under the conditions used in our study.1 This demonstration is one of the most important features of this work as it opens up the exploration of biological processes under intensity conditions relevant to probing the system response function (strongly coupled chromophore-protein). The noted phase dependence of the isomerization yield is relatively small; it contributes 5–7% to the overall ±20% effect of isomerization control. The dominant effect is clearly the spectral amplitude modulation of the pulse shapes. From a microscopic perspective, spectral modulation under the experimental constraint of constant actinic (absorbed) energy puts the maximum power possible into resonantly driving the key reactive modes in the reaction coordinate. For the same absorbed energy, it is possible to significantly increase the amplitude of the motions that are most strongly coupled to the reaction coordinate and by so doing control the chemical branching ratio. However, there is another contributing effect that cannot be separated from the observation of the shaped excitation field alone. The broad spectral bandwidth of the pulses excites a vibrational wave packet on the upper electronic surface (a linear process). This coherent superposition of vibrational modes will propagate along the upper state surface in which the coupling between modes in this highly nonstationary state necessarily must cascade to the reactive mode(s). The exact timing or phase profile of the excitation of this vibrational wave packet will affect the amplitude(s) of the reactive mode(s) as the system evolves to the conical intersection (a bifurcation point) where the chemical bran-

297

CI

0

b

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π

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+

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N

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Lys

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Fig. 1. (a) Schematic of a potential energy surface depicting the arrival of a wave packet on the excited state and subsequent evolution towards the conical intersection (CI), influenced by the timing of the excitation of the different vibrational components. The branching between trans and cis isomers occurs at this point and is denoted as a splitting of the vibrational wave packet. The branching ratio is very sensitive to the amplitude of the reaction mode (torsion) and initial conditions that affect its amplitude and the coupling mode (C–C stretch). (b) All-trans retinal is excited by a photon. The C@C stretch (yellow arrows) and torsional motion (red arrow) are highlighted to indicate the most relevant reaction coordinates. (c) 13-cis retinal in its fully isomerized form.

ching ratio is determined. The details of the coupling between modes and how the vibrational wave packet evolves in time depend on the excited state potential energy surface are shown schematically in Fig. 1. Within this context, it is expected that the phase dependence of the excitation should affect the reaction dynamics. This particular issue is specifically addressed in the present work where we investigated the effect of pulse shape on the ensuing reaction dynamics. 2. Phase dependence of reaction dynamics: direct experimental evidence

1

The field amplitude at the excitation conditions used is on the order of 1/4th–1/10th of that which saturates the S0 ! S1 transition. As we stated in [1], we cannot unequivocally rule out nonlinear contributions as there is a small degree of curvature to the intensity dependence. This saturation effect could mask fractional intensity dependencies that may arise from the excited-state resonant contributions; although the dominant contributions to the response must originate from the excited state levels of interest.

To specifically check how the phase of the excitation pulses affects the reaction dynamics, we performed a series of pump–probe experiments using the same pulse shapes obtained in [1] but with different phase profiles (for experimental details, see Appendix A). Note that the phase

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manipulations do not affect the spectral profiles of the excitation pulses and thus a direct comparison of results is possible. The measured time- and frequency-resolved transient differential spectra for the various excitation pulses are shown in Fig. 2 as 2D-plots; the difference in kinetics can readily be seen by eyes. However, a quantitative comparison of the data is required to determine the effect of phase modulation on the dynamics and underlying mechanism. This comparison can be made using a global analysis of the transient spectra, resulting in the decay-associated spectra (DAS; for details, see Appendix A). These spectra are shown in Figs. 3 and 4 and reveal a fairly strong dependence of the dynamics on the details of the phase of the excitation pulse. In this analysis, the full pump–probe spectrum is analyzed with the dynamics convolved to the instrument response function, i.e., the excitation pulse shape is explicitly taken into account. However, to avoid complications from the features of the shaped pulses

around zero delay times, the global analysis of the transient kinetics was conducted for time delays longer than 500 fs, where only free relaxation of the excited levels is present and thus a multi-exponential analysis is applicable. There may be some complications from the specific details of the shaped excitation pulse, however, the phase dependence of the dynamics is most pronounced for the anti-optimal pulse that has the shortest effective pulse width. The short time decay of approximately 500 fs associated with the formation of the J intermediate in the photo-excited bR relaxation pathway is most strongly affected for the anti-optimal case (both the lifetime and the spectral amplitude profile). If it was simply an effect associated with residual overlap with the excitation pulse, the optimal pulse should be the most sensitive to the phase, which is not the case. Also an important observation is that there are significant (up to 30%) changes in amplitudes of the 2-ps component, which can be attributed to formation of the K intermediate, and the 30-ps component. The latter decay component can

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Fig. 2. 2D-plot of measured pump–probe data DA(t, k) for excitation pulses without phase modulation (A), with phase modulation (B), and with flipped phase (C). Left panel – optimal pulse, right panel – anti-optimal pulse. All spectra were measured with the same actinic excitation energies. For better comparison, coloring of induced absorbance DA (in mOD) is identical for all spectra, as indicated by color bars. The logarithmic scale for large delays enables visualization of the kinetics in the full delay scan range.

