Lindblad parameters from high resolution spectroscopy to describe collision induced decoherence in the gas phase -- Application to acetylene
LLindblad parameters from high resolution spectroscopy to describe collision induceddecoherence in the gas phase – Application to acetylene
Antoine Aerts, Jean Vander Auwera, and Nathalie Vaeck a) Universit´e Libre de Bruxelles; Spectroscopy, Quantum Chemistry andAtmospheric Remote Sensing (SQUARES); 50 avenue F. Roosevelt, C.P. 160/09,B-1050 Brussels, Belgium (Dated: 23 February 2021)
Within the framework of the Lindblad master equation, we propose a general method-ology to describe the effects of the environment on a system in dilute gas phase. Thephenomenological parameters characterizing the transitions between rovibrationalstates of the system induced by collisions can be extracted from experimental tran-sition kinetic constants, relying on Energy Gap fitting laws. As the availability ofthis kind of experimental data can be limited, the present work relied on experimen-tal line broadening coefficients, however still using Energy Gap fitting laws. The 3 µ m infrared spectral range of acetylene was chosen to illustrate the proposed ap-proach. The method shows fair agreement with available experimental data whilebeing computationally inexpensive. The results are discussed in the context of statelaser quantum control. a) Email: [email protected] a r X i v : . [ qu a n t - ph ] F e b REDIT
The following article has been submitted to The Journal of Chemical Physics.
I. INTRODUCTION
With the increasing availability of computational power and promises of the quantumtechnologies, the impact of decoherence processes caused by the interactions with the envi-ronment is receiving more and more attention in studies of the dynamics of realistic opensystems. It is worth to mention the importance of decoherence in the context of molecularrotation control, which is particularly relevant in this case, but also in biological systems, charge transfer in molecules, vibration of adsorbed molecules or chemical reactions. In thecontext of quantum control, decoherence caused by the interaction of the system of interestwith its environment is a major issue for applications. Decoherence can act against thecontrol performance or induce a loss of “quantum” property (loss of coherence). In spite ofthis, theoretical control studies were mainly conducted in the gas phase without assessmentof environment-induced processes.The primary source of decoherence, inherent to the system, arises from intramolecularvibrational redistribution (IVR) and was discussed before. It is included in the simulationdynamics by design.
All environmental effects acting on the molecule are expected to causedecoherence that, in the gas phase, is well pictured by the alteration of the shape of thespectral lines. It can easily be shown that collisions are the main source of shape alterationat atmospheric pressure, giving rise in the time domain to population transfers from onerotational level to another.Rotational decoherence is particularly important in the understanding of dissipation oc-curring in gases. Its effects can be studied by a variety of experimental techniques includinginfrared-ultraviolet double resonance (IRUVDR) or molecular centrifuges more suitablefor studies of high angular momenta. The rotational relaxation can also be computed ab initio , however requiring a significant computational effort to obtain the intermolecu-lar potential and to calculate the S -matrix. An alternative method involves the study oflineshapes in the spectral domain. Collision-induced decays or transfers of rotational popu-lations indeed condition the shape of spectral lines, as modeled by the relaxation matrix. Energy Gap fitting laws arise from the intuitive picture that rotationalrelaxation should drop with the change of rotational quantum number J , or equivalentlywith the energy difference, induced by the collisions. This intuitive picture was well repro-duced in experimental studies since its first mention, and has spawned the developmentof numerous energy based scaling laws, which for example proved to be successful in thedescription