Vibrational relaxation and dephasing of Rb2 attached to helium nanodroplets
Barbara Grüner, Martin Schlesinger, Philipp Heister, Walter T. Strunz, Frank Stienkemeier, Marcel Mudrich
aa r X i v : . [ phy s i c s . a t m - c l u s ] F e b Vibrational relaxation and dephasing of Rb attached to helium nanodroplets B. Gr¨uner , M. Schlesinger , Ph. Heister ∗ , W. T. Strunz , F. Stienkemeier , M. Mudrich Physikalisches Institut, Universit¨at Freiburg, 79104 Freiburg, Germany and Institut f¨ur Theoretische Physik, Technische Universit¨at Dresden, 01062 Dresden, Germany (Dated: November 5, 2018)The vibrational wave-packet dynamics of diatomic rubidium molecules (Rb ) in triplet statesformed on the surface of superfluid helium nanodroplets is investigated both experimentally andtheoretically. Detailed comparison of experimental femtosecond pump-probe spectra with dissipativequantum dynamics simulations reveals that vibrational relaxation is the main source of dephasing.The rate constant for vibrational relaxation in the first excited triplet state 1 Σ + g is found to beconstant γ ≈ . − for the lowest vibrational levels v < ∼
15 and to increase sharply when excitingto higher energies.
I. INTRODUCTION
Helium nanodroplet isolation (HENDI) is a well-established technique for isolating molecules and form-ing clusters at low temperature (0.38 K) for spectroscopicstudies [1]. Shifts and broadenings of spectral lines ofmolecules embedded in the helium droplets are small dueto the weak dopant-host interactions as well as to the pe-culiar quantum properties of the superfluid helium nan-odroplets [2, 3]. Nevertheless, the details of the solute-solvent interactions that are at the origin of the observedline shapes are currently being studied with various ap-proaches [4–8]. However, to date no time-resolved stud-ies aiming at resolving the details of the interaction ofexcited molecules with helium nanodroplets have beenperformed.Vibrational relaxation of molecules and molecular com-plexes embedded in helium nanodroplets has been stud-ied by the group of R. Miller using high-resolution in-frared spectroscopy and bolometric detection [5, 9–11].It was found that systems having a large energy gap be-tween the molecular vibration and the excitations of thehelium ( e. g.
HF ( v = 1)) couple very inefficiently tothe helium environment, which leads to slow vibrationalrelaxation times t & . ∗ Present address: Fakult¨at f¨ur Chemie, Technische Universit¨atM¨unchen induced by the ultracold bath of helium atoms attractsan increasing amount of attention both from experimentand theory.The vibrational wave packet dynamics of alkali metaldimers attached to helium nanodroplets has been stud-ied using the femtosecond pump-probe technique in aseries of experiments in our group [3, 18–20]. Alkalimetal atoms and molecules represent a particular classof dopant particles due to their extremely weak bind-ing to the surface of He droplets in bubble-like struc-tures [21–24]. In particular, pump-probe measurementswith K diatomic molecules in singlet states attached tohelium droplets reveal a significant impact of the heliumenvironment on the vibrational dynamics, suggesting themanifestation of a Landau critical velocity for the vibra-tional motion of K on the surface of superfluid heliumnanodroplets [18, 25].In this work we present the detailed analysis of pump-probe measurements of the vibrational wave packet dy-namics of Rb molecules in triplet states attached to he-lium nanodroplets with regard to relaxation and dephas-ing induced by the helium environment. This system isparticularly well-suited for a quantitative study of themolecule-helium droplet interaction due to the preciseknowledge of the spectra and dynamics of gas-phase Rb molecules and due to the weak molecule-helium interac-tions which allow for an accurate theoretical description.The long-lasting wave packet oscillations that we ob-serve up to delay times t > ∼ . molecules desorb off the helium droplets on a shorttime scale t < ∼
10 ps and continue to vibrate freely inthe gas-phase [20]. This assumption was backed by thegood agreement between the measured and theoreticallypredicted vibrational frequencies. Besides, the measure-ment of beam depletion using a separate detector wasinterpreted as clear evidence that excited Rb moleculesdesorb off the droplets on the time scale of the flight timefrom the laser interaction region to the detector ( ∼ dimers aswell as theoretical simulations on K atoms attached tohelium nanodroplets indicated desorption times in rang-ing 3 – 8 ps and 10 – 30 ps, respectively [18, 26].A more detailed inspection of our data reveals, how-ever, that the Rb molecules are subject to continuousvibrational relaxation due to the constant coupling tothe bath of helium atoms on the time scale of the pump-probe measurements. The experimental signature of thecoupling of vibrating Rb to the helium is the decreas-ing contrast of wave packet oscillation signals as well aschanging amplitudes of individual Fourier frequency com-ponents due to the redistribution of populations of vibra-tional states. In particular, the pronounced dependenceof the dephasing time on the quantum number v of ex-cited vibrational levels points at system-bath couplingsbeing active. This observation is in line with earlier mea-surements of the fluorescence emissions of Na moleculesin triplet states, which indicated vibrational relaxationin the excited electronic state to take place on the timescale of the life time of the excited state due to sponta-neous emission ( ∼
10 ns) [27]. Recently, it was observedthat desorption upon electronic excitation may even becompletely inhibited in the case of Rb atoms excited ina particular laser wave length range [28].Pioneering experiments on the vibrational dephasingand relaxation of molecules ( I ) exposed to collisionswith rare gas atoms at high density were performed bythe Zewail group [29], motivating theoretical studies byEngel, Meier et al. [30, 31]. More recently, dephasingtimes as well as relaxation rates have been studied ex-tensively by the groups of Apkarian and Schwentner bymeans of femtosecond spectroscopy of the vibrational dy-namics of halide molecules isolated in cryogenic rare-gasmatrices [32–37]. Seminal work on time-resolved mea-surements of the dissipative fluid dynamics in bulk He-II has been performed using femtosecond pump-probespectroscopy of triplet He ∗ excimers created inside He-II [38]. Due to the strong coupling of the highly excitedHe ∗ to the surrounding He which forms an extended bub-ble around He ∗ the dynamics is fully damped after oneperiod of motion.Theoretical studies on collisional quenching of rota-tions and vibrations of alkali dimers and other smallmolecules by helium atoms at low temperatures have re-cently been stimulated by the prospects of creating sam-ples of cold molecules using buffer-gas cooling as well assympathetic cooling with ultracold atoms as a coolingagent [39–41]. In the system Li +He, for instance, thequenching rate constants in the approximation of van-ishing temperature are predicted to increase by aboutone order of magnitude with increasing vibrational levels v = 0 −