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Fig. 3. Decay-associated spectra of bR isomerization kinetics driven by the optimal pulse (for the pulse shape and characteristics, see [1]) with the phase modulation (blue), with flat phase (red), and with flipped phase across the pulse spectrum (black). Flipping of the phase leads to reversing of the pulse profile in the time domain (see in [23]). In all cases the excitation energy and spectra of pulses were the same (for details, see Appendix A and Fig. A1 there). (a) short decay component (500 fs) corresponding to formation of J-intermediate, (b) 2-ps component (K-intermediate), (c) 30-ps component, and (d) long-lived component (lifetime cannot be resolved within the delay window used).

be associated with the long-lived, wavelength-dependent component observed previously in time-resolved fluorescence studies [3,4]. These experimental results clearly demonstrate that the shaped excitation pulses are affecting the intrinsic molecular dynamics associated with the chemical branching ratio. The anti-optimal pulse in particular retards the relaxation to the J intermediate along the relaxation pathway. This relaxation pathway is in competition with non-radiative relaxation of the excited trans retinal back to the ground state on a very similar time and explains the reduced quantum yield for isomerization for the anti-optimal pulse shapes. The transform-limited pulse is an intermediate between the anti-optimal and optimal pulse shapes with respect to the observed dynamics and associated spectral amplitudes (a comparison of the DAS for the transform-limited pulse, and both optimal and anti-optimal is given in Fig. A4). These results are significant in that we now have a direct observation of the effect of the shaped excitation fields on the ensuing molecular dynamics and a direct correlation to the control of the chemical branching ratio. These findings clearly illustrate the high sensitivity of retinal photoisomerization to initial conditions (re: phase dependence) and the highly non-stationary response of the system.

3. Theoretical analysis of weak-field coherent control The question naturally arises how can one achieve control within the linear response limit? Here it is important to keep in mind that the conical intersection in the reaction coordinate makes the system very sensitive to initial conditions, as seen in the above experiments. The issue is related to the different time scales leading to the observable for closed and open quantum systems. For an open quantum system, strong coupling to the bath leads to fast dissipation processes that constrain the observable to the short time limit of the system dynamics; whereas in closed quantum systems the observable is made in the long time limit required to distinguish eigenstates. This difference greatly influences the degree of control and description of the system response. Specifically, for a closed, isolated, quantum system, the use of laser pulses excites a superposition of eigenstates (stationary states) of the system Hamiltonian. These states are orthogonal and by definition do not mix. In this case one obtains the same time averaged probability for observing the system in a particular eigenstate as long as the power spectrum of the applied field is the same at the corresponding frequency of the eigenstate of interest. This

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Fig. 4. Decay-associated spectra of bR isomerization kinetics driven by the anti-optimal pulse from [1] (color scheme and labelling of panels are the same as in Fig. 3). Excitation conditions are identical to those in the Fig. 3.

point for closed quantum systems was made earlier using the prospect for controlling the branching ratio between two dissociative channels of an isolated photo-excited molecule as a specific example [5]. It was shown that for a defined target state within a dissociative continuum one cannot control the branching ratio of a photochemical reaction within the linear excitation regime, even using spectral shaping. However, if two (or more) pathways with a fixed phase relationship are involved, interference effects can occur and it becomes possible to attain coherent control in the linear regime even for a closed system (the so-called bichromatic control [6,2]). The lesson from the theoretical work on bichromatic control with respect to weak-field control is that, in general, one needs to have more than one level coupled coherently to the reaction coordinate that can be optically addressed. The photoisomerization of retinal in bacteriorhodopsin is an example of an open quantum system in which the system (retinal) is coupled to the bath (protein). In solution phase, the photoisomerization of retinal involves several different isomers with quantum yields of only a few percent; basically all possible permutations of isomerization along the different double bonds in the retinal structure occur. In contrast, the protein environment leads to a high degree of specificity. The structure of the protein is orchestrated to drive the chemical reaction along a preferred pathway; that is, the trans to cis isomerization occurs exclu-

sively along the C13–C14 bond to give a single product with 65% quantum efficiency. The system is obviously very strongly coupled to the surrounding protein and highly biased by this interaction to direct the atomic motions along the coordinate that ultimately leads to proton translocation. Equally important, the protein relaxation/ response to the formation of the excited state occurs on similar timescales to the reaction dynamics and known decoherence times of both the electronic and vibrational degrees of freedom. The photoisomerization can be understood by using the open system approach where irreversible relaxation processes among the set of coupled states play a defining role. Indeed, the manifold of vibrational states of the ground state photoproduct (isomer) can not be populated directly from the manifold of vibrational ground states of the reactant – they appear as ‘‘dark’’ states. The only means to populate these levels is through the relaxation of the initially excited ‘‘bright’’ states. In general, the major difference between closed and open quantum systems is the description of the matter Hamiltonian. For a closed sysb is time-independent and tem, the matter Hamiltonian H can always be diagonalized and characterized by a set of stationary eigenstates; whereas the Hamiltonian of an open b ðtÞ, where the time dependence system is time-dependent H arises from the Brownian motions of an infinite number of bath degrees of freedom, and thus in principle can not be

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described by a set of stationary eigenstates. The dynamics of an open quantum system can be described in the frame^ðtÞ which obeys the work of the density matrix operator q Liouville von Neumann equation of motion: b ðtÞ þ Vb ðtÞ; q ^_ ¼ i½ H ^ q

kin

ð1Þ

b where the operator Vb ðtÞ ¼ ^ lðtÞ EðtÞ couples the external b to the entire system driving field given by the operator E ^. The dynamics of the system can via its dipole operator l be further explored using the reduced density matrix formalism [7]. For the weak-field limit Eq. (1) can be formally solved using perturbation theory.2 The corresponding physical processes giving rise to the linear response of matter (with respect to the number of absorbed photons and quadratic with respect to applied field) are schematically depicted in Fig. 5. The first Feynman diagram shows single-photon absorption (a stimulated process) which is governed by the second-order perturbative solution of Eq. (1) (two fields; two-time matter correlation function). The second and third diagrams describe the population of the system via spontaneous Raman processes (Ramaninduced population) where only part of the incoming photon is absorbed by the system. With respect to the incoming energy, this is also a linear process. The treatment of such population transfer requires the solution of Eq. (1) up to fourth order (four fields; four-time matter correlation function). However, it is clear that the first process will be dominant in transferring population to the desired ‘‘target state’’ – i.e., in the creation of the photoisomer. In the previously reported experiments [1] it was found that from the achieved control of the isomerization yield (40%) the phase dependence accounts for only 5–7%. The major effect is the spectral modulation in the pulse shape. Therefore, we can assume that the first process (Fig. 5a) corresponding to the second-order solution of Eq. (1) describes the main features of the experimental observations; whereas the Raman-induced population (Fig. 5b and c) might contribute to the phase dependence of isomerization. However, before proceeding with the solution of Eq. (1), it is necessary to bring to the reader’s attention the complexity of the problem: (i) A rigorous solution of Eq. (1) in the weak-field limit requires a description of the external field in quantum form, especially for the Raman-induced population. The pulsed field can be described as a superposition of Glauber coherent states [10]; however, while each coherent state separately can be given a classical interpretation in terms of an optical field with a certain complex amplitude, the superimposed coherent state has, in general, no classical interpretation [11]. 2