of line mixing effects in collisionally-broadened CO infrared branches. The aim of the present work is to couple high resolution spectroscopy with quantum dy-namics in the context of collision dynamics in the gas phase. High resolution spectroscopicdata on acetylene (effectively its main isotopologue C H ) were used to illustrate the pur-pose. The time evolution induced by collisions of the population of a rovibrationally excitedlevel of acetylene in a dilute gas phase is calculated using the Lindblad master equation,the required (Lindblad) parameters being obtained from either pump-probe experiments orcollisional line broadening coefficients, relying on Energy Gap fitting laws. Approaches usedin this work are well established in their respective communities; this article tries to presentthem in an unified manner. The present work is part of our effort to better understandthe dynamics and the control of the vibrational population of acetylene to possibly accessthe vinylidene isomer. It was shown that a specific vibrational mode of acetylene (C H )in dilute gas phase can be populated using a single shaped laser pulse. In other words, astate well isolated from others with respect to IVR could be specifically populated via thecontrol of the population of vibrational levels of C H . This study was later extended byinclusion of the rotational structure in the description of the controlled system state, whichproved to be essential for the determination of the control pulse shape. Another importantaspect of the proposed protocol is to establish a methodology to allow taking into accountthe effect of the environment, which is the purpose of this work.In the next sections, the vibration-rotation structure of the acetylene molecule is brieflyintroduced followed by the Lindblad master equation that describes the quantum dynamicsin a system, including the effects of its environment. The determination of the Lindbladparameters from pump probe experiments and collisional line broadening coefficients is thendetailed. This article concludes with some perspectives.3 I. THE VIBRATION-ROTATION STRUCTURE OF ACETYLENE
The vibrational motion of the main isotopologue of the acetylene molecule, C H , canbe described in terms of the 5 modes of vibration presented in Table I. The zero-order vibra-tional levels associated with the excitation of these modes are identified by | v v v v(cid:96) v(cid:96) (cid:105) ,where v i is the vibrational quantum number associated with the mode of vibration ν i ( i = 1 . . .
5, see Table I) and (cid:96) and (cid:96) are the quantum numbers associated with thevibrational angular momenta generated by the excitation of the doubly degenerate bendingmodes ν and ν , respectively. These vibrational levels interact mostly through anharmonicresonances, leading to the formation of so-called polyads. The present work focuses on the polyad of vibrational levels of the ground electronicstate of C H that gives rise to the strong 3 µ m infrared absorption of this molecule whenaccessed from the vibrational ground state. This polyad involves 3 zero-order levels, namely | (cid:105) , | − (cid:105) and | (cid:105) . Allowed one-photon transitions from the ground stateonly involve the | (cid:105) level. It is therefore identified as a “bright” level, while the othertwo are identified as “dark” levels. Anharmonic resonances strongly mix the | (cid:105) and | − (cid:105) levels, while the | (cid:105) level essentially keeps its “dark” character. As thislatter level was shown to not being involved in the collisional processes considered here, it will not be considered any further. The eigenstates resulting from the mixing of the formertwo levels are identified from now on as Γ and Γ , Γ having the highest energy of the two.Transitions from the ground state to Γ and Γ result into two strong bands observed near3 µ m, which are called experimentally the ν + ( ν + ν ) and ν bands, respectively. Theexponent 0 appearing for the former band is the value of k = (cid:96) + (cid:96) and the “+” signrefers to the symmetry of the vibrational wavefunction with respect to the infinity of planescontaining the molecule. In section