10 [39].In the case of weak couplings, which applies to oursystem, the concept of perturbations of the vibrationallevels of the molecules by fluctuations in the bath modesis well-established. It leads to vibrational energy relax-ation along with the decay of vibrational coherences.There may well be additional pure dephasing mecha-nisms that are expected to vanish in the low temper-ature limit [42, 43]. In the related energy gap picturethe populations of individual vibrational energy levels v relax stepwise to the next lower vibrational energy lev-els v − v [34, 43, 45]. The evolution of coherences andpopulations of vibrational levels in the weak-couplinglimit is often being modeled with the master equationdescription, obtained from the anharmonic molecular os-cillator coupled to a harmonic bath. Various couplingterms are used for describing different interaction mech-anisms [34, 43, 46–48].Following established practice in the chemical literature,in this contribution we use “dephasing” to describe thegeneral mechanism of loss of coherence between quantumstates. Note that in other fields, this process is prefer-ably referred to as “decoherence” while “dephasing” isthen used to describe that special occurrence of decoher-ence, where no dissipation is involved. II. VIBRATIONAL WAVE PACKET DYNAMICS
The experimental arrangement used for recording fem-tosecond pump-probe photoionization transients is iden-tical to the one described previously [20]. In short, a con-tinuous beam of helium nanodroplets of the size of about8000 He atoms is produced by expanding high-purity Hegas out of a cold nozzle (T ≈
17 K, diameter d = 5 µ m) athigh pressure (p ≈
50 bar). The helium droplets are dopedwith two Rb atoms on average per droplet by passingthrough a pick-up cell that contains Rb vapor at a pres-sure p Rb ≈ × − mbar. Alkali atoms and moleculesare peculiar dopants in that they reside in bubble-likestructures on the surface of He droplets. Upon formationof a Rb diatomic molecule, the binding energy is dissi-pated by evaporation of helium atoms from the dropletsand occasionally by desorption off the droplets of thenewly formed molecule itself. This leads to an enrich-ment of droplet-bound Rb molecules in weakly boundtriplet states.Further downstream, the doped He droplet beam in-tersects the laser beam inside the detection volume of acommercial quadrupole mass spectrometer. Due to thelimited mass range, only bare Rb +2 photoions are detectedmass-selectively. The laser beam consists of pairs ofidentical pulses produced by a commercial mode-lockedTi:sapphire laser and a Mach-Zehnder interferometer toadjust the time delay between the pulses. The pulseshave a duration of ≈
160 fs and a spectral bandwidthat half maximum of ∆ ω las ≈
80 cm − and peak pulseintensity ∼ .A pronounced oscillatory photoionization signal isobserved in the pump-probe transients for laser wavelengths in the range λ = 960 nm – 1032 nm. A typicalmeasured pump-probe transient recorded at λ = 1006 nmis depicted in Fig. 1 (a). On the time scale of picosec-onds, the transient signal is modulated by wave packet(WP) oscillations with a period T Σ g ≈ .
95 ps (see inset).In addition, this oscillation is amplitude- and frequency-modulated due to dispersion and subsequent revivals ofthe WP motion in an anharmonic potential. On the long
190 200 210 220 230 240
190 200 210 220 230 240
SimulationExperiment R b + i on s i gna l ( c oun t s / s ) Delay time (ps) a) b) FIG. 1: Experimental (a) and simulated (b) pump-probe tran-sients of Rb formed on helium nanodroplets recorded at thelaser wave length λ = 1006 nm. time scale of hundreds of ps, the contrast of WP oscil-lations degrades monotonically and eventually vanishesat delay times & molecules to the helium droplets.Other sources of dephasing are conceivable: gas phasecollision with evaporated gas and clusters of helium ordephasing due to the influence of rotations. However, anestimate of the Rb -He gas phase collision rate gives avalue far too small to account for the observed data. Asfor the influence of rotations, it is clear that an initialthermal population of rotational levels leads to a sim-ilar decay of signal contrast as observed in the experi-ment [49]. Again, in our case a detailed analysis showsthat in order to account for the observed decay rates, un-physically large temperatures would have to be assumed.Moreover, the observed functional dependence of decayrates on the laser wavelength cannot easily be explained.Taking into account all these findings, our picture of vi-brational damping of the dimer through the interactionwith the helium droplet allows for the most consistentexplanation of all observed phenomena. A. Free gas phase dynamics
Let us first briefly review the fundamental aspects ofpump-probe spectroscopy of diatomic molecules isolatedin the gas-phase. A first pump pulse excites a coher-ent superposition of vibrational states (WP) in an ex-cited electronic state. After some time delay a secondprobe pulse projects the WP to a final ionic state whichis detected as a function of time delay between the twopulses. Fig. 2 shows the relevant potential energy curves
10 15 20 2510 15 20 250123 v=5 v=0 Rb+Rb + Internuclear distance [a ] P o t en t i a l ene r g y [ c m - ] Rb +2 P r obe P u m p a FIG. 2: (Color online) Selected triplet potential-energy curvesof neutral Rb and of the Rb +2 ionic ground state relevantto the present study. The arrows indicate the creation of avibrational wave packet in the first-excited state followed byresonant two-photon ionization. of the Rb molecule in the triplet manifold. The straightarrows symbolize the pump and probe pulse excitationpathways.Due to the cold helium environment, at time t = 0 onlythe vibrational ground state on the lowest triplet state a Σ + u is occupied. Detailed experimental and theoret-ical analysis reveals that rotational degrees of freedomneed not be taken into account (see also Sec. III). Asoutlined in Sec. II C, we fully solve the time-dependentSchr¨odinger equation involving all relevant potential en-ergy surfaces. Here, we want to point out that the maineffect of the pump pulse is to generate a coherent su-perposition of vibrational eigenstates | v i on the excitedelectronic state surface (1) Σ + g . More specifically, thecreated WP can be written as | ψ e ( t = 0) i = P Nv =0 c v | v i ,where c v denotes complex expansion coefficients and v =0 , , , . . . is the vibrational quantum number. CoherentWPs will also be created in higher lying electronic statesor, by resonant impulsive stimulated Raman scattering(RISRS) in the triplet ground state, with a significantlysmaller amplitude, though [50]. Thus, all linear and non-linear processes are fully taken into account in our simu-lations. The respective populations of vibrational levels | c v | depend on the pump pulse parameters and on theFranck-Condon factors of the excitation transition. Thepump pulse wave length determines the central vibra-tional level, while the pulse width, the pulse energy andthe Franck-Condon factors determine the number andthe relative populations of vibrational levels. The createdWP in the (1) Σ + g -state propagates in the region betweenthe classical inner and outer turning points. Direct inte-gration of the Schr¨odinger equation i ~ ∂ t | ψ e i = H e | ψ e i yields | ψ e ( t ) i = N X v =0 c v e − i Evt ~ | v i , (1)where E v is the energy of the v -th vibrational level. Asdiscussed below, the measured signal allows to extract in-formation about the density matrix of vibrational states ρ e ( t ). For the isolated dimer, ρ e ( t ) = | ψ e ( t ) ih ψ e ( t ) | de-scribes a pure state at all