Pump–probe spectroscopy with phase-locked pulses in the condensed phase: decoherence and control of vibrational wavepackets

Abstract

Electronic and vibrational coherences of Cl₂ embedded in solid Ar are investigated by exciting to the B state with a phase-locked pulse pair from an unbalanced Michelson interferometer, where the chirp difference matches the B state anharmonicity. Recording the A′ → X fluorescence after relaxation is compared to probing to charge transfer states by a third pulse. The three-pulse experiment delivers more details on the decoherence processes. The signal modulation due to phase tuning up to the third vibrational round-trip time indicates that the electronic coherence in the B ← X transition is preserved for more than 660 fs in the solid Ar environment where many body electronic interactions take place. Vibrational coherence lasts longer than 3 ps according to the observed half revival of the wavepacket. Control of the coupling between wavepacket motion and lattice oscillation is demonstrated by tuning the relative phase between the phase-locked pulses, preparing wavepackets predominantly composed of either zero-phonon lines or phonon side bands.

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1. Introduction

The transfer of various techniques from isolated molecules in the gas phase to multidimensional systems in the condensed phase in order to study coherence properties and to implement coherent control schemes is at present an active field in spectroscopy. Dihalogens in rare gas matrices serve as well suited test beds for the required modifications in the experiments and a development of the theoretical framework. The spectral properties are flexible enough for the use of the available ultra short light sources. An important feature is that the coupling of vibrational wavepackets in electronically excited chromophore states with the bath of the vibrational degrees of freedom in an environment varies by orders of magnitude between an excitation deep in a potential well and that close to or above a dissociation limit. This property allows us to compromise between a sufficiently long preservation of coherence on the time scale of several vibrational round-trips and a rather strong coupling which is characteristic for many more complex chromophore–bath systems. The transfer of sophisticated CARS techniques to I₂ in solid Ar¹ and of wavepacket interferometry to Br₂ in solid Ar² demonstrates this model character. The method to analyse decoherence in rovibronic transitions by pairs of phase-locked pulses was originally developed by Scherer et al. and applied to free I₂ molecules.^3,4 Variations of the optical phase between the pulse pairs in a controlled³ or statistical way (COIN)⁵ have been investigated. In the condensed environment, the method has been introduced besides other techniques^6,7 to study predominantly electronic coherence in photoemission from surfaces,⁸ excitons in semiconductors⁹ and site effects in molecular crystals.¹⁰ In this contribution, we emphasize the original idea to study coherence properties of vibrational wavepackets in electronically excited states and we chose Cl₂ in an Ar matrix as the analogous system of a dihalogen in a crystalline rare gas environment. Chlorine molecules are theoretically easier to handle compared to I₂ molecules and, indeed, simulations of the vibrational dynamics of Cl₂ in solid Ar following the B ← X transition have been carried out.^11,12 Solid Ar is the preferred matrix because Cl₂ fits well in a double substitutional site of Ar crystals according to several investigations.^11–15 Recently, Ar cage motions induced by the B ← X transition have been followed by femtosecond pump–probe spectroscopy combined with simulations.¹⁶

First, we present the intensity of the electronically and vibrationally relaxed A′ → X fluorescence of Cl₂versus the time delay and phase difference between the pulse pair in the B ← X excitation in full analogy to the original experiment.³ The first pulse prepares a wavepacket in a coherent superposition of the molecule in its ground and excited states. A further wavepacket on the same molecule from the second pulse interferes constructively or destructively with the still coherent part from the first pulse depending on the chosen phase 0 or π, respectively, for time delays being multiples of one vibrational round-trip.¹⁷ Thus, the population in the B state and the resulting relaxed population in the emitting A′ state which is displayed in the fluorescence intensity, will be modulated accordingly. The experimental spectrum shows that the method is applicable also to the condensed phase. The interference contrast, however, becomes low already after the first round-trip, suggesting quick dephasing in this setup. The experiment is rather passive, waiting for relaxation and the radiative decay on a 76 ms time scale,¹³ and accumulates decoherence in all excited components. To emphasise coherent contributions, we investigate conventional pump–probe spectra which reflect vibrational coherence and demonstrate that at least vibrational coherence of wavepackets in the excited B state is maintained for some 15 round-trips.