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Note that application of perturbation theory for open systems with dissipative dynamics is not always valid. The interplay between the coherent driving field and dissipation may introduce a large cooperative effect (the so-called stochastic resonance) that breaks down the weak-field response theory no matter how weak the field is [8,9].

i

f kout

kin

k f

i kout

kin k

i

f

Fig. 5. Feynman diagrams showing light–matter interactions accounting for the linear response (with respect to the number of absorbed photons). (a) one-photon (stimulated process, absorption), two-field interaction. System, being initially in state |ii, goes to state |fi after absorption of a photon. (b) and (c) two-photon, four-field interaction. Between absorption of a photon and emission of a Stokes photon, the system is in an intermediate state |ki. In this process only part of the incoming photon is transferred to the energy in system.

At the lowest excitation used in [1] (0.2 nJ), the number of photons per mode corresponds to 10, and thus the applicability of the classical description of the field as E(t) is questionable. (ii) The solution of Eq. (1) requires knowledge of the density operator at the initial time t = t0, before a perturbation (the external field) is applied. At this ^ðt0 Þ can be factorized to the product of the mattime q ^ðt0 Þ ¼ q ^m ðt0 Þ  q ^f ðt0 Þ. ter and field density operators: q Conventional quantum relaxation theory uses the socalled equilibrium density operator (for matter) which is assumed to be time-independent. The corresponding matter correlation functions are stationary and obey the time-translation invariance (in other words, ‘‘they are functions of only differences among their time arguments’’ [12]). In particular, for the second-order matter correlation function that will appear in the second-order solution of Eq. (1), this condition means that v(2) / h(t1  t2)f(t1  t2) where the Heaviside step-function h is required by the causality principle. However, for open quantum systems such time-independent equilibrium density operators do not exist – there are no stationary solutions to Eq. (1) [without the perturbation term Vb ðtÞ] except ^m ¼ 0. This means that for for the trivial solution q the matter correlation functions of open systems,

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the time-translation invariance will be locally violated. On the other hand, the initial density operator for the field is essentially non-equilibrated and contains off-diagonal terms [13] [only for chaotic or ^f ðt0 Þ is purely diagonal]. Therefore, thermal fields q the solution of Eq. (1) depends on the choice of initial conditions at t0 and in general, there are no t0-independent stationary solutions. Rigorous analysis of weak-field coherent control requires resolving the above questions and will require intense theoretical work. While the matter–field interactions have been analyzed by means of quantum optics for some simple fields (thermal field, single-mode or twomode fields), the population due to spontaneous Raman processes (Fig. 5b and c), to the authors’ knowledge, has never been considered. However, the main features of weak-field coherent control can be explored using appropriate simplifications and approximations to the problem. The simplest analysis can be performed using well-known Redfield theory [14] which operates with the classical field Vb ðtÞ ¼ ^ lEðtÞ; both the system and bath Hamiltonians are time-independent, and the initial density operator is also time-independent and factorized to the product of the system and bath equilibrium operators (thus the matter correlation functions are invariant with respect to time translation). In this context, Redfield theory fails to describe the phase-controlled Liouville-space pathways interference, observed experimentally in heterodynedetected stimulated photon echo [15]. In the Redfield approach, the reduced Liouville von Neumann equation Eq. (1) is represented in the basis set of the time-independent system Hamiltonian, and the system–bath interaction is considered as a white-noise stochastic process with an infinitely short bath memory (the Markov approximation). In the weak excitation limit and using the so-called secular approximation [7,14], the expression for the populations of system levels qii can be derived in analytical form (see Appendix B): Z 1 Z t dt1 dt2 Eðt1 ÞEðt1  t2 ÞM i ðt; t1 ; t2 Þ ð2Þ qii ðtÞ ¼ q0ii þ 1

q0ii

0

where are the initial populations at t0 !  1, and M is ~ t1 ; t2 Þ ¼ ðeW ðtt1 Þ Þ  ~ the matter response function Mðt; F ðt2 Þ. Here the parentheses contain the exponential functions of the decay matrix W describing the relaxation of population between system levels, and Fj(t) corresponds to the Fourierimages of spectral profiles Fe j ðxÞ of the system states j. From this expression, it immediately follows that at any fixed time t51 [including also delay times of 20 ps where the phase sensitivity of isomerization was monitored in [1]] the induced population will have some phase sensitivity since the matter response function due to non-commutativity of the system and system-bath Hamiltonians in presence of dissipation cannot be reduced to a function depending on the difference of time variables t2  t1, or single variable tn, and thus the integral part in Eq. (2) will not be equal to