times. Its time evolution isgiven by ρ e ( t ) = X v,v ′ c v c ∗ v ′ e − i ( Ev − Ev ′ ) t ~ | v i h v ′ | . (2)The diagonal elements of the density matrix ρ vv = | c v | represent the populations that are constant in time, whilethe off-diagonal elements ρ vv ′ ( t ) ≡ h v | ρ e ( t ) | v ′ i with v = v ′ oscillate with Bohr frequencies ω vv ′ = ( E v − E v ′ ) / ~ andrepresent the coherences between the vibrational eigen-states | v i , | v ′ i .The probe pulse produces photoions through a reso-nant 2-photon-transition from the excited state to theionic ground state Σ + g of Rb +2 . Transitions preferablytake place when the WP is located around a well-definedtransition region, where the transition dipole matrix ele-ment is maximal (Franck-Condon window). Even thoughour simulations are numerically exact, it is instructive toconsider the perturbative dependence of the ion signal onthe WP density matrix [51–54], S ( t ) = X vv ′ A vv ′ ρ vv ′ ( t ) . (3)The coefficients A vv ′ in Eq. (3) contain products of tran-sition moments and field parameters and provide infor-mation about the vibrational populations in the finalstate. Through the dependence on the density matrix,the signal S is composed of beat frequencies ω vv ′ betweenall pairs of energy levels that contribute to the WP. Themost prominent oscillation in the signal originates fromcomponents ω vv +1 and reflects the circulation of the WPon the potential energy surface. From the Fourier spectraof the signal information about higher-order frequencycomponents ω vv +∆ v with ∆ v > E n = ~ ω e n ,where n = 0 , , , . . . denotes the number of eigenstate | n i , Eq. (1) yields a periodic oscillation with classicalperiod T c = h/ ∆ E = 2 π/ω e , where ∆ E denotes the con-stant energy spacing between adjacent levels n . The sig-nal S in Eq. (3) features a periodic oscillation with period T c . In the anharmonic Morse potential with energy spec-trum E v = ~ ω e ( v − x e v ), however, initially well-localizedWPs spread out due to dispersion on the characteris-tic time scale T disp = ~ ˆ ω/ ( ω e x e ∆ E pump ) [37, 55, 56].Here, x e is the anharmonicity constant, ˆ ω denotes thecentral vibrational frequency of the WP and ∆ E pump is the spectral energy width of the pump laser pulse.Considering the parameters of our experiment we obtain T disp ∼
100 ps. Around t = T disp , all contributions on theright-hand side of Eq. (3) appear uncorrelated, whichmeans that the oscillatory signal collapses [57]. There-fore, dispersion of the WP leads to a decay of the pump-probe signal contrast. Note, however, that due to disper-sion neither populations ρ vv nor the absolute values ofcoherences | ρ vv ′ | , v = v ′ , change in time.In the Morse potential, a revival of the initial WPtakes place at certain times, i. e. the original phase cor-relation in the WP is restored and the WP partly orfully revives. At the full revival time, the signal ampli-tude ideally reaches its initial height, which underlinesthat coherence is preserved. Full revivals occur at times t = k × T rev /
2, where T rev = 2 π/ ( ω e x e ), when all vi-brational eigenstates have accumulated a phase of 2 πk with k = 1 , , , . . . . At fractions of the revival time, t = p/q × T rev where p/q is an irreducible fraction of inte-gers, the WP consists of a superposition of q copies of theoriginal WP (fractional revivals) [58, 59]. For instance, athalf-period revivals ( p/q = 1 /
2) the initial well-localizedWP evolves into a highly quantum mechanical state thatconsists of two counter-propagating partial WPs that in-terfere with each other when colliding. In the electronicstate of relevance for the present analysis, 1 Σ + g of Rb ,which perfectly matches the shape of the Morse poten-tial in the accessible range of v -states, the first full revivaltime is T rev / ≈
160 ps [20].
B. Wave packet dynamics with dissipation
The previous discussion was devoted to isolated vibrat-ing diatomic molecules in the gas-phase. Let us nowconsider Rb molecules (M) coupled to the dissipativeenvironment realized by helium nanodroplets (HND), towhich the molecules are attached.A Rb molecule attached to a HND is a closed butcomplicated system which can be described by the Hamil-tonian H = H M + H HND + H M ↔ HND . H M denotes theisolated molecule as discussed before, H HND is the Hamil-tonian for the pure helium nanodroplet and H M ↔ HND contains the interaction between the two. As we are onlyinterested in the dynamics of the molecule we call thisour ”system” and the helium nanodroplet our ”bath”.In our experiments the WP dynamics of the coupled Rb molecule is mostly very similar to that in the gas phase,which means that we see the same fast oscillations, WPdispersion with time constant T disp , and (fractional) re-vivals at times T rev . However, on the long time scale ofthe experiment (nanoseconds) the oscillatory signal ex-ponentially decays due to slow system dephasing. Thus,a description in terms of a weak system-environment cou-pling is justified.In the experiment, we observe a decay of the revivalamplitudes with a rate γ D . As we will show, this decay isrelated to the environment-induced dephasing of the WPdue to dissipation. Dephasing can also be caused by aprocess which only affects the off-diagonal elements of thedensity matrix (no dissipation). This process is referredto as ”pure dephasing”. For the well-known two-levelsystem, the overall dephasing time constant T is relatedto relaxation ( T ) and pure dephasing without dissipa-tion ( T ∗ ) by 1 /T = 1 / (2 T ) + 1 /T ∗ . For multi-leveloscillators, which we consider here, the relation betweendephasing and dissipation is more subtle. No simple gen-eral expression connecting the corresponding time scalesexists: depending on the shape of the WP, dephasing maytake place on a much shorter time scale [60–63]. We re-call that it is important to distinguish between contrastdecay due to dephasing – which is an irreversible pro-cess – and the reversible drop of the observed oscillationamplitude due to dispersion in an anharmonic potential.The dissipative vibrational dynamics is described us-ing the framework of Markovian master equations. Atthis stage, we do not aim at deriving such an equationfrom a microscopic Hamiltonian, which would require de-tailed knowledge of the helium “bath” and interactionHamiltonians, H HND and H M ↔ HND , respectively. In-stead, we choose a well-established Markovian quantumoptical master equation [64] for a weakly coupled environ-ment. The density operator ˆ ρ ( t ) of the (reduced) systemthat describes dissipation in near-harmonic systems atzero temperature is given by ∂ t ˆ ρ = 1 i ~ h ˆ H M , ˆ ρ i + X j (cid:18) ˆ L j ˆ ρ ˆ L † j − { ˆ L † j ˆ L j , ˆ ρ } (cid:19)| {z } coupling to bath . (4)This equation is of Lindblad form [65]. To describe dissi-pation, we use L i = √ γ i ˆ a i , where ˆ a i is the usual quantummechanical ladder operator, defined through the harmon-ically approximated potential energy curve i . This Lind-blad operator induces vibrational dissipation on the timescale 1 /γ i . More specifically, independently of the initialconditions, to good approximation, the mean energy ofthe excited WP decreases exponentially with a respectiverate γ i .The relaxation rate constants γ i in Eq. (4) are takenas fit parameters to match the experimental data. Thechosen dissipative Lindblad operator also affects the off-diagonal elements of the