Now, we merge together the method of phase-locked pulse pairs for excitation with pump–probe spectroscopy in order to monitor directly the dynamics resulting from the interference of the generated two wavepackets. These three-pulse experiments have been applied to free molecules to control the ionisation probability¹⁸ and relative populations of vibrational eigenstates.¹⁹ Here, we focus on the chromophore-lattice interaction. Separation of dephasing into different components and even a control of the coupling to the lattice will be treated. Probing the interfering wavepackets with a third pulse can distinguish a very quickly dephasing part which shows up as a background from the part which remains coherent for several round-trips. The phase modulation contrast in this coherent part is now sufficiently enhanced that it is possible to discern at least two contributions. The excitation spectrum of Cl₂ in solid Ar at 4 K shows a vibrational progression of zero-phonon lines which are accompanied by phonon side bands.¹³ The spectroscopic signature of the zero-phonon line indicates that the transition occurs without exciting lattice modes while the phonon side band corresponds to the simultaneous excitation of lattice modes and a chromophore vibration.²⁰ Tuning the phase in the exciting pulse pair enables us to enhance the zero-phonon versus phonon side band contribution in the wavepacket after the interference as is demonstrated in the measured spectral patterns. This decomposition leads to the control aspect in the phase-locked excitation.

Pump–probe spectra of the wavepackets predominantly composed of zero-phonon lines display the intuitively expected weaker coupling to the lattice via a reduced vibrational relaxation rate compared to that of phonon side bands. Thus, this new technique employing two phase-locked pulses for wavepacket preparation and a third pulse for detection allows us not only to monitor the decoherence but also to steer the coupling of a chromophore transition to the bath.

2. Experimental

Argon crystals containing 0.2% of chlorine molecules were prepared by spraying premixed gas onto a cold CaF₂ window kept at 18 K, mounted on the cold finger of a He flow cryostat. The pressure in the vacuum chamber was maintained at 5.0 × 10⁻⁵ mbar during the deposition for 1–4 hours, depending on the thickness of the crystals. The crystals were cooled down gradually to 5 K and then all measurements were performed.

The visible pump pulse in the range of 470–700 nm with a typical full width at half maximum (FWHM) of 50 fs was obtained by introducing the fundamental pulse (λ = 778 nm, FWHM = 150 fs, rep. 1 kHz) from a commercial Ti:sapphire regenerative amplifier laser system (CPA2001, Clark MXR) into a home-made noncollinear optical amplifier (NOPA).²¹ An unbalanced Michelson interferometer²² with a 5 mm thick BK7 beamsplitter was adopted to generate phase-locked pulse pairs. The beamsplitter induces a chirp difference corresponding to a group velocity dispersion of 1177 fs² (β′ = 1.39 fs cm) to the second pulse with respect to the first pulse. This chirp difference between the pulses was designed to produce spectral interferences matching the anharmonicity of the Cl₂ vibration. Note that a balanced interferometer would produce equally spaced spectral fringes. The second harmonic of a second NOPA pulse tunable between 250 and 350 nm serves as the probe pulse with approx. 80 fs duration. The laser induced fluorescence (LIF) of Cl₂ around 800 nm was detected by using an optical multichannel analyser (OMA) system (SD2000, Ocean Optics) with a glass filter (RG 630, Schott). For the ultraviolet LIF from the charge transfer D′ state, a photomultiplier was used with a combination of an interference filter centered at 360 nm (Δλ=10.6 nm, Coherent) and a bandpass filter (UG11, Schott).

3. Experimental results

3.1 Fluorescence induced by a phase-locked pulse pair

Excitation of Cl₂ molecules in an Ar matrix to the B state leads to vibrational and electronic relaxation to the v′ = 0 level of the lowest lying excited electronic state A′, and the well known A′ → X fluorescence is emitted in the spectral range around 800 nm²³ with a life time of 76 ms.¹³ The intensity in three prominent bands at 792.7, 825.9, 861.4 nm is recorded in the OMA. The excitation pulse is tuned to a central wavelength of 512 nm and split in the Michelson interferometer. The integrated fluorescence intensity as a function of the delay between the pulses is shown in Fig. 1, where the intensity induced by light from only one arm of the interferometer is normalized to unity. The light pulse from this arm will be called reference pulse in the following and its arrival time at the sample t_r will be taken as the time origin indicated by an arrow at Δτ = 0 in Fig. 1. The anharmonicity in the molecule leads to longer round-trip times in the high energy parts of the wavepacket which are prepared by the blue components of the light pulse. The second pulse is positively chirped with respect to the first pulse in all experiments due to the intentionally unbalanced Michelson interferometer. The chirp causes a delay of the blue components compared to the red components. Thus, this chirp serves to match the anharmonicity in the B state as will be pointed out in Section 4.2. Tuning the translation stage in the second arm in a way that this second pulse at t coincides in time with t_r, i.e. once more Δτ = 0, leads to an increase or decrease of the fluorescence intensity as displayed by the set of points in Fig. 1. These points correspond to a fine tuning of the relative phase ϕ between the two electric fields E(t_r) and E(t) by the piezo driven mirror which occurs on a subfemtosecond time scale according to the central light angular frequency ω_c = 3.68 × 10¹⁵ Hz. Both fields have equal amplitudes and an optimal superposition with ϕ = 0 would double the field at the sample and quadruple the intensity I since it corresponds to the square of the field. The fluorescence is expected to scale with I and it would increase to a value of 4 in the optimal case according to the normalisation. For ϕ = π, on the other hand, both fields would cancel each other, the sample remains in the dark and the fluorescence in Fig. 1 should drop to zero in the optimal case. For each time delay, the phase was scanned in 16 steps from 0 to 2π and in the experiment a reduced contrast with a maximum of 3 and a minimum of 1 is achieved as is shown in Fig. 1 (still at Δτ = 0). This deviation from the optimal value range between 4 and 0 originates mainly from the chirp difference between the pulses mentioned above. A calculation of the maximal variation of the photon number transmitted by the unbalanced Michelson interferometer at Δτ = 0 yields this reduced modulation between 3 and 1. A statistical phase relation between E(t_r) and E(t) leads to a summation of the intensity of each of the individual pulses and results in a value of 2 indicated by a dotted line in Fig. 1. The contrast in the observed modulation of the LIF compared to the statistical case is still large and allows to display the coherent contributions in the molecular dynamics in the time range Δτ covered in Fig. 1.