V, self broadening coefficients of vibration-rotation lines of the ν band areneeded. However, they have never been directly measured for this band, most probablybecause of its strength. Fortunately, the vibrational dependence of self broadening coeffi-cients is generally small and has never clearly been evidenced for acetylene. In the frameof the present work, we therefore relied on the extensive set of self broadening coefficientsmeasured for the 3 ν , (2 ν + ν ) I and (2 ν + ν ) II bands observed in the 5 µ m region(1860 − − ). The Roman numerals I and II discriminate the two k = 1 vibrational4 ormal mode H C C H Harmonic frequency / cm − ν ν ν ν ν C H in its ground electronic state and theirharmonic frequencies. The first three are non degenerate stretching modes and the last two aredoubly degenerate bending modes. levels with the same symmetry that arise from the simultaneous excitation of 2 quanta in ν and one quantum in ν , the Roman numeral I identifying the level characterized by thehighest energy. Acetylene is a linear molecule in its ground electronic state. Therefore, the descriptionof its rotational motion mainly involves the quantum number J associated with the angularmomentum of the molecule. III. LINDBLAD EQUATION
The evolution of a system coupled to the environment in the Born and Markov approxi-mation can be described by the Lindblad master equation: d ˆ ρ ( t ) dt = − i (cid:126) (cid:104) ˆ H ( t ) , ˆ ρ ( t ) (cid:105) (1)+ (cid:88) ij (cid:18) ˆ L ij ˆ ρ ( t ) ˆ L † ij − (cid:104) ˆ ρ ( t ) , ˆ L † ij ˆ L ij (cid:105) + (cid:19) where ˆ H ( t ) = ˆ H S − ˆ µ e (cid:15) ( t ) is the total Hamiltonian in the absence of interaction withthe environment and ˆ H S the Hamiltonian of the system. If a laser field is applied, ˆ µ e is the electric dipole moment of the system and (cid:15) ( t ) is the time-dependent amplitude ofthe laser pulse. [ A, B ] + is the anti-commutator of arguments A and B , ˆ ρ ( t ) is the time-dependent density operator and ˆ L ij are operators representing the transitions induced by5he environment between states i and j of the system. A phenomenological representationof the transition operator was used: ˆ L ij = (cid:112) θ ij | i (cid:105)(cid:104) j | . (2)With this form, decoherence in the system is characterised by the Lindblad parameters θ ij ,which are the frequencies of the transitions induced by non-radiative processes betweeneigenstates | i (cid:105) and | j (cid:105) of the system Hamiltonian. Comments on the implementation of theLindblad master equation are given in the Supplemental Material. The determination ofthe Lindblad parameters θ ij from experimental data published in the literature is discussedin the next two sections. IV. LINDBLAD PARAMETERS FROM PUMP-PROBE EXPERIMENTS
Frost and Henton et al. studied experimentally the dynamical effects of collisions inthe Γ and Γ interacting states using infrared ultraviolet double resonance (IRUVDR). Aninfrared “pump” laser of fixed wavelength was used to populate a specific rotational levelof the eigenstate Γ or Γ and an ultraviolet “probe” laser induced a transition from theexcited rovibrational level to the first excited electronic state. Collision-induced transitionswithin the eigenstates Γ and Γ were monitored from the observed evolution of the laserinduced fluorescence with the probe laser wavelength. The corresponding state-to-staterate constants for rotational relaxation were determined. Frost initiated the study for self-relaxation (C H in an environment of C H ), while Henton et al. refined his measurementsand extended this research to the effects of foreign gases (Ar, He and H ) on the dynamics.Frost and Henton et al. considered four relaxation processes caused by collisions. Theyare described as follows :C H [ Γ ; J i ] + M −−→ C H [ Γ ; J j ] + M + ∆E (R1)C H [ Γ ; J i ] + M −−→ C H [ Γ ; J j ] + M + ∆E (R2)C H [ Γ ; J i ] + M −−→ C H [ Γ ; J j ] + M + ∆E (R3)C H [ Γ ; J i ] + M −−→ C H [ Γ ; J j ] + M + ∆E (R4) The use of θ is a deliberate choice; γ is generally used in the literature. i.e. other C H molecules in the present case), J i and J j arerespectively the initial and final rotational levels