the product of the classical field correlation function R1 CðtÞ ¼ 1 dsEðsÞEðs  tÞ and F(t). The amount of absorbed energy (for thin optical layers) corresponds to the overlap integral between spectral profiles of the P sample R1 absorption and the excitation pulse Eabs ¼ j 1 dxIðxÞ Fe j ðxÞ; the quantum yield Yn is defined as the ratio between the induced population of the desired state qnn(t)  qnn(1) and the absorbed energy, and can be controlled in the weak-field limit, as follows from Eq. (2). In real open systems, the stationary states correspond to equilibrium states; the induced population of any excited state will irreversibly relax to its equilibrium value. However, if the relaxation time of some state is much longer than all other states, it can be considered as a quasi-stationary state. Population of such a state (or set of quasi-stationary states) can be obtained from Eq. (2) by extending the integration upper limit to infinity and setting the corresponding relaxation rate (eigenvalue of the decay matrix W) to zero. In this case, the quantum yield of the quasi-stationary state n is: Z 1 XX Yn ¼ T na ðT 1 Þak dxIðxÞ Fe k ðxÞ=Eabs ð3Þ k

a

1

where T is the matrix of eigenvectors of W, and summation over a corresponds to the number of quasi-stationary states [if there is only one, a = (1, 2) since the ground state is a stationary state as well]. As can be seen, the yield can be controlled – optimized or anti-optimized – even for quasi-stationary states (except for two-level systems that do not have any relaxation pathways). In the absence of relaxation between system levels, all N system states are true stationary states so that {a}  {N};P the summation over a degenerates to the Kronecker d ð Na¼1 T na T 1 ak ¼ dnk Þ, and the value of the induced stationary population is simply proportional to the energy absorbed in this state. That is, in a closed system (no dissipative dynamics) the product yield cannot be controlled, as shown previously [5]. The yield of the quasi-stationary state is phase-insensitive in this approximation to the multilevel problem. This is not a surprising result since Redfield theory is based on b tot ¼ H b sys þ the time-independent total Hamiltonian H ^ bath þ H ^ s–b (here H ^ s–b is the system–bath interaction) H which can be always diagonalized to the eigenstate representation and thus represents, in general, a closed quantum ^ sys system which is conditionally separated to the system H ^ bath . Control of stationary states is only possible and bath H through spectral modulation of the excitation light (note that we have not included the Raman-induced population even in this simple approximation). However, this revelation is one of the most important outcomes from this analysis as it illustrates that control of chemical processes can be accomplished through crafting the power spectrum of the excitation source. Generally, quantum yields are made using relatively narrow band monotonic sources; the above treatment illustrates that for certain systems it will be possible to increase the desired photoproduct states by

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matching the light source to the level couplings to the process of interest and could have practical applications. It should be noted that the above derivation is based on Redfield theory in the simplest fashion (secular approximation), the bath memory is completely ignored (Markov approximation), and the field-induced dissipation in the system [16] is also ignored. Thus, Eq. (3) can explain the observed spectral sensitivity only in part. Taking into account these time-dependent effects, and mixing of the state dynamics beyond the secular approximation, will considerably increase the degree of possible control for any given system. Using Eq. (2), the induced population at the end of the excitation pulse can be roughly estimated as the convolution of the pulse profile and the exponential decay function of the corresponding excited superposition state (assuming fast dephasing). In other words, the degree of population control depends on the ratio between characteristic pulse duration (stretched due to shaping) and characteristic lifetime of the prepared state of interest. Taking into account that the optimal pulse in our pump–probe experiments is localized to 400 fs [1], it is not surprising that the magnitude of the 500-fs DAS (Fig. 3) is affected by the phase modulation since the ratio between the pulse duration and the relaxation time is close to 1. For long-lived components (2-ps and 30-ps) the magnitude of the induced population should be insensitive to the pulse profile in the framework of the Markov approximation, and depend only on the integrated pulse area (i.e., the excitation energy). In particular, the 30-ps DAS-component is very well satisfied by the quasi-stationary conditions with respect to the excitation time (400 fs) and should be insensitive to the pulse profile. However, these conclusions, based on Eq. (2), are in vast disagreement with the experimental observations showed in Figs. 3 and 4. Actually, the same statement is equally valid for the 500-fs DAS-component in the case of excitation by the anti-optimal pulse which is only stretched over 150 fs range. Thus, one can assume that these prominent changes in amplitudes of the DAS-components are caused by another process that is clearly missed in the above simple theoretical analysis, based on Redfield theory. The apparent phase sensitivity of the reaction dynamics, monitored by means of time-resolved transient spectra and DAS (Figs. 3 and 4), can be qualitatively understood by taking into account the enhancement of the oscillation magnitude q(t) of the Raman-active torsional mode xR. The driving force of a Raman mode F(t), according to the Placzek model [17], is proportional to the squared amplitude of the electrical field, F(t) = E2(t). The oscillation magnitude can be found by solving the second-order _ þ x2R qðtÞ / differential equation [12,18,19] € qðtÞ þ 2cqðtÞ aF ðtÞ; where c is the friction damping rate, and a is the molecular polarizability. It is known that for the isomerization process the motion around the C13–C14 bond requires displacement of the torsional reactive mode by the corresponding angle for overcoming the potential barrier

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between the trans and cis isomers [20,21]. The reaction probability is directly related to the magnitude of the oscillation of the torsional mode. As follows from the equation for q(t) given above, the magnitude of the oscillations will be explicitly phase-sensitive since the Fouriertransform of the driving force is proportional to the convolution integral between spectra of electric fields: R1 e e  XÞ, and thus: Eðx Fe ðxÞ ¼ ð2pÞ1 1 dX EðXÞ qðtÞ / a