Fig. 1 Intensity of A′ → X fluorescence (792.7, 825.9 and 861.4 nm) versus time delay Δτ of two phase-locked pulses at 512 nm (pulse duration 50 fs) arriving at t_r and t and preparing interfering wavepackets in the B ← X excitation. The phase is varied for each Δτ in 16 steps between 0 and 2π (circles). Solid lines indicate the maximum and minimum envelopes. The recurrence at 260 fs correspond to the central round-trip time at 512 nm.

The consequence of a time delay Δτ between the coherent pulse pair is the following; the reference pulse (λ = 512 nm) with a pulse duration of 50 fs is short with respect to the vibrational round-trip time and prepares a coherent superposition of several vibrational eigenstates. This coherent superposition launches a spatially rather sharp wavepacket in the B-X Franck–Condon region near the inner turning point(Fig. 2). The spectral envelope covers vibrational levels from about v′ = 10 up to v′ = 13 within its FWHM according to the literature spectrum¹³ for the vibrational progression in Fig. 3c. This wavepacket generated from the first pulse at t_r propagates forward to the outer turning point and leaves the Franck–Condon region. Thus, the overlap with the similar wavepacket from the second pulse started at t decreases with increasing Δτ = t − t_r in Fig. 1. This is displayed in the drop of the phase sensitive part to the statistical limit within 100 fs. Now, the first wavepacket is located around the outer turning point while the second one is being formed on the opposite side at the inner turning point. The spatial separation prohibits interference and thus effects of the phase ϕ. For even larger Δτ, however, the first wavepacket returns to the Franck–Condon region. That part conserving full vibrational and electronic coherence can interfere once more with the wavepacket from the second pulse. Indeed, phase sensitivity shows up again around Δτ = 260 fs despite of the large drop of the intensity. The time delay of 260 fs is in full accordance with the vibrational frequency spacing around v′ = 12 of 126 cm⁻¹ which corresponds to 264 fs. Thus, the vibrational round-trip is observed in full analogy to the previous free molecule case³ and the analysis can be applied accordingly. It should be noted that even in the free molecule case the decrease in contrast from Δτ = 0 to that after one round-trip has been observed. The origin is nevertheless quite different. Rotational dispersion provides the dominant contribution in the free molecule⁴ while rotation is blocked in the rigid double substitutional lattice site.^12–15 The interesting dephasing by the coupling to the lattice prevails in the solid state case and a more detailed characterisation is highly desirable. The second recurrence around 520 fs has not been measured, but it could be hardly discernable from the background in Fig. 1. This first recurrence delivers the coherence properties of the phase-locked pulse pair sequence. To enhance the signal to noise ratio and to emphasize coherent contributions, a probe pulse will be added.

Fig. 2 Scheme for wavepacket preparation by two phase-locked pulses at t_r and t_i and recording either A′ → X fluorescence according to Fig. 1 or probing the wavepacket at the outer turning point by a third pulse at 282 nm and recording the charge transfer(CT) emission. The center wavelength of 521 nm corresponds to v′ = 9 and is used in Figs. 3–6.

Fig. 3 Trace (a) shows the envelope of a single pulse with the lock wavelength of 519 nm indicated by the arrow. It corresponds to the phonon sideband of v′ = 9 in the excitation spectrum from ref. 13 shown in trace (c). Interference fringes for ϕ = π and ϕ = 0 are presented in traces (b) and (d), respectively. The maxima in trace (d) coincide with the phonon side bands for v′ = 8–11 according to the arrows.