density matrix. The latter de-cay with time due to dephasing which implies a transi-tion from an initially pure state to a state mixture. For t → ∞ , only the ground vibrational state is occupied, Time (ps) (0,1)(1,2) (5,6)(2,3) (4,5)(3,4) (5,6)(4,5)(3,4) (2,3)(1,2) (b) Morse potential O ff - d i agona l m a t r i x e l e m en t s v v + (a) Harmonic potential (0,1) FIG. 3: (Color online) First-order coherences (absolute val-ues of the off-diagonal elements of the density matrix, ρ v v +1 ), v = 0–6, for vibrational wave packet dynamics in the 1 Σ + g -state potential of Rb (b) in comparison with the harmoni-cally approximated potential (a). which is, of course, a pure state again. For the masterequation (4) it is well known that localized WPs are “ro-bust” in the sense that they suffer only little dephasing.In contrast, two such WPs separated by a (dimensionless)distance D in phase-space loose their coherence with anaccelerated rate D γ [66–69].This effect is visualized in Fig. 3, in which the time evo-lution of the first-order coherences | ˆ ρ vv +1 | of the dynam-ics in the Morse-type potential of the 1 Σ + g -state of Rb (b) is compared with the dynamics in the harmonicallyapproximated potential (a). The shown initial distribu-tion of coherences is obtained when exciting a WP formedof vibrations v ≈ λ = 1025 nm. The matrix elementsˆ ρ vv ′ are computed by solving Eq. (4) numerically and byprojecting onto the eigenstates | v i at every time step. Inthe case of a harmonic potential (Fig. 3 (a)), beats be-tween low-lying levels n < ∼ ,
2) and (2 ,
3) incontrast to the faster decay of (3 ,
4) and (4 ,
5) is partlyreminiscent of the vibrational redistribution mentionedbefore. In addition, higher excited levels decay faster.Thus, for the Morse oscillator the overall dephasing ap-pears to be accelerated with respect to the harmonic os-cillator. Moreover, beats (2 ,
3) and (3 ,
4) in Fig. 3 (b) areperiodically modulated with period T rev /
2. In particu-lar, a slow loss of coherence or even a momentary increaseof coherence is apparent at times close to the full vibra-tional recurrences when the WP is well-localized again.Around the half-period fractional revivals, when the WPsplits into partial WPs that are maximally delocalized(large D ), however, dephasing is fastest. C. Numerical simulation
In order to reproduce the ion yield in the gas phase, wecalculate the final state probability after the interactionwith the laser field. For the isolated dimer, we here followthe approach of [70] and fully numerically solve the timedependent Schr¨odinger equation ∂ t | Ψ( t ) i = − i ~ H M | Ψ( t ) i (5)for the full state vector Ψ = ( ψ g , ψ e , . . . ). The Hamiltonoperator H M now also contains the field interaction withthe molecule. In particular, we take into account thatthe final state consists of the bound ion plus an ejectedelectron with energy E . Following the approach of [71],we use a discretization of the electronic continuum. Wedetermine the final state probability | ψ f ( E, τ ) | for dif-ferent pump-probe delays τ and electronic energies E .Adding contributions with different energies E , we ob-tain a signal S ( τ ), which is proportional to the gasphase ion yield. For the isolated molecule, Eq. (5) alsodirectly allows to obtain the density operator throughˆ ρ ( t ) = | Ψ( t ) ih Ψ( t ) | . Potential energy surfaces and tran-sition dipole moments were provided by O. Dulieu [72].For our phenomenological description of the helium in-fluence on the dimer dynamics, we switch to the densitymatrix description, Eq. (4). Our aim is to ascribe cer-tain damping parameter values γ i ( λ ) to the measuredpump-probe signal at wave lengths λ . From a numericalpoint of view, the evolution of the density matrix canbecome very costly, in particular, if one considers manypotential energy surfaces and/or many vibrational states.We therefore return to an equation for the state vector, i. e. to a Schr¨odinger-type equation including relaxation(and thus, dephasing). The density matrix of the masterequation (4) is recovered from the stochastic Schr¨odingerequation on average, ˆ ρ ( t ) = | Ψ( t ) ih Ψ( t ) | [73]. In practice,one has to determine many realizations of state vectorsΨ SSE ,i , which can then be used to extract the densitymatrix (Fig. 3), coordinate, momentum, energy expecta-tion values , or the final state probability (Fig. 1 (b)) tocompare with the experiment. III. PUMP-PROBE SPECTRA
Upon laser excitation of Rb molecules formed on he-lium nanodroplets at wave lengths in the range λ =960 nm – 1032 nm, coherent vibrational WPs are created in the first excited triplet state 1 Σ + g as well as in thelowest triplet state a Σ + u by RISRS with varying rela-tive intensity. Around λ = 1010 nm, the pump-probesignal as shown in Fig. 1 (a) is dominated by WP motionin the 1 Σ + g -state. The amplitude modulation resultsfrom dispersion of the WPs and half-period as well asfull recurrences are observed with high contrast at revivaltimes T rev / ≈
80 ps and T rev / ≈
160 ps, respectively.The nearly exponential decrease of the signal contrast isattributed to relaxation-induced dephasing and will beinvestigated in detail in the following. The simulatedtransient ( γ Σ g = 0 .
45 ns − ), depicted in Fig. 1 (b), nicelyreproduces both the vibrational recurrences as well as theoverall damping due to vibrational dephasing.At short delay times t < ∼
50 ps we observe slight devia-tions between the simulated and experimental transientsignals, which we attribute to the influence of rotations,as seen previously for iodine molecules [74]. As men-tioned in the introduction, a dephasing influence of ro-tations on pump-probe spectra is conceivable [49], yetrequires unphysically high temperatures & K in ourcase.Gas phase simulations of the vibrational WP dynamicsincluding free rotations in the Rb system feature notablerotational recurrences at half and full rotational periods T rot / ≈
575 ps and T rot ≈ moleculespresumably with the molecular axis being oriented par-allel to the surface [75], we expect the rotation to de-compose into weakly perturbed in-plane rotation andstrongly hindered out-of-plane rotation which more likelyresembles a pendular motion. The latter may efficientlycouple to surface modes of the droplets causing fast relax-ation. Couplings between vibration, rotation, and libra-tion may therefore induce intricate relaxation dynamics,which, however, would require more expanded simula-tions that lie beyond the scope of the present workThe experimental pump-probe signal of Fig. 1 (a) isanalyzed by Fourier transforming the time trace inside atime window of a width of 5 ps and a Gaussian apodiza-tion function with full width at half maximum (FWHM)of 2 . λ = 1006 nm is displayed in Fig. 4 (a). Inthis representation, the individual WP oscillations areno longer resolved, but the full, half-period and evenone third-period revivals are clearly visible and can beattributed to frequency-beats between vibrational statesseparated by ∆ v = 1, 2, and 3 vibrational quanta, re-spectively [59]. The fact that the WP recurrences areseen with such an extraordinarily high contrast even atlong delay times is a consequence of the shape of the1 Σ + g -potential that nearly perfectly matches that of theMorse potential in combination with weak system-bathcouplings [20].
20 30 40 50 60 70 80 90 100 110 12020040060080010001200300400500600700800900 exp. sim.