Z

1

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Fe ðxÞ eixt  x2 þ icx

ð4Þ

This equation only illustrates the possibility to control a particular Raman-active mode by pulse shaping. In this regard the experimental results, obtained in a recent study [22], are very important to the problem considered here. The authors demonstrated experimentally that phase modulation only, without any changes in the excitation spectrum, leads to an enhancement of population transfer in the weak excitation limit (maximal population of the excited state was 5% of initial absorbance; see Fig. 3 in [22]). This work is in agreement with an earlier study of the coherent control of population transfer in complex molecules at weak excitations [23]. From Eq. (4) it follows that the magnitude of the oscillations of the key torsional mode can be controlled by both spectral and phase modulation of the excitation pulse. 4. Discussion The above treatment implicates the importance of Raman processes as well as the details of the time evolution along the excited state surface in the presence of timedependent system–bath interactions. The occurrence of a conical intersection in the reaction coordinates makes the system very sensitive to initial conditions and amplitudes in the reaction modes. The importance of Raman processes in the overall mechanism of control is seen by the presence of the approximately 200 cm1 Raman-active mode in the dynamics of the excited state as revealed in the pump– probe kinetics driven by transform-limited excitation pulse (19 fs FWHM). Fig. 6 shows residuals obtained by multiexponential fitting of transient spectra in the whole delay range, and the corresponding 2D Fourier-transform. At least two vibrational components, located principally at 535 nm and 550 nm, can be clearly distinguished. As can be seen from the Fourier spectra, the dominant frequencies occur between 200 cm1 and 400 cm1. In this respect, the application of Eq. (4), based on a simple damped oscillator, is very limited. However, it gives a clear understanding of the nature of the phase sensitivity in the coherent control of the isomerization yield in bR, as well as explaining why the spectrum of the optimal pulse is shifted to this spectral region and has a prominent spectral modulation (basically three components separated by 7 nm; see in [1]). The role of the experimental constraint of constant actinic energy needs to be fully appreciated at

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wavelength [nm]

500 550 600 650 0

500 1000 delay [fs]

1500

wavelength [nm]

500 550 600 650 0

500 frequency [cm–1]

1000

Fig. 6. Residuals (a) and corresponding Fourier-transforms (b) of transient kinetics of bR-isomerization, driven by transform-limited (19 fs FWHM) pulse. Two components in the excited state (535 nm and 550 nm), having frequencies distributed in the range 200–400 cm1, can be clearly resolved.

this point. For constant absorbed energy, spectral amplitude modulation matched to the bandwidth and frequency of the Raman mode puts the maximum possible power into the induced polarization to selectively drive the target mode, simultaneously at the spectral position for maximum Franck–Condon factors to further enhance the resonant coupling. It also needs to be noted that the theoretical consideration discussed above is based on a perturbative approach to the field-dependent matter Hamiltonian. However, the isomerization process leads to a strong distortion of the initial Hamiltonian (actually, trans and cis forms are two different molecules), and as such, a perturbative treatment is not completely applicable. The straightforward theoretical approach will be a direct numerical solution of the corresponding wave equation (or the Liouville von Neumann equation) with a time-dependent Hamiltonian. The most relevant work in this regard is the recent theoretical calculations of Flores and Batista [20] that studied the photoisomerization of retinal in rhodopsin under the same excitation conditions as our experiment. They conducted a fully time-dependent quantum mechanical calculation of the response of retinal in rhodopsin, including 25 vibrational modes of retinal, under excitation conditions set to excite 4% of the population (5% in our reported experiments). They found that the reaction probability, or branching ratio, was very sensitive to the relative phase and amplitude modulation of the excitation field in good agreement with the experimental findings shown in Figs.

3 and 4. Flores and Batista took an educated guess at a relative phase relationship that might affect the reaction. The fact that the phase dependence is observed at all with a single assumed phase profile for such a complex multivariable problem shows how sensitive the system is to initial conditions. Ideally, one would like to calculate the true optimal pulse shape for a given potential energy surface profile and use this procedure in conjunction with experimental results to refine the potential energy surfaces. In this regard, the work of Abe et al. [21] for the photoisomerization of rhodopsin is instructive. They explored high field conditions for the coupling between the potential energy surfaces of the ground and first excited states. With no constraints on the field amplitude, they found the optimal pulse shape for maximum isomerization within this two surface model.3 The interesting observation is that the optimal fields under these conditions were comprised of a complex train of pulses, similarly to observed experimentally in [1]. The period separating the sub-pulses corresponds to the C–C stretching frequency or coupling mode in the reaction coordinate. However, the optimal fields were 40 times larger than those used in our previously reported experiments in order to achieve effectively 100% control of the reaction coordinate. If the calculation explored a longer time window to drive the reactive mode (80 cm1 for rhodopsin), as well as the coupling mode, significantly smaller amplitude fields would be needed. The above fully quantum treatments of the isomerization of retinal show how sensitive the system response to an applied field is to the initial conditions. This aspect of the theoretical modelings is confirmed by the present experiments (Figs. 3 and 4). The high degree of sensitivity can be attributed to interference effects at the conical intersection between the excited state surface and ground state reactant and product surfaces. Conical intersections are very narrow seams in the potential energy landscape describing different nuclear configurations that connect the reactant and product surfaces at a degenerate point. These intersections occur in photochemical reactions and behave as bifurcation points; small motions of the primary reactive mode displace the system from the reactant to product surfaces at this point and determine the chemical branching ratio. There are very steep potential energy gradients near such critical points in a reaction coordinate associated with motion of the reactive mode that act to funnel the reaction through this point. Control of the reactive mode cedes control of the chemical branching ratio and is the underlying mechanism for the coherent control of isomerization. The specific microscopic motions involved can be readily understood by simply considering what motions must

3 The upper excited states would be accessed under high field conditions. Since these upper states were not included, the calculation effectively treats the coupling between only the ground and first excited state surfaces, a process that only occurs under weak field conditions experimentally.