3.2 Pump–probe spectroscopy with phase-locked pairs of pump pulses

To develop the technique systematically we start with the reference pulse, now, at 521 nm and we add a probe pulse with a wavelength of 282 nm to induce a transition from the B state to a charge transfer state. The pump–probe scheme is illustrated in Fig. 2. The charge transfer emission of Cl₂ was studied by several groups^21,24 and we tune the fluorescence recording accordingly to the center wavelength at 360 nm by a filter and photomultiplier combination. The Franck–Condon region for transitions from the ground state¹⁴ and from the B state²¹ to the charge transfer states are rather different. In a previous study on ClF in solid Ar,²¹ the Cl₂ pump–probe spectroscopy has been carried out, however unintentionally, since ClF contains an amount of 1% of Cl₂. The time courses marked by an asterisk in Fig. 3 of ref. 21 covering a probe wavelength range from 286 nm to 276 nm belong to the Cl₂ contribution,²⁵ but were mistakenly assigned to excited Ar_nF centers which have a similar emission centered also at 360 nm. From these data and more extensive studies which will be published elsewhere, it is evident that we can probe the B state vibrational wavepacket with a pulse at 282 nm. We convinced ourselves by a variation of the probe wavelength that with a pulse at 282 nm we probe the center of the wavepacket (close to v′ = 9) near the outer turning point as indicated in Fig. 2. Under this condition a large contrast can be obtained in the pump–probe spectrum by applying negatively chirped pulse as discussed in ref. 2. The Cl₂ pump–probe spectrum optimised in this way is displayed in Fig. 4. The wavepacket is generated with the reference pulse at t_r, and the fluorescence intensity from the charge transfer state (360 nm) probed at T is recorded with a variable time delay Δt = T − t_r. The oscillation maxima correspond to the largest detection sensitivity with the wavepacket at the outer turning and the minima to the inner turning point. A period of 220 fs is observed for the full time course and is equivalent to the round-trip time which corresponds to the spectral frequency spacing (150 cm⁻¹) between the v′ = 9 and v′ = 10 levels.¹³ The signal rides on a background Z which will be discussed later on. The modulation depth is partially restricted by the probe pulse duration of 80 fs originating from the limited bandwidth in the frequency doubling. The modulation around the averaged value Y dies out between the 8th and 12th oscillations (Δt ≈ 1.8 ps) and it recovers from the 12th to 17th oscillation (Δt ≈ 3.3 ps) with about half of the contrast near Δt ≈ 0.5 ps. Anharmonicity among the contributing vibrational levels of v′ = 7–11 is responsible for the dip in the contrast around 1.8 ps according to the discussion section and affects also the other regions. Obviously the modulations can last for more than 3 ps; the pure vibrational dephasing time, which is relevant to the contrast in the pump–probe spectra, exceeds the dephasing visible in Fig. 1 considerably. This suggests that more information on the coherence properties can be obtained by adding a probe pulse to a phase-locked pulse pair excitation.

Fig. 4 Charge transfer emission intensity versus time difference Δt of the probe pulse (282 nm at T) relative to a single pump pulse (521 nm at t_r). A background Z and the signal Y are indicated by arrows. The round-trip numbers 1, 4, 8, 12 and 16 are marked together with the decaying mean value (solid line) and with the half revival time T_rev/2. The timing t₀–t₃ of the phase-locked pulse pair and T_m of the probe pulse used in Fig. 5 is illustrated.

The timing in the following three-pulse experiment is outlined on the basis of the observed vibrational dynamics in Fig. 4. The delay line is adjusted in a first run to a time coincidence of the phase-locked pulse pair with Δτ = t₀ − t_r = 0. The probe pulse is set either to the fifth oscillation maximum at T_m indicated by an arrow in Fig. 4 or to the background at T_BG for the time delay Δt = T_BG − t_r = −3 ps. The relative phase ϕ between the pulse pair is tuned by several π via the piezo on the Michelson interferometer and the fluorescence (360 nm) intensity from the charge transfer state is recorded for T_BG (Fig. 5a) and T_m (Fig. 5b). Both signals are well modulated. This is not surprising because the modulation for Δτ = 0 reflects just the interference contrast of the phase-locked pulses from the unbalanced Michelson setup like the modulation at Δτ = 0 in Fig. 1. The modulation amplitudes are collected in Table 1. The amplitude for T_m contains that for Z + Y while at T_BG only the background Z contributes; the difference amplitude represents the modulation of the signal Y, which displays the B state wavepacket dynamics. Now, in a second run the time difference between the phase-locked pulse pair is set to one vibrational period, i.e. Δτ = t₁ − t_r = 220 fs. The probe pulse positions at T_BG and T_m are kept as in Fig. 4 and the phase ϕ is tuned again. A rather noise free modulation recorded at T_m is shown in Fig. 5c while the background part at T_BG is constant on the same noise level and therefore not shown. The amplitudes are included in Table 1 also. The lack of modulation in the background is remarkable; it shows that the corresponding population displayed in Z originates from a contribution which is dephased already within 220 fs or one vibrational period. The contribution in the signal part Y, on the other hand, preserves a large fraction of coherence within one period. The normalisation of the modulation amplitude ΔY at t₁ to that at t₀ indeed confirms that it amounts to more than 50%. The possibility to separate out these contributions Z and Y demonstrates a significant advantage of the three-pulse experiment with respect to the pulse pair experiment from Fig. 1. There, all contributions merge together in the recurrence and its modulation contrast relative to Δτ = 0 is much weaker compared to the more coherent fraction Y in the three-pulse experiment.