and ∆E is the energy released/absorbedduring the collision. It is worth noting that these processes are subject to the parity selectionrule that forbids the change of the total wavefunction parity, i.e. ortho ↔ para transitionsare not observed. Ortho and para label the levels with the highest and lowest nuclearspin statistical weights, respectively. In C H , ortho ( resp. para) labels levels with odd( resp. even) J . Consequently, non-radiative transitions involving even ∆J = | J j − J i | aresolely observed in experiments.Frost and later Henton et al. determined the transition kinetic constants k ( j ← i )of processes R1 to R4 using the IRUVDR technique. They fitted their measurements toExponential Gap Laws (EGL) of the form: k ( j ← i | K , η ) = K exp (cid:18) − η | E j − E i | k B T (cid:19) with E j > E i (3)where K (in units of frequency per pressure as k ) and η (dimensionless) are fitted parametersand E i is the energy of level i . Their reported parameters and uncertainties are reproducedin Table II, together with the identification of the level populated by the pump laser. TableII shows that the η parameters of the 5 different processes have the same value within thestated uncertainties. On the other hand, comparisons of the magnitude of the K parametersshow that transitions within the same vibrational eigenstate (R3 and R4) are an order ofmagnitude faster than transitions involving a change of eigenstate (R1 and R2) and thattransitions with ∆J = ± Γ eigenstate are greatly favoured. In this later case, the η parameter was fixed by Henton et al. to the value reported for the ∆J ≥ To express the Lindblad parameters θ ij (see Eq. 2) from this data set, we assumed thatthey are related to the transition kinetic constants k ( j ← i ) by: θ ij = P k ( j ← i ) (4)The linear dependence with pressure should be suitable for P up to 1 atm. A correctionto the collision frequency may however be needed at higher pressure, or even a differentapproach, because the IRUVDR measurements could not be extrapolated to that regimeand equations given in the present work would not hold.The experimental results and the choice of the EGL somewhat validate the Lindbladmaster equation in this case as the decays induced by the master equation are exponential. rocess K η InitialR1 0 . . Γ , J = 12R2 0 . . Γ , J = 10R3 0 . . Γ , J = 12R4, | ∆J | = 2 0 . . Γ , J = 10R4, | ∆J | ≥ . . Γ , J = 10TABLE II. Experimental parameters ( K in cm − atm − and η is dimensionless) of the EGL law(Eq. 3) from Frost and Henton et al. used in this work to calculate the kinetic constants ofprocesses R1 to R4. Numbers between parentheses are the standard deviations in the units of thelast digit quoted. The last column identifies the level initially populated by the pump laser. It can indeed be shown that, for an eigenstates system dynamics driven solely by interactionwith its environment (cid:16) − i (cid:126) (cid:104) ˆ H ( t ) , ˆ ρ ( t ) (cid:105) = 0 (cid:17) , Eq. 1 simplifies to a sum of products of scalarsand density matrices when the phenomenological representation of the transition operatorsis used (see Eqs. S3 and S4 in the Supplemental Material).To illustrate the collision-induced dynamics, Fig. 1 shows the evolution of the relativepopulation of selected levels of the polyad of interest at P = 1 atm, as described by theLindblad master equation (Eq. 1) without interaction with an external laser field. The tran-sitions frequencies θ ij are obtained from Eqs. 4 and 3, and the experimental EGL parametersgiven in Table II. The H S hamiltonian of acetylene itself is described in the state space bya global effective hamiltonian, built from the extensive set of experimental high-resolutionspectroscopic studies carried out in the ground electronic state of the molecule. More detailscan be found in our previous work, and a thorough and pedagogical introduction to thestrategy applied to build such global hamiltonians was given by Herman. Working in thestate space connects the relevant quantities obtained from high resolution spectroscopy, i.e. eigenstates energies from line positions and transition dipole moment operators from lineintensities, to the system dynamics. The state basis used for the simulation includes levelsfrom J =0 to J =100. The initial population lies in the Γ , J = 12 rovibrational level as anillustration of the dynamics probed by the experiments of Frost. The dynamics is solelydriven by population transfers between rotational levels induced by inelastic collisions, i.e R e l a t i v e popu l a t i on Time (ps)