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occur to enable the motion of the heavy atoms about the C13–C14 bond of retinal in the isomerization process. The formally double bond at this point must stretch to permit the rotation of the carbon atoms about the axis defined by the carbon backbone and this motion must occur in concert with the rotational/torsional motion. These motions are the very same modes referred to as the coupling mode and reactive modes found in high level calculations [20,21]. The configuration of cis-retinal is highly strained within the protein to strongly bias the displacements along this isomerization coordinate, making the system even more sensitive to motion of the torsion mode than in the absence of coupling to the protein. The concept of controlling chemical branching ratios through control of reactive modes modulating conical intersections has been proposed previously (see, e.g., [24– 26]). There have been a number of theoretical papers on this problem, however, only the recent work of Flores and Batista [20] treats control under the weak field conditions relevant to probing biological systems. Similarly, only the experimental work of Herek et al. [27] has approached the some experimental conditions of the present work for the study of a biological problem under which weak field conditions apply [28]. There is, however, a growing body of experimental evidence that illustrates that it is possible to selectively excite vibrational modes under linear intensity conditions consistent with control of reactive mode coupling, as outlined above. The problem reduces to finding a system with a conical intersection that has a reactive mode that can be optically addressed and has a rate of passage through the branch point on time scales comparable to decoherence of the relevant degrees of freedom. In this context, the photoisomerization of retinal in bR (and related opsin moieties) represents an ideal system. With the above correlation of the molecular dynamics, this system has provided a clear example of coherent control of a conical intersection. 5. Summary and conclusions From both a theoretical perspective and additional experimental analysis we now have a clearer picture of the mechanism for coherent control in the weak-field limit using retinal isomerization in bacteriorhodopsin as a model system. Weak-field control requires experimental constraints in which the actinic energy is kept constant to avoid trivial solutions. In this case, the observable is the chemical branching ratio. The dominant effect is the amplitude modulation of the spectral components. The optimal wavelengths selectively excite various vibrational states in the excited state surface that are coupled to varying degrees to the key reactive mode that controls the isomerization branching ratio at the conical intersection. The optimal pulse shape is amplitude modulated in a manner that also puts the maximum power into resonantly driving the Raman-active torsional mode around 200 cm1 that has been implicated as the reactive mode (cf. Fig. 1). Both

305

the modulation of the field amplitude, and subsequent relaxation from excited vibrations, couple to the reaction mode(s) and will constructively drive the amplitude(s) of this mode(s) under the conditions of the optimal pulse shape. In contrast, the specific pulse shape and spectral position of the anti-optimal pulse is constructed in which the driving field itself is off resonance from the key torsional mode coupled to the reaction/relaxation coordinate, and subsequent relaxation to this mode would be out of phase and attenuates the amplitude of this key mode. This specific effect of the anti-optimal pulse shape was shown experimentally to retard the motion along the reaction coordinate through a detailed investigation of the transient spectra. There are two effects that contribute to the noted phase dependence: time-dependent propagation of the vibrational wave packet on the excited state surface, and resonant coupling to Raman active modes most strongly coupled to the reaction coordinate. The observed phase dependence can be understood solely from the perspective of a resonant Raman process. Under actinic control, the maximum power to resonantly drive a Raman mode is to have the spectrum modulated at this frequency and to be spectrally positioned at the maximum Franck–Condon overlap for this mode. The maximum modulation of the induced polarization at this frequency is obtained in the case of amplitude modulation if there is a zero phase difference between the different frequency components of the field. Within this interpretation for the optimal pulse, the relatively small deviations of the phase profile from this condition reflects the relaxation of the driven vibrational wave packet along the upper electronic state surface. Both effects contribute to the phase sensitivity. The specific details of motion along the excited state surface and how the torsional mode controls the chemical branching ratio will require detailed fully quantum mechanical treatments of the problem similar to those conducted for retinal in rhodopsin [20,21]. Here we note that the potential energy surface for bacteriorhodopsin is significantly different from rhodopsin and may contain more than one conical intersection connecting the ground and excited state surfaces. There is, however, a great deal of high quality timeresolved transient data available to construct these surfaces. The importance of these studies is that they illustrate the fundamental distinction between closed and open quantum systems. The relaxation along the reaction coordinate for an open quantum system can not be described within an eigenstate basis but must specifically include the system– bath coupling that truncates the observable, the chemical branching ratio, onto the short time limit of observation. There is another important consideration not treated in depth here. The degree of coherent control depends strongly on the relevant decoherence times associated with the electronic and vibrational coherences. The relaxation along the reaction coordinate must occur on the same time scale as decoherence – the time scale under which the cau-

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sal connection between the phase of the forcing function and system response is lost. There are currently no direct determinations of the relevant electronic decoherence time scales; the electronic polarization is dominated by nuclear dephasing due to the rapid motion of the vibrational wave packet out of the Franck–Condon region [29]. Seminal work using photon echo spectroscopy by Wiersma and co-workers have taught us that coherences can persist even for systems as complex as protein [30,31]. The problem is accessing information on the dark states coupled to the initially prepared states. The best estimates are from theoretical considerations that place the relevant decoherence time at 200 fs [20] that is consistent with collisionally induced decoherence of related organic chromophores that occur on 100 fs time scales [23]. Interestingly enough, light activated biological processes have evolved to the point that the optimized function occurs on exactly this time scale in order to compete with the rapid non-radiative relaxation processes inherent to complex molecules [32]. Coherent control in the weak-field limit offers the prospect of gaining new insight into the specific details of the highly evolved potential energy surfaces that form the basis of the structure–function relationship of biological systems as well as resolve important issues related to decoherence in complex systems.