Fig. 5 Three-pulse experiment with two phase-locked pulses (521 nm) at t = t₀ for traces (a) and (b), t = t₁ for (c), t = t₂ for (d), and t = t₃ for (e). The symbol t_i stands for ith round-trip time. The modulation in the normalized charge transfer emission intensity with the relative phase ϕ in the pulse pair is presented. The trace (a) corresponds to a probe pulse time at T_BG ahead of t_r and displays background Z(open circles) while the traces (b) to (e) correspond to T_m and Z + Y (closed circles) from Fig. 4. The phase ϕ is absolute for the traces (a)–(c) and modulo an unrecorded constant for the traces (d) and (e). The modulation in Y is given as ΔY, and the normalized modulation with time difference Δτ in the phase-locked pair, i.e. round-trips 0 to 3 is displayed in trace (f).

Table 1 Time separation Δτ of phase-locked pulses and corresponding phase modulation constant of background ΔZ, signal and background Δ(Y+Z), signal ΔY and decay of normalized signal (see Figs. 4 and 5)

t i	t 0	t 1	t 2	t 3
Δτ/fs	0	220	440	660
ΔZ at TBG	0.35	<0.01	—	—
Δ(Y + Z) at Tm	0.63	0.15	0.11	0.12
ΔY at Tm	0.28	0.15	0.11	0.12
ΔY(ti)/ΔY(t0)	1.0	0.55	0.40	0.43

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Due to this enhanced contrast in the Y contribution it is possible to follow the coherence decay for larger delays Δτ in the phase-locked pulse pair. The variation in the fluorescence intensity with phase ϕ for a coincidence of the second pulse with the second round-trip time t₂ (Δτ = 440 fs) and the third one t₃ (Δτ = 660 fs) is presented in Figs. 5d and 5e, respectively, at the same probe pulse position T_m as in Fig. 5b. The modulations are still well above noise and the normalized amplitudes are included in Table 1. The survey on the amplitudes in the Y contribution versus the number of round-trips in Fig. 5f indicates a rather moderate decay and more than 40% of the initial fraction survive even at the third round-trip. It seems that after a drop by a factor of two in the first round-trip, the decay is even slowed down in the next round-trips.

The three-pulse scheme is flexible and offers modifications in order to learn more on the evolution of coherence in the wavepackets. An example is demonstrated in Fig. 6. The time delay Δτ in the phase-locked pair is adjusted to the first round-trip, i.e. Δτ = 220 fs. Now the delay of the probe pulse with respect to the reference pulse Δt = T − t_r is tuned with a delay line, and the fluorescence intensity of the charge transfer emission is once more recorded. In this experiment the definition of lock frequency ω_L (or lock wavelength λ_L) and relative phase ϕ in the phase-locked pulse pair is essential. The NOPA pulse is tuned with its maximum of the envelope near the desired lock frequency ω_L = 3.63 × 10¹⁵ Hz (λ_L = 519 nm) indicated by the arrow in Fig. 3a. The precision in setting this maximum is not sufficient for our purposes. The lock frequency ω_L and the phase ϕ can be defined, however, relative to each other much more precisely and the accuracy can be probed with high sensitivity in a spectroscopic way. We set that exactly at ω_L the pulse pair should have the identical phase to fulfil ϕ = 0. The two pulses at the sample are well separated in time due to the time delay of 220 fs and can not interfere in this place. Passing them together through the grating monochromator of the OMA setup leads to a prolongation of each of them according to the different path lengths along the grating and they interfere in this device. The OMA displays directly the spectral distribution of the resulting interference fringes and two examples for ϕ = π and ϕ = 0 are presented in Figs. 3b and 3d, respectively. Equal phase of the pulses at ω_L leads to a constructive interference at ω_L. Thus, to set ϕ = 0 we tune the piezo in the Michelson interferometer, position the central maximum of the fringes to ω_L and obtain the pattern in Fig. 3d. The phase of ϕ = π corresponds to destructive interference at ω_L and a minimum like in Fig. 3b. The pattern remains stable during the time required for a scan of Δt in Fig. 6, preserving the phase in such a scan. Constructive interference for ϕ = 0 means that at ω_L just multiples n of 2π have been acquired during the time delay of Δτ_vib = t₁ − t_r = 220 fs. Moving away in frequency (or wavelength) from ω_L in Fig. 3 leads to an increasing deviation from the integer number n. The intensity decreases, reaches a minimum at n + 1/2 and rises to the next maximum at n + 1. Thus, the frequency spacing Δω in Figs. 3b and 3d is determined by the pulse delay with

Δω = 2π/Δτ_vib.