34 34.2 34.4 34.6 34.8 35 35.2 35.400.20.40.60.8
Frequency [cm -1 ] Frequency [cm -1 ] T i m e [ p s ] T i m e [ p s ] FT a m p . [ a r b . u .] a) c) b) FIG. 4: (Color online) Sliding window Fourier spectra (spec-trograms) of the pump-probe transients recorded at λ =1006 nm. The width of the time window is 5 ps in (a) and400 ps in (b). (c) displays the power spectrum of the integralexperimental and theoretical transients. Fig. 4 (b) displays a magnified view of the spectro-gram of the same data when using a time window ofwidth 400 ps and an apodization function with FWHM209 ps in the spectral range ν = 34–35 . − . In thisrepresentation of the data, the frequency resolution iscomparable to the one obtained by transforming the in-tegral data set (Fig. 4 (c)) while still retaining the dy-namics on the long time-scale. The individual frequencycomponents reflect beats between coherently excited ad-jacent vibrational states that are unequally spaced dueto the anharmonicity of the potential. By comparing tothe Fourier spectrum of the simulated data in Fig. 4 (c)we conclude that the WPs excited at λ = 1006 nm arecomposed of vibrational states v = 6–11 with relativeamplitudes determined by the spectral intensity profileof the fs laser. (b) D ( v = , ) / D ( v = ) Mean vibrational level < v > v=1 v=2 v=3Laser wave length (nm) 98599510061015 D e c a y r a t e D [ n s - ] (a) FIG. 5: (Color online) Exponential decay time constants ob-tained by fitting the maxima of the full (∆ v = 1) as well asfractional (∆ v = 2 ,
3) revivals plotted against the average vi-brational quantum number of vibrational states initially pop-ulated by the pump pulse h v i . The filled symbols refer to thewave packet dynamics in the excited 1 Σ + g -state, the opensymbols refer to the a -state. A. Analysis of dephasing dynamics
In a first attempt to analyze the loss of contrast of thecoherent WP oscillation signal the envelopes of the beatsignals corresponding to ∆ v = 1 , , ∝ exp( − γ D t ) to infer the characteristic decay rate γ D .The fit results for γ D are depicted in Fig. 5 as a func-tion of the mean vibrational quantum number h v i deter-mined by the central laser wave length. The horizontalerror bars reflect the width of the distribution of excitedvibrational levels due to the laser band width, the verticalerror bars depict the fit errors. Strikingly, the decay ofcontrast is strongly dependent on the level of vibrationalexcitation and features rapidly increasing decay rates γ D with increasing v . The solid lines in Fig. 5 (a) representmodel curves obtained by fitting quadratic functions tothe data.Similar behavior was observed in time-resolved coher-ent anti-Stokes Raman-scattering measurements of theWP dynamics of molecular iodine I in the groundstateisolated in rare-gas cryo-matrices [32–34]. There, thetransition from a linear v -dependence of γ to a quadraticdependence with increasing temperature of the matrixwas observed. Linear v -dependence at low temperatureswas interpreted in terms of dephasing induced only byvibrational energy relaxation whereas at higher matrixtemperatures pure elastic dephasing also contributed.Coherences between vibrational states spaced by ∆ v = n > γ ∆ v =2D /γ ∆ v =1D ≈ . γ ∆ v =3D /γ ∆ v =1D ≈ γ ∆ v = n D with the order n of the beat has beendiscussed in the context of different mechanisms of puredephasing [48]. Depending on the collision model consid-ered in that study, a scaling behavior ranging from ze-roth to second order with n was expected. Our approachwould require the inclusion of pure dephasing terms toaccount for ∆ v -dependent decay times whereas in thecurrent model, where dephasing is solely induced by dis-sipation, the revival decay times are found to be inde-pendent of ∆ v .The open symbols in Fig. 5 depict decay rate constants γ a D for the WP dynamics in the lowest triplet a Σ + u -state.While γ a D of the first order coherence (∆ v = 1) is similarto those of the excited 1 Σ + g -state for small h v i , γ a D forthe higher order coherences are significantly higher thanfor the 1 Σ + g -state dynamics at low h v i . This result isreproduced by our assumption of vibrational relaxationand does not imply additional dephasing mechanisms.Note, that the initial distribution of v -state populationsin the a -state is very different from that in the 1 Σ + g -state. While in the 1 Σ + g -state several v -levels are popu-lated with similar intensities, in the a -state the popula-tion is peaked at v = 0 and higher v -levels are much lesspopulated by RISRS. Therefore, the γ a D values to goodapproximation reflect dephasing rates between individual v -levels, since the beat signal of n -th order coherence ismainly composed of just one beat frequency.As we will see later, in the present case of Rb cou-pled to helium nanodroplets, vibrational relaxation islikely to be the main source of dephasing. Yet, puredephasing without population transfer does contributeto some extent. In a more complete description bothdissipative as well as additional pure dephasing termsshould be included to account for the observed v - and∆ v -dependences. For the sake of restricting the modelto the essential features of the problem, however, in thefollowing discussion we focus on the model calculationsthat are based on vibrational relaxation. B. Numerical simulation
In order to obtain a more quantitative description ofthe observed dynamics, the experimental data are mod- eled using the method outlined in Sec. II C. The onlyadjustable parameters entering the simulation are the en-ergy relaxation rate constants in the triplet ground andfirst excited states, γ a and γ Σ g , respectively, as well asrelaxation constants for the two probe states 3 Π u and4 Σ + u . The resulting spectrograms of the best fits to theexperimental data are displayed in Fig. 6 (bottom row)for the selected laser wave lengths λ = 1025 , , and970 nm. For comparison, the top row depicts the exper-imental data and the middle row shows the simulationwhen relaxation is absent ( γ a = γ Σ g = 0).The transient recorded at λ = 1006 nm (center col-umn in Fig. 6), already shown in Fig. 4, is dominated bythe fundamental as well as by the first and second over-tone beats of the 1 Σ + g -state. The experimental data(Fig. 6 (d)) are very well reproduced by the numeri-cal simulation for a damping constant γ Σ g = 0 .
45 ns − (Fig. 6 (f)), whereas the agreement is clearly worse whenno damping is assumed (Fig. 6 (e)). At laser wavelengths λ = 1025 nm and λ = 970 nm, WP oscillations inboth ground a Σ + u and excited states 1 Σ + g are present.At λ = 1025 nm, the excited state-dynamics clearlyfades away more slowly than at λ = 1006 nm, whichis in agreement with the simulated data when setting γ Σ g =0 .