ps delay range, linearly spaced (from 30 fs to 100 fs) in the [2, 5] ps delay range, and logarithmically spaced between 5 ps and 600 ps. The spectral profiles of the excitation optimal- and anti-optimal pulses are shown in Fig. A1; the corresponding control phase profiles are also shown for comparison. As can be seen, change of phase modulation across the spectrum does not affect the spectral profiles; however, the changes in the temporal profiles are significant (Fig. A2). In order to avoid the influence of coherent dynamics caused by the shaped pulses within the area of temporal overlap between the pump- and probe pulses, the measured transient spectra were fitted in the delay range [0.5, 600] ps using a multi-exponential fit function: X An ðkÞet=sn DAðt; kÞ ¼ IðtÞ  n



X

An ðkÞ n

Z

t

dxIðxÞeðxtÞ=sn

ðA:1Þ

1

where I(t) is the instrument response function [cross-correlation of pump- and probe pulses, measured independently using sum frequency generation in a 100 lm nonlinear crystal (BBO)], the symbol  denotes a convolution integral, and An(k) are the amplitudes of the decay-associated spectra (DAS). As an example, Fig. A3 shows decay traces

Appendix A. Materials and methods The sample was prepared according to the standard procedure, described in detail in [1]. Before measurements, the sample was illuminated with a 75 W halogen lamp for 2 h, and during measurement the sample was kept under continuous illumination to insure that the protein was lightadapted throughout the experiment. Pump–probe measurements of the kinetics of bR-isomerization were performed using the laser set-up as described in detail elsewhere [23]. Briefly, the pump- and probe beams were focused into the flow cell (optical path 400 lm, window thickness of 100 lm) with a spherical mirror having 150 mm focal length (beam diameter in the sample of 150 lm). Duration (and shape) of the probe pulse (white-light, compressed by chirped mirrors) was estimated to be 25–30 fs from the measured cross-correlation with a short (19 fs FWHM) pulse with known shape (obtained from independently measured FROG-traces). An initial OD of 0.9 at 568 nm was used, and the excitation energy was kept equal for all excitation pulses (15 nJ). Transient differential absorption spectra DA(k,s) = A(k,s)  A0(k) (where A and A0 are the optical densities of the excited and non-excited sample, respectively) at delays s = 5 to +600 ps were measured in the 485–670 nm wavelength (k) window using different delay steps: 15 fs in the [1, +2]

20

20

10

10

0

0

–10

–10

–20

520 540 560 580 600 620

–20

520 540 560 580 600 620

intensity (norm.)

This work was supported by the National Sciences and Engineering Research Council of Canada.

phase [rad]

Acknowledgement

520 540 560 580 600 620 wavelength [nm]

520 540 560 580 600 620 wavelength [nm]

Fig. A1. Spectral profiles of the optimal (b) and anti-optimal (d) pulses having different phase modulations (a and c, respectively), used for excitation of bR. From top to bottom: pulses with phase modulation (blue), without (flat phase; red), and with flipped phase (black). Flipping of phase modulation leads to time-reversing of pulse, as it was directly measured in [23]. For better comparison, the spectra are shifted along the Y-axis. As can be seen, the spectra are identical within experimental accuracy and are not affected by the phase modulation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

intensity [a.u.]

V.I. Prokhorenko et al. / Chemical Physics 341 (2007) 296–309

1

1

0.5

0.5

0

–500

0

0

500

As it follows from Eq. (A.1), for delays longer than the duration of the excitation pulse tmax (more precisely, longer than the cross-correlation function), the decay behavior is independent of the pulse shape and is given by: X DAðt > tmax ; kÞ ¼ En  An ðkÞet=sn ; ðA:2Þ –500

0

n

500

intensity [a.u.]

1

0.5

0.5

0

–500

0

0 500 time [fs]

–500

R tmax

dxIðxÞex=sn is a constant factor. The where En ¼ e 1 standard deviation of the fitted and measured transient spectra, using 4 lifetime components [s1  500 fs, s2  2ps, s3  30 ps, s4 = 1 (not resolved)], was 0.4 mOD that corresponds to the RMS of the pump–probe signal (for exact numbers of s13, see Figs. 2 and 3). Fig. A4 shows a comparison of the DAS, obtained by excitation with transform-limited, optimal and anti-optimal pulses. tmax =sn

1

307

0 500 time [fs]

Fig. A2. Temporal intensity profiles of excitation pulses retrieved from the FROG-measurements. (a, b) Optimal- and anti-optimal pulses with phase modulation, respectively. (c, d) Corresponding temporal intensity profiles of the optimal and anti-optimal pulses without phase modulation (flat phase). Intensity profiles of pulses with flipped phase modulation (cf. Fig. A1) correspond to those shown in A and B, but reversed in time.

at several wavelengths recorded by excitation with different pulses, and the region fitted (on the right from the vertical line; note nonlinear scale of the time delay axes).

Appendix B. Semi-classical theory of weak-field coherent control (Markov limit) In the framework of Redfield theory [14], the bath dynamics are fully ignored (infinitely short bath memory; corresponds to the Markov limit), and bath correlation functions are approximated by the Dirac delta-functions and thus the relaxation tensor R is time-independent. The Redfield equation of motion for the system density operator q is [14]:

30

485 nm

20 10 0

dA [mOD]

–1

0

1

2

10

100

500

0 –20 –40

558 nm

–60 –80 –1

0

1

2

10

100

500

20 0

625 nm –20 –40

–1

0

1

2 delay [ps]

10

100

500

Fig. A3. Decay traces at different wavelengths (as indicated) recorded by excitation with transform-limited (blue), optimal- (red) and anti-optimal (black) excitation pulses. Transient kinetics were fitted in the delay range from 500 fs (indicated by a black vertical line) to 600 ps delay. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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optimal pulse

dA [mOD]

transform–limited pulse (19 fs)

anti–optimal pulse

40

40

40

20

20

20

0

0

0

–20

–20

–20

–40

–40

–40

–60

–60 470 fs 2.1 ps 30 ps Inf

–80

–80 –100

–100

500

550

–60 490 fs 2.1 ps 30 ps Inf

600

650

500

700 fs 2.1 ps 30 ps Inf

–80 –100

550 600 650 wavelength [nm]

