## Abstract

Prokhorenko *et al*. (Research Articles, 1 September 2006, p. 1257) reported that, in the weak-field regime, the efficiency of retinal isomerization in bacteriorhodopsin can be controlled by modulating the spectral phase of the photoexcitation pulse. However, in the linear excitation regime, the signal measured in an experiment involving a time-invariant, stationary process can be shown to be independent of the pulse spectral phase.

Prokhorenko *et al.* (*1*) recently reported on the optimal control of retinal isomerization in bacteriorhodopsin. One of their most striking results is the experimental observation that, in the linear excitation regime, the pump-pulse spectral phase has a measurable effect on the pump-probe signal recorded for time delays longer than the characteristic time scales of the system. This result contradicts a general property of the interaction of a light pulse with a material system. Indeed, a stationary, time-invariant signal measured in an experiment performed in the linear excitation regime cannot depend on the spectral phase of the exciting pulse. In other words, coherent control is impossible in this excitation regime.

Let us consider an experiment where a stationary signal *S* is produced by a material system long after its excitation by an ultrashort pulse associated with electric field *E*(*t*). We assume that the interaction between the ultrashort pulse and the system can be described within the semiclassical approximation (quantum system and classical electromagnetic field) and that perturbation theory can be applied. We further assume that the signal *S* varies linearly with respect to the pulse energy. Within perturbation theory, this assumption implies two field interactions only, corresponding to the following general expression involving a quadratic form (1) where *R*(*t*_{1},*t*_{2}) is the system response function (*2*) at the relevant order of perturbation theory. Finally, we make use of the hypothesis of time invariance, which means that a time-shifted electric field *E*(*t* + *T*) produces the same signal as the original field *E*(*t*), where *T* is an arbitrary time delay. Following the reasoning about the linear polarization found in most optics textbooks (*3*), we replace *E*(*t*) with *E*(*t* + *T*) in Eq. 1 and change variables in the integral (*t*_{1}→ *t*_{1} – *T* and *t*_{2}→ *t*_{2} – *T*). We thus obtain (2) The equality between Eqs. 1 and 2, valid for any arbitrary function *E*(*t*), implies that the respective response functions are identical: *R*(*t*_{1},*t*_{2}) = *R*(*t*_{1} – *T, t*_{2} – *T*). This relation holds for any values of *t*_{1}, *t*_{2}, and *T*, and can thus be applied to the particular case where *T* = *t*_{2}. Thismeansthat the response function depends only on the difference between the two interaction times: *R*(*t*_{1},*t*_{2})= *R*(*t*_{1} – *t*_{2}). Therefore, (3) Expressing *R*(*t*_{1} – *t*_{2}) as a function of its Fourier transform *R*(ω), we immediately obtain (4) Hence, the signal is independent of the exciting-field spectral phase. The stationary signal *S* can be, for example, the total amount of absorbed energy, which is well known to be independent of the spectral phase (*4*). Another example is the branching ratio between two photoproducts, also independent of the spectral phase in this excitation regime, as previously discussed by Brumer and Shapiro (*5*, *6*). Photodetection based on detectors quadratic in the electric field, such as photodiodes, is also known to be phase insensitive.

The above general property also applies to the experimental situation considered by Prokhorenko *et al*. (*1*), where *S* is the pump-probe signal recorded for a large value of the time delay between the shaped pump pulse and the probe pulse. First, the linear dependence of *S* on pump-pulse energy has been carefully checked [figure 6A in (*1*)]. Second, although our hypothesis of time invariance is only approximately true in this case, it can still be used as long as the time delay *T* defined above remains smaller than the characteristic time scales. This situation of local time invariance is illustrated in Fig. 1, which shows an example of the two-dimensional response function *R*(*t*_{1},*t*_{2}) associated with a pump-probe measurement in a two-level system. This response function is obviously not time invariant (see dotted-line box), which is not surprising, because the very purpose of a pump-probe experiment is to monitor the variation of the probe absorption change as a function of the delay between the pump and probe pulses. This lack of time invariance also explains why Eq. 4 does not apply to coherent transients, which have been reported to exhibit a strong dependence on the pump spectral phase (*7*, *8*). However, the present experiment in bacteriorhodopsin deals with a quite different regime, associated with large values of the pump-probe delay, after such transient processes have settled down so that the signal *S* takes its stationary value. This means that the pump field will explore a restricted region of the (*t*_{1},*t*_{2}) space, for example, inside the solid-line box in Fig. 1, where *R*(*t*_{1},*t*_{2}) is time invariant (i.e., it depends only on *t*_{1} – *t*_{2}). Indeed, Prokhorenko *et al*. mention that “the absorption changes were constant within the 20- to 600-ps delay window,” which is exactly the condition required to write Eq. 2. Therefore, a spectral-phase change of moderate amplitude (small enough to rule out a trivial broadening or time shift of the pump pulse outside of the time window where time invariance holds) should not affect the result, in disagreement with the reported phase sensitivity. We stress that our reasoning is rather general because it is based on the system response function rather than on a particular microscopic theory. It requires only two hypotheses: (i) the signal *S* is assumed to be bilinear in the pump electric field (Eq. 1) and (ii) within the time window explored by the pulse-shaping apparatus, the measured signal *S* is independent of the arrival time of the shaped pump pulse. In particular, the vanishing effect of the spectral phase is not limited to quantum systems isolated from their environment, in contrast with the suggestion of Prokhorenko *et al*. on p. 1260.

To summarize, the results reported by Prokhorenko *et al*. (*1*) raise many questions. Does perturbation theory break down despite the reported linear dependence with respect to the pulse energy? Is the pump-probe delay of 20 ps too short, compared with the duration of the shaped pulse, for a true time invariance? Considering possible experimental artifacts, is the measured value (5 to 7%) large enough to support the claim of this unexpected phase sensitivity? These important issues warrant further discussion.