36 ns − (Fig. 6 (c)). In contrast to the 1 Σ + g -stateWP-dynamics, the a -state beats feature less visible dis-persion and recurrences of the WP motion. This is due tothe fact that predominantly the vibrational groundstate v = 0 is populated by RISRS. Consequently, the funda-mental spectral component ω a ≈
13 cm − is mainly com-posed of the beat frequency ( E v =1 − E v =0 ) / ( h c ), with lit-tle contributions of ( E v =2 − E v =1 ) / ( h c ) and higher levelbeats [20]. Best agreement with the experimental data isobtained for γ a = 3 ns − (Fig. 6 (c)). When no dampingis assumed, the simulation clearly severely deviates fromthe experimental data (Fig. 6 (b)). We attribute the ad-ditional spectral features to the dynamics in the higher-lying electronic state 3 Π u which has a similarly shapedpotential curve as the 1 Σ + g -state. In all simulations in-cluding vibrational relaxation, the damping constants ofthe 3 Π u and of the 4 Σ + u -states are set to 0 . − toachieve fast damping of the corresponding WP dynamics,no direct WP signal related to these states is observedin the experiment. However, the inclusion of the twostates is crucial in order to reproduce the experimentallyobserved WP signals in the a -state. The excited state dy-namics at λ = 970 nm is only visible in the time range 0– 20 ps, after which the a -state dynamics prevails. Thisbehavior is reasonably well reproduced by the simula-tion when assuming very fast relaxation ( γ Σ g = 0 . − ,Fig. 6 (i)). In contrast, the same simulation with γ Σ g = 0shows a dominant contribution of the excited 1 Σ + g -statedynamics (Fig. 6 (h)). At λ = 980 nm (not shown inFig. 6), 1 Σ + g -state components are still visible duringdelay times 0 – 100 ps, which implies fast relaxation at arate γ Σ g = 0 .
01 ps − .An even more detailed verification of the numericalmodel is achieved by comparing the experimental and FIG. 6: (Color online) Comparison between experimental (top row) and theoretical (middle and bottom row) data in spec-trogram representation at selected laser wave lengths; the middle row shows the simulation of undamped vibration withoutcoupling to the helium droplets; the bottom row shows the simulation including damping. (0,1)(1,2)(2,3)(3,4)(4,5)(5,6) I n t en s i t y [ a r b . u .] R e l a t i v e i n t en s i t y [ a r b . u .] Experiment Simulationa) c)d)b) 600 1000 200 600 1000
FIG. 7: (Color online) Time evolution of individual beatsbetween adjacent vibrational levels (∆ v = 1) of the 1 Σ + g -state extracted from spectrogram analysis of the transientat λ = 1025 nm with a 400 ps-time window ((a) and (b)) incomparison with simulated density matrix elements | ρ vv +1 | ((c) and (d)). Plots (b) and (d) show the same data as (a)and (b) where the beats are normalized to the sum of all beatsat each time step. theoretical data in the spectrogram representation usinga long time window of width 400 ps, as shown in Fig. 4(b). The high spectral resolution retained in this anal-ysis allows to compare the time evolution of individualbeats between adjacent vibrational states. The ampli- tudes of individual frequency components are extractedfrom vertical cuts through the spectrograms at maximumpositions and are plotted in Fig. 7 for λ = 1025 nm. Pan-els Fig. 7 (a) and (b) represent the experimental data,where in (b) each amplitude component is normalized tothe sum of all contributing beat amplitudes. Althoughall of the frequency components except the lowest one( v = 0, v = 1) decay in absolute amplitude (Fig. 7 (a)),the relative amplitudes only decrease in the case of thehigh-lying level beats (4, 5) and (5, 6), whereas the lowerbeats (3, 4) remain constant or even rise [(0, 1), (1, 2),and (2, 3)] in amplitude in proportion to the sum of all.The numerical simulations (Fig. 7 (c) and (d)) showthe evolution of the first-order coherences of the densitymatrix. The good agreement justifies the assumed modelbased on vibrational relaxation and highlights the pos-sibility of extracting information about the density ma-trix by appropriately analyzing the measured ion yields.The general decay and oscillatory behavior of individ-ual beats is well reproduced by the numerical simulationexcept for the (0, 1)-beat which is extraordinarily promi-nent in the experimental data. A slight increase of theabsolute beat amplitude of the (0, 1)-component can onlybe explained by a redistribution of population of higher-lying vibrational levels into lower-lying ones. The weakperiodic modulations of both experimental and theoret-ical curves are reminiscent of the revival structure thatbecomes more pronounced as the Fourier time windowis reduced. The different decay rates for the individualvibrational beats have been discussed in terms of vibra-tional redistribution in the harmonic and anharmonic os-cillators, (Sec. II B and Fig. 3).Since all of the beats are subject to dephasing the vi-0
123 40
Time [ps]00.10.20.3 P opu l a t i on < v > a)b)
234 80100120140160 < E > [ c m - ] FIG. 8: (Color online) (a) Time evolution of the populationsof individual vibrational states extracted from the simulationat λ = 1025 nm. (b) Evolution of the vibrational level v pop-ulated on average and of the mean vibrational energy h E v i . brational redistribution is masked by an overall decay inthe representation of absolute amplitudes in Fig. 4 (a)and (c). However, from the simulation we can extractinformation about the evolution of populations of the in-dividual vibrational states. To this end, the diagonal el-ements of the density matrix are computed for each timestep by projecting the wave function onto the vibrationaleigenfunctions. The resulting populations of levels v = 0– 6 are depicted in Fig. 8 (a). Fig. 8 (b) shows the timeevolution of the quantum number of the vibrational statethat is populated on average as well as the correspondingaverage vibrational energy. Accordingly, at λ = 1025 nmthe vibrational populations relax down by about 1.8 vi-brational quanta during 1.5 ns. The corresponding vi-brational energy is reduced by about E diss = 73 cm − .Note that at shorter wave lengths the amount of de-posited vibrational energy into the droplets in this timeinterval is considerably larger, e. g. E diss = 157 cm − at λ = 1006 nm and E diss = 656 cm − at λ = 970 nm.At such high rates of energy transfer to the heliumdroplets one has to consider the droplet response in termsof heating, superfluidity and cooling by evaporation ofhelium atoms. Dissipation of vibrational energy up to E diss = 656 cm − into the droplets leads to a significantrise in droplet temperature, which may locally exceed thetransition temperature to the superfluid phase (2 .