It clearly resembles the spacing in the vibrational progression (Fig. 3c). The anharmonicity is considerable and the matching from v′ = 8 to v′ = 11 indicated by the arrows can only be achieved with the chirp difference in the phase-locked pulses mentioned before. With this spectral definition of ϕ it is possible now to scan Δt of the probe pulse and to record the fluorescence intensity with ϕ as a stable parameter. The result for the two extreme cases ϕ = 0 and ϕ = π is shown in Fig. 6. It is fully consistent with the phase tuning data from Fig. 5. The difference in LIF intensity between ϕ = 0 and ϕ = π in Fig. 6 for the indicated probing position T_m chosen for Fig. 5 is equal to the amplitude in the corresponding modulation in Fig. 5c. The background in Fig. 6 ahead of t_r shows once more no phase sensitivity. The progress in Fig. 6 is that we can follow now the dynamics of wavepackets prepared with different ϕ. First of all, both traces display the same oscillation period of 220 fs in the signal part Y. Thus, both traces originate from a vibrational wavepacket in the B state of Cl₂ around v′ = 9. The mean values for the two phases ϕ = 0 and ϕ = π indicated by Y₀ and Y_π in Fig. 6 differ significantly near Δt = 0. The larger value of Y₀ will be explained in the discussion by using the spectral properties from Fig. 3. With increasing Δt both traces approach each other because the average value of Y_π rises slightly while that of Y₀ drops. Thus, by variation of ϕ we succeeded in preparing two types of wavepackets with a significantly different time evolution apparent in the pump–probe spectrum. In the discussion the difference will be assigned to a selection in the coupling of the Cl₂ vibration to the lattice by the phase variation, showing that a control of this property can be achieved.

Fig. 6 Charge transfer emission intensity in three-pulse experiment with one round-trip separation(Δτ = t₁ − t_r) of the phase-locked pulses versus the time delay Δt of the probe pulse at T with respect to t_r. The phase ϕ is fixed to 0 (solid) and π (dotted), respectively and the time courses of the mean values Y₀ and Y_π are indicated together with T_m from Fig. 5.

4. Discussion

4.1 Dispersion and dephasing times

For clarification we want to specify our use of the phrases “dispersion”, “dephasing” and “decoherence” in this contribution. A wavepacket composed of a frequency distribution is going to broaden in time. This broadening remains reversible as long as no external phase or frequency distortions occur and will be attributed to “dispersion”. A signature of “dispersion” are revival phenomena²⁶ and in special cases “dispersion” can be compensated at specified times by an excitation with chirped pulses.² “Dephasing” and “decoherence” are used synonymously for statistical and irreversible distortions of phase relations. The time evolution of a wavepacket of a free molecule on a time scale shorter than radiative decay is the model case of “dispersion”. Therefore, it was possible to explain the phase-locked spectra of free and isolated I₂ molecules completely with the absorption spectrum which delivers the vibrational and rotational frequency components.⁴ This supported the development of the methodology. The phase-locked data deliver in this case, on the other hand, no additional information on the basic molecular properties. The statistical and irreversible impact of background gas adds “dephasing”, and a theoretical frame work is under construction using again I₂ experimental data.²⁷

The dihalogen chromophore in the condensed phase dealt with here is on the side of the other extreme of a multidimensional interaction for which typically a distinction of system (here Cl₂) and bath(Ar) is used. The fixed lattice positions in the solid phase and the high symmetry due to the fcc structure of the surrounding matrix, however, reduce to a large extent the statistical, i.e. “dephasing” components; rather long decoherence times have been observed for Br₂ in solid Ar.² The relevant correlation functions were worked out in a basic simulation study for I₂ in a Kr matrix²⁸ and we rely on the systematics of system–bath correlations in our discussion.

All correlation functions contribute to the phase-locked spectrum of Fig. 1 and it averages over all excited states in the excitation spectrum of Fig. 3c. In this case the spectrum is dominated by the fastest “dispersion” and “dephasing” processes in the system–bath and bath–bath correlations.²⁸ In this respect the spectrum indicates a remarkable high degree of coherence after one vibrational period. Unfortunately, the fluctuations are too large in order to follow the decay of coherence for further vibrational periods. Therefore, it is difficult to sort out the time courses of the different correlations exclusively with this type of spectroscopy.