500

550

600

650

Fig. A4. Comparison of decay-associated spectra derived from time-resolved transient kinetics of bR-isomerization, driven by different excitation pulses (as indicated).

q_ nm ¼ ixnm qnm þ i

X

ðlnk qkm  qnk lkm ÞEðtÞ þ

X

Rnmij qij ;

ij

k

ðB:1Þ where xij = xi  xj are energy differences between the system states i and j, lij are transition dipole moments, E(t) is the external classical electric field, and the elements of the relaxation tensor R contain both relaxation and dephasing rates for the system levels xk. In the weak excitation limit, an analytical expression for the populations qii can be derived using the so-called secular approximation [14,7]. In this case, the equations for the diagonal terms of q (populations) and off-diagonal terms (coherences) can be separated: q_ ij ¼ ixij qij  cij qij þ ilij ðqjj  qii ÞEðtÞ; X X W ij qjj þ i ðlik qki  qik lki ÞEðtÞ: q_ ii ¼ j

ðB:2Þ ðB:3Þ

k

In Eq. (B.3) it is assumed that initially the system is in an equilibrium state, thus the initial coherences qij(t !  1) are zero. Here cij are dephasing rates, and Wij is the relaxation matrix which contains the relaxation rates between corresponding levels. Under weak excitation conditions, this set of equations can be solved sequentially, and the populations can be obtained to second-order with respect to the applied field. The first-order solution of Eq. (B.2) is: Z 1 qij ¼ ilij P 0ij dsEðt  sÞeixij scij s ; ðB:4Þ

Here the dephasing rates are separated into the real and imaginary parts, i.e., cij ¼ c0ij þ ic00ij , and x0ij ¼ xij þ c00ij . Note that the matrix W of the relaxation rates is Hermitian and real. Eq. (B.5) can be also given in vector form: Z 1 ~ q_ ¼ W ~ q þ EðtÞ dsEðt  sÞ~ F ðtÞ; ðB:7Þ 0

of which the solution is: Z Z t W ðtt1 Þ ~ qðtÞ ¼ ~ q0 þ dt1 Eðt1 Þe 1

where ¼ qii ð1Þ  qjj ð1Þ are the differences in initial populations. By substituting Eq. (B.4) into Eq. (B.3) we get: Z 1 X q_ ii ¼ W ij qjj þ EðtÞ dsEðt  sÞF i ðsÞ; ðB:5Þ 0

j

with the time-dependent coefficients: X 0 2 F i ðtÞ ¼ 2 jlim j P 0mi ecim t cosðx0im tÞ: m

ðB:6Þ

dt2 Eðt1  t2 Þ~ F ðt2 Þ;

0

ðB:8Þ where ~ q0  fqii ð1Þg are initial populations. The elements of the exponential function eWt can be expressed using eigenvalues k and eigenvectors T of matrix W as: X T aa eka t T 1 ðB:9Þ ðeWt Þab ¼ ab : a

Now, considering the stationary values for the populations (after long enough time to ensure that all relaxations between the system levels have occurred), the upper limit in the first integral in Eq. (B.8) can be extended to infinity, and only terms with ka = 0 will contribute: Z 1 XX qnn ð1Þ ¼ qnn ð1Þ þ T na T 1 dt1 ak

0

P 0ij

1



k

Z

a

1

1

dt2 Eðt1 ÞEðt1  t2 ÞF k ðt2 Þ:

ðB:10Þ

0

Note that the number of zero eigenvalues {a} corresponds to the number of stationary states in the system (if there are no excited stationary states, only one zero value of k corresponding to the lowest ground state of the system will, however, be present). Using the correlation function of R1 the electric field CðtÞ ¼ 1 dsEðsÞEðs  tÞ, Eq. (B.10) can be recast as:

V.I. Prokhorenko et al. / Chemical Physics 341 (2007) 296–309

qnn ð1Þ ¼ qnn ð1Þ þ

XX

T na T 1 ak

Z

a

k

1

References

dtCðtÞF k ðtÞ; 0

ðB:11Þ or, by applying the Fourier-transform, it becomes: Z 1 1 XX T na T 1 dxIðxÞ Fe k ðxÞ; qnn ð1Þ ¼ qnn ð1Þ þ ak 2p k a 1 ðB:12Þ e e e ðxÞ  CðxÞ is the spectrum of the where IðxÞ ¼ EðxÞ E applied laser pulse, and Z 1 Fe k ðxÞ ¼ dtF k ðtÞeixt : ðB:13Þ 0

is the spectral profile of the corresponding state k. The amount of absorbed energy DW for the thin optical layer is proportional to the overlap integral between the spectrum of the laser pulse and the absorption spectrum of the system (sum of spectral profiles of all states): XZ 1 DW / dxIðxÞ Fe k ðxÞ: ðB:14Þ k

1

The ‘‘product yield’’ Yn, defined as the ratio between the population of the desired state n and the amount of absorbed energy, is: Y n ðtÞ ¼

qnn ðtÞ  qnn ð1Þ : DW

ðB:15Þ

Using Eq. (B.8), we have: Y n ðtÞ R1 ¼

0

dt1

R1 0

dt2 Eðt  t1 ÞEðt  t1  t2 Þ DW

P

i;k T ni e

ki t1

T 1 ik F k ðt 2 Þ

:

ðB:16Þ As can be seen, the product yield can be controlled (optimized and anti-optimized) in the weak-field limit, even using the Markov approximation to the bath dynamics. In the case of a stationary population: R PP 1 1 e k a T na T ak 1 dxIðxÞ F k ðxÞ Y n ðt ! 1Þ ¼ : ðB:17Þ P R1 dxIðxÞ Fe k ðxÞ k

1

309

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