17 K forbulk helium) of even the boiling point. Note, however,that cooling of the droplets due to evaporation of he-lium atoms may counteract the heating process. In theconsidered energy range, effective cooling is expected toset in on a time scale of ∼
100 ps [76]. Thus, slow en- -3 -2 -1 R e l a x a t i on r a t e [ p s - ] Mean vibrational quantum number
FIG. 9: (Color online) Dependence of the damping parameterobtained from fitting the simulation to the experimental dataas a function of the average vibrational level populated atdifferent laser wave lengths. Filled and open symbols refer tothe 1 Σ + g - and the a -states, respectively. ergy transfer from the molecules to the droplets at excita-tions to low-lying v -levels could be partly compensatedby evaporation of helium atoms ( E diss ≈ − − perevaporated atom), whereas fast energy transfer may leadto effective heating, to subsequent local disequilibriumstates and even to the breakdown of superfluidity.However, after the time of flight of the droplets tothe beam depletion detector ( ∼ from the droplets, as assumed previously.This assumption is supported by the observed significantblue-shift of the beam depletion signal with respect to thepump-probe-photoionization spectrum (Fig. 11 in [20]).In the spectral range λ & γ i obtained by fitting the model to the ex-perimental data. Fig. 9 shows γ a and γ Σ g as a functionof the average vibrational quantum number correspond-ing to WPs created at different laser wave lengths (topscale). Interestingly, in the range v = 2 – 14 the damp-ing parameter remains nearly constant, γ Σ g ≈ . − ,even though significantly varying decay rates γ D havebeen measured (Fig. 5). This discrepancy reflects thescaling behavior of dephasing times with v , as discussed1in Sec. II B. The nearly v -independent values of γ Σ g arequite unexpected considering various model predictionsof strongly v -dependent relaxation rates [39, 44, 77]. Athigher vibrational excitations v &
15, though, the exper-imental data can only be modeled when assuming drasti-cally increased values of γ Σ g . The ground state relaxationrate γ a (open symbol in Fig. 9) is found to be higher byabout a factor 6 as compared to the excited state rate γ Σ g .The nearly constant low values of γ Σ g in the range ofsmall values of v may be related to significant mismatchbetween energy spacings ( ∼
35 cm − ) and excitation en-ergies of the helium bath modes. The ripplon energiesare in the range 0 . − , phonon modes have energies ∼ − , and the roton energy is about 10 cm − [76, 78].Consequently, the ground state vibration with lower levelspacing ( ∼
13 cm − ) more efficiently couples to the he-lium environment, which would explain the higher re-laxation rate γ a . The sharp rise of γ Σ g at h v i > ∼ -He interface induced by fast heating. New cou-pling channels, e. g. the excitation of collective modes ofthe helium droplets that may be related to their super-fluid character ( i. e. rotons) may also be at the originof increasing relaxation rates [25]. At this stage, how-ever, this assumption seems unlikely, given the fact thatthe vibrational energy quanta ( ∼
35 cm − ) largely exceedthe elementary excitations of superfluid helium droplets.Possibly, intra-molecular couplings or more complex ex-citation pathways leading to the ionic continuum mayalso be involved. IV. CONCLUSION
In conclusion, femtosecond pump-probe measurementsof the vibrational wave packet dynamics of Rb moleculesattached to helium nanodroplets are analyzed using dis-sipative quantum simulations. In contrast to earlier in-terpretations, Rb excited to triplet states are found toremain attached to the helium droplets on the time scaleof the pump-probe experiments and reveal slow dampingof the vibrational wave packet signal due to the inter-action with the helium droplet environment. The weaksystem-bath coupling results in slow damping dynamics compared to the periods of vibration, which is proto-typical for the applied master equation. Thus, heliumdroplets provide a versatile test bed for studying relax-ation dynamics and cooling processes induced by a highlyquantum environment.From the detailed comparison of the experimental datain the time- and frequency domains with model calcula-tions it is possible to deduce the evolution of the densitymatrix describing the vibrating Rb molecules. Whilerotational-vibrational coherences as well as pure dephas-ing of the vibrational wave packet dynamics may play aminor role, good agreement is achieved under the modelassumption of vibrational relaxation-induced dephasing.We extract damping constants for the vibrational relax-ation in the lowest triplet state a Σ + u and in the firstexcited state 1 Σ + g , γ a ≈ − and γ Σ g ≈ . − , re-spectively. The pronounced dependence of γ Σ g on thevibrational quantum number v may be related to the in-terplay of effective heating of the droplets by fast energytransfer from the molecules and cooling due to evapo-ration of helium atoms. High heating rates may inducephase transitions in the droplets that affect the dynamicsof attached molecules.Further experiments as well as modeling of the re-sponse of the helium droplets to vibronic excitations ofembedded atoms and molecules are needed. In particu-lar the dependence of dephasing and relaxation dynam-ics of the vibrational state quantum number may pro-vide detailed information about the solute-solvant cou-pling mechanisms and may contribute to interpreting theshapes of spectral lines. Moreover, the dynamics of thedesorption process of alkali atoms and molecules off thedroplet surface in dependence of the atomic species andthe particular vibronic state excited is still mostly unre-solved. Acknowledgments