The internal Cl₂ vibrational correlations are projected out in the pump–probe spectrum of Fig. 4 and it represents an interesting example of “dispersion” and “dephasing” contributions. The anharmonicity ω_ex_e (=5.45 cm⁻¹) of gaseous Cl₂ in the B state is rather large among the dihalogens and a similar value of 5.47 cm⁻¹ for Cl₂ in solid Ar was derived from the excitation spectrum¹³ reproduced in Fig. 3c. The “dispersion” in the vibrational wavepacket leads to a broadening and causes the loss of modulation depth around T_disp = 1.8 ps as is shown in Fig. 4. Without any “dephasing” the original wavepacket would be restored after a revival time of T_rev = 2π/ω_ex_e. The full amplitude, however, is known to recover already at a half revival T_rev/2, but with opposite phase.²⁶ The value of T_rev/2 = 3.0 ps is indicated in Fig. 4 and it is consistent with the recovery of the modulations around 3 ps. The experiment in Fig. 4 was thus designed in the following way. By using a negatively chirped pulse, the optimal amplitude was delayed² to about 0.5 ps in order to display the best contrast in the modulation depth for the first period. The modulations are washed out around 1.8 ps, the time when the quarter revival with a half of the original oscillation period is expected. The recurrence around 3 ps demonstrates that the largest contribution to the damping at 1.8 ps originates indeed from “dispersion” which is reversible. The irreversible decrease in amplitude amounts to about 50% around 3 ps, indicating a pure vibrational “dephasing” time of at least 3 ps. The exceptionally large anharmonicity speeds up “dispersion”. A combination with the large vibrational frequency spacings and the probe pulse duration of 80 fs limits the modulation depth at early times of Fig. 4 and inhibits the visibility of partial recurrences in the damping region.

The electronic B ← X transition can be expected to be more sensitive to the system-bath correlations than the internal B state vibrational dynamics. This corresponds to the general notion of shorter electronic “dephasing” times than vibrational ones, and it is confirmed by the phase-locked pump–probe data shown in Fig. 5. The phase sensitivity in the pulse pair for the B ← X transition imposes coherence onto the electronic and vibrational degrees of freedom. By setting the phase-locked pulse pair separation as multiples of the B state vibrational period, we care for the vibrational coherence, and in view of the long vibrational “dephasing” time as is shown in Fig. 4, it is obvious that we monitor predominantly the electronic coherence.

An important result is the separation of a background Z, which shows “dephasing” within one vibrational period, from the phase sensitive part Y. Accumulation of a steady state population in the A′ state after the electronic and vibrational relaxation has to be expected. The A′ state has a 76 ms lifetime.¹³ On account of the 1 kHz repetition rate of our laser system, a steady state population is acquired to which each subsequent laser pulse contributes only about 2%. The probe pulse photon energy was chosen to be the minimal one which just suffices to lift B state population from the outer turning point in Fig. 2 to the accessible charge transfer state. In this way lifting of the energetically lower lying accumulated A and A′ state population to charge transfer states is suppressed. This strategy is successful obviously for the phase sensitive signal part which displays also the B state dynamics. The background corresponds to another route which cannot be identified at present. A reinvestigation of the excitation spectrum is planned in this context to clarify the origin of the smooth background on which the zero-phonon and phonon side bands ride (Fig. 3c). It may belong to a different transition, e.g. to a continuum which leads to a phase insensitive contribution. Most important is, however, that in contrast to spectra like Fig. 1 we can sort out this phase insensitive part Z in the pump–probe spectra and concentrate on the phase sensitive signal part Y. Inspection of Fig. 5f shows that the electronic coherence in the B ← X transition is quite long and conserved up to the third vibrational round-trip(660 fs). This observation is analogous to that for the B ← X transition of Br₂ in solid Ar,² where a similar coherent spectroscopy in the condensed phase was used. A first quick decay is followed by a much slower decrease. A more quantitative discussion of this behavior has to wait once more for a clarification of the excitation spectrum.