J=4J=6J=8J=10J=12J=14J=16J=18J=20
FIG. 1. Collision induced dynamics in Γ ( continuous ) and in Γ ( dashed ) from unitary popula-tion in the Γ , J = 12 (cid:105) state (shown in red and on the right axis) under P = 1 atm. State-to-statetransition frequencies are calculated from Eqs. 4 and 3, and the parameters reported by Frost and Henton et al. (reproduced in Table II). transfers from/to another polyad or vibrational level are neglected as are parity-violatingtransitions that happen on a longer timescale. The simulation shows a rapid populationtransfer from the initially populated state to other states (almost complete within 100 ps),with a larger propensity for transfers within the same vibrational eigenstate. The statisticallimit is almost reached within a few hundreds of ps. This means that any successful state-control targeting either the Γ or Γ eigenstates or any combination would therefore rapidly( i.e. within hundreds of ps) be lost due to collision-induced transitions.9 . LINDBLAD PARAMETERS FROM LINE BROADENINGCOEFFICIENTS As detailed in the previous section, the Lindblad parameters θ ij required to model pop-ulation transfers induced by collisions between molecules in the gas phase in laser andenvironment-driven dynamics simulations can be taken from direct kinetic measurements.However, this kind of experimental data may not be available. In this section, we describethe alternative methodology we used to determine the parameters of the EGL law, namely K and η , from self broadening coefficients.In this perspective, we opt for the impact and binary collision approximations, relyingon the construction of the real part of the relaxation matrix and statistically based onEnergy Gap fitting laws. The approximations imply that the resulting perturbation onthe spectrum is proportional to the gas density (binary collisions) and that the duration ofthe collisions is negligibly short, so that the perturbation is independent of the frequencyover the spectral range considered (impact approximation).Using the relaxation matrix ( W ) and including the relevant factors for intensities, theabsorption coefficient α (in cm − ) is given by the imaginary part of the (unnormalised) lineprofile: α (˜ ν, P, T ) = 8 π hc π(cid:15) n L T Q ( T ) T [1 − exp {− hc ˜ ν/k B T } ]˜ ν × π (cid:88) (cid:96) (cid:88) (cid:96) (cid:48) ρ(cid:96) ( T ) d(cid:96)d(cid:96) (cid:48) (cid:8) [ Σ − L a − iP W ( T )] − (cid:9) (cid:96) (cid:48) (cid:96) (5)where 8 π / (3 hc )(1 / (4 π(cid:15) )) ≈ . × − D − cm , n L = 2.686780111 × cm − atm − is the Loschmidt constant, T = 273 .
15 K and Q ( T ) is the total internal partition sumwith Q ( C H ) = 412 .
45 at 296 K, ˜ ν is the wavenumber (in cm − ), P is the pressureof the perturber (in atm), T is the temperature (in K), ρ(cid:96) ( T ) is the equilibrium relativepopulation of the initial level of line (cid:96) , d(cid:96) is the tensor that couples radiation and matter(electric dipole moment in this case, in Debye) for line (cid:96) and i is the imaginary number. Σ and L a are defined in the line space as: Σ(cid:96) (cid:48) (cid:96) = δ(cid:96) , (cid:96) (cid:48) × ˜ ν (6) { L a } (cid:96) (cid:48) (cid:96) = δ(cid:96) , (cid:96) (cid:48) × ˜ ν(cid:96) (7)10ith ˜ ν(cid:96) the position of line (cid:96) and δ(cid:96) , (cid:96) (cid:48) = δ i (cid:96) ,i (cid:96) (cid:48) δ f (cid:96) ,f (cid:96) (cid:48) where i(cid:96) and f(cid:96) are the initial andfinal rotational or rotation-vibration levels of the transition associated with line (cid:96) . Both arediagonal matrices of size N × N in the line space, where N is the number of lines.The photon interacting with the molecules undergoing collisions is eventually dissipatedin the system with a resulting broadening and shifting of the resonance wavenumber. This