We thank G. Stock, F. Mintert, and B. v. Issendorfffor valuable discussions. Support by the DeutscheForschungsgemeinschaft (DFG) is gratefully acknowl-edged. Computing resources have been provided by theZentrum f¨ur Informationsdienste und Hochleistungsrech-nen (ZIH) at the TU Dresden. M. S. is a member of theIMPRS Dresden. [1] J. P. Toennies and A. F. Vilesov, Angew. Chem. Int. Ed. , 2622 (2004).[2] S. Grebenev, J. P. Toennies, and A. F. Vilesov, Science , 2083 (1998).[3] F. Stienkemeier and K. Lehmann, J. Phys. B , R127(2006).[4] C. Callegari, I. Reinhard, K. K. Lehmann, G. Scoles,K. Nauta, and R. E. Miller, J. Chem. Phys. , 4636(2000). [5] K. Nauta and R. E. Miller, J. Chem. Phys. , 45084514(2001).[6] B. Dick and A. Slenczka, J. Chem. Phys. , 10206(2001).[7] O. B¨unermann, G. Droppelmann, A. Hernando,R. Mayol, and F. Stienkemeier, J. Phys. Chem. A ,12684 (2007).[8] R. Lehnig, P. L. Raston, and W. J¨ager, Faraday Discuss. , 297 (2009). [9] K. Nauta and R. E. Miller, J. Chem. Phys. , 3426(1999).[10] K. Nauta and R. E. Miller, J. Chem. Phys. , 9466(2000).[11] D. T. Moore and R. E. Miller, J. Chem. Phys. , 9629(2003).[12] M. Koch, G. Aub¨ock, C. Callegari, and W. E. Ernst,Phys. Rev. Lett. , 035302 (2008).[13] A. Braun and M. Drabbels, Phys. Rev. Lett. , 253401(2004).[14] A. Przystawik, S. G¨ode, T. D¨oppner, J. Tiggesb¨aumker,and K.-H. Meiwes-Broer, Phys. Rev. A , 021202(2008).[15] O. Kornilov, C. C. Wang, O. B¨unermann, A. T. Healy,M. Leonard, C. Peng, S. R. Leone, D. M. Neumark, andO. Gessner, J. Phys. Chem. A , 1437 (2010).[16] G. Droppelmann, O. B¨unermann, C. P. Schulz, andF. Stienkemeier, Phys. Rev. Lett. , 0233402 (2004).[17] M. Mudrich, G. Droppelmann, P. Claas, C. Schulz, andF. Stienkemeier, Phys. Rev. Lett. , 023401 (2008).[18] P. Claas, G. Droppelmann, C. P. Schulz, M. Mudrich,and F. Stienkemeier, J. Phys. B , S1151 (2006).[19] P. Claas, G. Droppelmann, C. P. Schulz, M. Mudrich,and F. Stienkemeier, J. Phys. Chem. A , 7537 (2007).[20] M. Mudrich, P. Heister, T. Hippler, C. Giese, O. Dulieu,and F. Stienkemeier, Phys. Rev. A , 042512 (2009).[21] R. Mayol, F. Ancilotto, M. Barranco, O. B¨unermann,M. Pi, and F. Stienkemeier, J. Low Temp. Phys. ,229 (2005).[22] F. Dalfovo, Z. Phys. D , 61 (1994).[23] F. Ancilotto, G. DeToffol, and F. Toigo, Phys. Rev. B , 16125 (1995).[24] F. Stienkemeier, J. Higgins, W. E. Ernst, and G. Scoles,Phys. Rev. Lett. , 3592 (1995).[25] M. Schlesinger, M. Mudrich, F. Stienkemeier, and W. T.Strunz, Chem. Phys. Lett. , 245 (2010).[26] T. Takayanagi and M. Shiga, Phys. Chem. Chem. Phys. , 3241 (2004).[27] F. R. Br¨uhl, R. A. Trasca, and W. E. Ernst, J. Chem.Phys. , 10220 (2001).[28] G. Aub¨ock, J. Nagl, C. Callegari, and W. E. Ernst, Phys.Rev. Lett. , 035301 (2008).[29] Q. Liu, C. Wan, and A. H. Zewail, J. Phys. Chem. A , 18666 (1996).[30] V. A. Ermoshin, A. K. Kazansky, and V. Engel, J. Chem.Phys. , 7807 (1999).[31] C. Meier and J. A. Beswick, J. Chem. Phys. , 4550(2004).[32] M. Karavitis, D. Segale, Z. Bihary, M. Pettersson, andV. A. Apkarian, Low Temp. Phys. , 814 (2003).[33] M. Karavitis and V. A. Apkarian, J. Chem. Phys. ,292 (2004).[34] T. Kiviniemi, J. Aumanen, P. Myllyperki¨o, V. A. Ap-karian, and M. Pettersson, J. Chem. Phys. , 064509(2005).[35] M. G¨uhr, H. Ibrahim, and N. Schwentner, Phys. Chem.Chem. Phys. , 5353 (2004).[36] M. Fushitani, M. Bargheer, M. G¨uhr, and N. Schwentner,Phys. Chem. Chem. Phys. , 3143 (2005).[37] M. G¨uhr, M. Bargheer, M. Fushitani, T. Kiljunen, andN. Schwentner, Phys. Chem. Chem. Phys. , 779 (2007).[38] A. V. Benderskii, J. Eloranta, R. Zadoyan, and V. A.Apkarian, J. Chem. Phys. , 1201 (2002).[39] E. Bodo, F. A. Gianturco, and E. Yurtsever, Phys. Rev. A , 052715 (2006).[40] E. Bodo and F. Gianturco, Int. Rev. Phys. Chem. ,313 (2006).[41] D. Caruso, M. Tacconi, E. Yurtsever, and F. A. Gi-anturco, Phys. Rev. A , 042710 (2010).[42] A. H. Zewail and D. J. Diestler, Chem. Phys. Lett. ,37 (1979).[43] M. Karavitis, T. Kumada, I. U. Goldschleger, and V. A.Apkarian, Phys. Chem. Chem. Phys. , 791 (2005).[44] A. Nitzan, S. Mukamel, and J. Jortner, J. Chem. Phys. , 200 (1975).[45] R. Englman, Non-Radiative Decay of Ions and Moleculesin Solids (North Holland, Amsterdam, 1979).[46] J. S. Bader, B. J. Berne, E. Pollak, and P. H¨anggi, J.Chem. Phys. , 1111 (1996).[47] P. F¨oldi, M. G. Benedict, A. Czirj´ak, and B. Moln´ar,Fortschr. Phys. , 122 (2003).[48] E. Gershgoren, Z. Wang, S. Ruhman, J. Vala, andR. Kosloff, J. Chem. Phys. , 3660 (2003).[49] M. Schlesinger and W. T. Strunz, Phys. Rev. A ,012111 (2008).[50] J. Chesnoy and A. Mokhtari, Phys. Rev. A , 3566(1988).[51] M. Gruebele, G. Roberts, M. Dantus, R. M. Bowman,and A. H. Zewail, Chem. Phys. Lett. , 459 (1990).[52] V. Engel, Chem. Phys. Lett. , 130 (1991).[53] M. Seel and W. Domcke, Chem. Phys. , 59 (1991).[54] M. Seel and W. Domcke, J. Chem. Phys. , 7806 (1991).[55] I. S. Averbukh and N. F. Perelman, Phys. Lett. A ,449 (1989).[56] S. I. Vetchinkin and V. V. Eryomin, Chem. Phys. Lett. , 394 (1994).[57] M. O. Scully and M. S. Zubairy, Quantum Optics (Cam-bridge University Press, 1997).[58] I. S. Averbukh, M. J. J. Vrakking, D. M. Villeneuve, andA. Stolow, Phys. Rev. Lett. , 3518 (1996).[59] M. J. J. Vrakking, D. M. Villeneuve, and A. Stolow, Phys.Rev. A , R37 (1996).[60] E. Joos and H. D. Zeh, Z. Phys. B , 223 (1985).[61] W. H. Zurek, Physics Today , 36 (1991).[62] D. Giulini, Decoherence and the Appearance of a ClassicalWorld in Quantum Theory (Springer, Heidelberg, 1996).[63] M. Tegmark and J. A. Wheeler, Scientific American(2001).[64] H. Breuer and F. Petruccione,
The theory of open quan-tum systems (Oxford University Press, 2002).[65] G. Lindblad, Commun. Math. Phys. , 147 (1975).[66] A. O. Caldeira and A. J. Leggett, Phys. Rev. A , 1059(1985).[67] D. F. Waals and G. J. Milburn, Phys. Rev. A , 2403(1985).[68] M. Brune, S. Haroche, J. M. Raimond, L. Davidovich,and N. Zagury, Phys. Rev. A , 5193 (1992).[69] D. Braun, F. Haake, and W. T. Strunz, Phys. Rev. Lett. , 2913 (2001).[70] R. de Vivie-Riedle, K. Kobe, J. Manz, W. Meyer, B. Reis-chl, S. Rutz, E. Schreiber, and L. W¨oste, J. Phys. Chem. , 7789 (1996).[71] R. de Vivie-Riedle, B. Reischl, S. Rutz, and E. Schreiber,J. Phys. Chem. , 16829 (1995).[72] R. Beuc, M. Movre, V. Horvatic, C. Vadla, O. Dulieu,and M. Aymar, Phys. Rev. A , 032512 (2007).[73] N. Gisin and I. C. Percival, J. Phys. A: Math. Gen. ,5677 (1992). [74] M. Gruebele and A. H. Zewail, J. Chem. Phys. , 883(1993).[75] S. Bovino, E. Coccia, E. Bodo, D. Lopez-Duran, andF. A. Gianturco, J. Chem. Phys. , 224903 (2009).[76] D. M. Brink and S. Stringari, Z. Phys. D , 257 (1990). [77] S. A. Egorov and J. L. Skinner, J. Chem. Phys. , 7047(1996).[78] S. A. Chin and E. Krotscheck, Phys. Rev. B52