4.2 Control of wavepacket–bath couplings with phase ϕ

The phase-locked pump scheme in combination with a sufficiently long coherence time of a wavepacket bears the perspective to control the coupling of wavepackets to the multidimensional environment, and to monitor the control effect in the probe step. The experiment displayed in Fig. 6 was designed in this respect. The vibrational modulation and the significant Y part sensitive to phase ϕ indicate the prerequisite of coherence in the vibrational and electronic degrees of freedom. The essence lies in the choice of the lock frequency relative to the vibrational progression shown in Fig. 3 and we recall shortly the strategy. The first pulse at t_r prepares a wavepacket which is composed of all absorptions covered by the envelope of Fig. 3a and the center is close to v′ = 9. Those parts in the wavepacket which preserved coherence after one round-trip interfere with the second phase-locked pulse at t₁ in the same way as the two interference patterns in Fig. 3b and 3d display. The lock frequency (and wavelength 519 nm) was chosen to coincide with phonon side band of v′ = 9. The constructive interference for ϕ = 0 covers the phonon side bands from v′ = 8 up to v′ = 11. More importantly, the minima for ϕ = π indicating destructive interference are situated also just on top of the phonon side bands and especially for v′ = 8–11 (compare Fig. 3b and 3c). Thus, the coherent interaction with the second pulse diminishes the phonon side band contribution and enhances the zero-phonon contribution for ϕ = π. For ϕ = 0 we have just the opposite case and the side bands coincide with the maxima from constructive interference and they are enforced (compare Figs. 3c and 3d). In summary, the phase sensitive part in Fig. 6 reflects wavepackets which are composed for ϕ = π predominantly of zero-phonon lines and vice versa of phonon side bands for ϕ = 0. The spectral weight or area of the zero-phonon lines is lower in the relevant region than that of the side bands. Indeed the initial mean amplitude Y_π for ϕ = π is lower than Y₀ for ϕ = 0. The difference is remarkably large, indicating that the control of zero-phonon wavepackets versus phonon side band wavepackets via the phase ϕ works encouragingly well. The incoherent parts give an indifferent background. The selection quality of these two types of wavepackets is very sensitive to a precise positioning of the constructive or destructive sequence with respect to each one of the bands over the spectral width of the pulse. It covers about v′ = 7 to v′ = 11 according to Fig. 3. The large anharmonicity leads to a variation of the round-trip time of 181 fs for the v′ = 6–7 spacing to 246 fs for the v′ = 10–11 spacing. An interference pattern with a constant fringe spacing Δω would obviously fail to select out a sequence of zero-phonon lines and to suppress all the side bands sufficiently. The measured interference fringes in Fig. 3b and 3d in comparison with Fig. 3c demonstrate that the chosen chirp difference compensates the anharmonicity very well, which is a prerequisite to this control scheme.

The time evolution with Δt in Fig. 6 displays the further fate of these two types of wavepackets. There is in fact a qualitative difference. The mean value for ϕ = π stays constant or increases slightly while that for ϕ = 0 falls for larger Δt. The mean value is a signature of the detection sensitivity and correlated with the energetic position of the wavepacket relative to the energy of the probe window (see Fig. 2). A reduction in amplitude indicates that a wavepacket slides out of the probe window to lower energies due to vibrational energy relaxation. A change in coherence properties is irrelevant in this respect, not only because we take the mean value being insensitive to vibrational coherence, but also because electronic coherence is not probed any more after completion of the pulse pair. Obviously, the Y₀ wavepacket is prone to more efficient vibrational energy relaxation compared to the Y_π wavepacket. It is a generally observed phenomenon that an increase in temperature stimulates vibrational relaxation of a chromophore in the condensed phase due to the large amplitudes in the bath coordinates.²⁹ In zero-phonon transitions no bath mode is affected while for each electronic transition with phonon side bands, the corresponding bath mode quantum number is increased by one. The typical phonon side band energy corresponds to about 50 K in temperature units. The Y₀ wavepackets for phonon side bands are born inherently with larger bath amplitudes compared to the Y_π wavepackets for zero-phonon lines. Thus, in Fig. 6 we demonstrate the control of the coupling to the bath via the phase ϕ of the phase-locked pulse pair and the resulting variation in vibrational energy relaxation. A broader application of this control and detection scheme can be anticipated. Bath-induced nonradiative electronic transitions are abundant and their efficiency is enhanced in general also with temperature.³⁰ Coherent control of photochemical reactions become feasible in this way.

5. Concluding remarks

We have shown that phase-locked pulses are applicable to chromophores in the presence of a condensed environment. The excitation of Cl₂ by phase-locked pulses leads to a signal modulation resulting from the interference of vibrational wavepackets which can be steered with the phase, and distinguishes this signal from a spectrally broad background which is essentially insensitive to the phase. It turns out that the wavepacket keeps its electronic coherence for more than 660 fs while the vibrational coherence extends even beyond 3 ps. Perhaps surprisingly, the simultaneous excitation of a chromophore vibrational wavepacket and a lattice mode provided the higher coherent signal than the zero-phonon excitation; the disturbance by the lattice motions does not destroy the electronic coherence within 220 fs. We have also demonstrated that the strength in coupling between wavepacket and lattice motions can be controlled by tuning the relative phase between the phase-locked pulses.

Acknowledgements

This work is financially supported by the Deutsche Forschungsgemeinschaft via the Sonderforschungsbereich 450.

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Footnote

† Present address: Max-Born Institut, Max-Born-Str. 2a, 12489 Berlin, Germany.

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