ismodeled using the relaxation matrix W , which adds a complex perturbation to the resonancewavenumber within the impact approximation. The relaxation matrix is constructed withline broadening (real part) and line shift (imaginary parts) coefficients on its diagonal.The real part of the non-diagonal elements model line mixing effects on the line shape,their imaginary part being usually small and neglected. Only considering the diagonalelements of W , the spectrum calculated with Eq. 5 is a sum of Lorentzian. The term[ Σ − L a − iP W ( T )] is a complex matrix of size N × N ( N is the number of lines), which mustbe inverted. Details on how to simulate a spectrum using Eq. 5 along with an illustrationfor the ν band of C H are provided in the Supplemental Material.As demonstrated by Fano, the line broadening coefficients γ(cid:96) (at half width at halfmaximum, HWHM) of the (spectral) line (cid:96) are related to the state-to-state kinetic constants k by: Re[ W(cid:96)(cid:96) ] = γ(cid:96) = 12 (cid:88) i (cid:96) (cid:48) (cid:54) = i (cid:96) k ( i(cid:96) (cid:48) ← i(cid:96) ) + (cid:88) f (cid:96) (cid:48) (cid:54) = f (cid:96) k ( f(cid:96) (cid:48) ← f(cid:96) ) . (8)The first sum describes collision-induced transitions between rotational levels in the lowervibrational level (for cold bands, i(cid:96) levels belong to the ground vibrational state) and thesecond sum describes collision-induced transitions in the excited vibrational level(s). Bothcontributions are often assumed to be of similar magnitude. Again, the kinetic constants k can be parametrized using a number of “gap” laws. One of the simplest is the ExponentialGap Law (EGL) (Eq. 3). Alternatively, a third parameter can be introduced using theExponential Power Gap Law (EPGL) given by: k ( j ← i | K , η, β ) = K (cid:18) | E j − E i | k B T (cid:19) − β exp (cid:18) − η | E j − E i | k B T (cid:19) with E j > E i (9)where K (same units as k ), β and η are the fitted parameters and E i is the energy of level i .The downwards transition kinetic constants are deduced from Eq. 3 (or 9) and the detailed11alance (to ensure conservation of total population): k ( i ← j ) = ρ i ρ j k ( j ← i ) = 2 J i + 12 J j + 1 exp (cid:18) E j − E i k B T (cid:19) k ( j ← i ) (10)with ρ i the relative population of level i , proportional to (2 J i + 1) exp ( − E i /k B T ).The interacting system Γ − Γ considered here requires some care as it may not beas straightforward to calculate the broadening coefficients from the state-to-state kineticconstants using Eq. 8 as it may seem. The last term of the right part of Eq. 8 must indeedinclude the relevant transfer channels in the excited vibrational levels described by reactionsR1 to R4, e.g. R1 and R3 for the Γ eigenstate. In addition, the selection rule that forbidsparity change of J in collision-induced transitions must be taken into account: (cid:88) f (cid:96) (cid:48) (cid:54) = f (cid:96) k ( f(cid:96) (cid:48) ← f(cid:96) ) = N c (cid:88) c =1 (cid:88) f (cid:96) (cid:48) (cid:54) = f (cid:96) k c ( f(cid:96) (cid:48) ← f(cid:96) ) (11) ∆J = J f (cid:96) (cid:48) − J f (cid:96) = evenwhere k c is the state-to-state kinetic constant within channel c and N c is the number ofchannels ( N c = 2 for each vibrational eigenstate here; see Eqs. 13 and 14 for the explicitsums).Broadening coefficients have never been directly measured for the ν band of acetylene,most probably because of the large transition dipole moment associated with this band.However, vibrational dependence of the broadening coefficients (HWHM) is generally small,especially in acetylene. Therefore, we used the extensive set of experimental broadeningcoefficients measured for cold bands in the 5 µ m region of acetylene. They are presentedin Fig. 2. The error bars associated with these data correspond to the upper limit of theirreported precision of measurement, i.e. and Henton etal. (see Table II) are also presented in Fig. 2. They were calculated using Eq. 8 restrictedto non radiative transitions within the excited vibrational level, i.e. γ(cid:96) = (cid:88) f (cid:96) (cid:48) (cid:54) = f (cid:96) k ( f(cid:96) (cid:48) ← f(cid:96) ) , (12)due to the lack of measurements reported for the ground vibrational state, together withEqs. 3, 10 and 11. The labels Γ and Γ in Fig. 2 identify the nature of the excitedvibrational level. The energies of the vibrational eigenstates of interest were calculated12sing the effective Hamiltonian, with rotational quantum number up to J = 100. Energiesagree with reported experimental line positions with a root mean square deviation of0.0007 cm − . γ 𝓁 / c m –1 a t m –1 | m | P (2 ν + ν ) I Q (2 ν + ν ) I EGL prediction Γ P (2 ν + ν ) II Q (3 ν ) EGL prediction Γ P (3 ν ) R (2 ν + ν ) II EGL fit Γ R (3 ν ) EGL fit Γ EPGL fit Γ EPGL fit Γ FIG. 2. Dependence of line broadening coefficients γ(cid:96) (HWHM) with the absolute value of therotational quantum number ( | m | = J i l + 1 for R branch lines and | m | = J i l for P and Q branchlines). The experimental data measured for cold bands in the 5 µ m region of acetylene arecompared with values predicted and fitted using the EGL and EPGL (see text for details). Thelines are guides for the eyes. Fig. 2 shows that the predictions underestimate the broadening coefficients of most linessuggesting that elastic collisions, not considered by Frost and Henton et al. , contributesignificantly to line broadening. Additionally, the predicted rotational dependence does notmatch the observation, probably because it involves extrapolation of an undersized set ofmeasurements (only from one J to others).In view of these disagreements, the values of the parameters K and η involved in the EGLmodeling of the transition kinetic constants k ( j ← i ) were determined by fitting Eqs. 3 and10 to 12 to the measured broadening coefficients presented in Fig. 2. Additionally, Eq. 1113as adapted in two ways. Orr suggested that the contributions of processes such as R1 andR2 should correspond to at most 10% of the total transition frequency from one rotationallevel. In Eq. 11, the sum of state-to-state transition frequencies of processes R1 and R2 wastherefore constrained to contribute to 10% of the broadening coefficient (processes R3 andR4 thus contribute to 90%). Although the contributions of R1 and R2 were later refinedto approximately 21% in the polyad of interest using direct measurements of state-to-statekinetic constants, confirming Orr’s suggestion that the intramolecular couplings enhancevibration – vibration transfers, significant uncertainties remain. To highlight the ability ofthe present methodology to describe the environment-induced processes in generic problems,this work relied on as few specific parameters as possible; this refined measurement wastherefore ignored. Additionally, to keep the problem tractable and avoid strong correlationsbetween parameters, the number of fitted parameters was restricted to two sets identifiedby the upper level of the transition belonging to either the Γ (R2 and R4) or Γ (R1 andR3) eigenstate. Taking these two constraints into account, Eqs. 12 and 11 become in theEGL: γΓ (cid:96) = 0 . (cid:88) f (cid:96) (cid:48) (cid:54) = f (cid:96) k R ( f Γ (cid:96) (cid:48) ← f Γ (cid:96) | KΓ , ηΓ ) + 0 . (cid:88) f (cid:96) (cid:48) (cid:54) = f (cid:96) k R ( f Γ (cid:96) (cid:48) ← f Γ (cid:96) | KΓ , ηΓ ) (13) γΓ (cid:96) = 0 . (cid:88) f (cid:96) (cid:48) (cid:54) = f (cid:96) k R ( f Γ (cid:96) (cid:48) ← f Γ (cid:96) | KΓ , ηΓ ) + 0 . (cid:88) f (cid:96) (cid:48) (cid:54) = f (cid:96) k R ( f Γ (cid:96) (cid:48) ← f Γ (cid:96) | KΓ , ηΓ ) (14)where the transition kinetic constants are given by Eq. 3. All broadening coefficients re-ported by Jacquemart et al. for cold bands in the 5 µ m region of acetylene were included inthe fitting procedure. The coefficients reported for the same value of | m | were averaged andassigned to transitions independently of their upper level belonging to the Γ or Γ eigen-state. The fitting was performed using the optimization library of SciPy (scipy.optimize)with all tolerance parameters set to machine epsilon. The best-fit values of the EGL param-eters obtained are given in Table III and the corresponding rotational dependences of thebroadening coefficients are presented in Fig. 2 (dashed lines identified by “EGL fit Γ ” and“EGL fit Γ ,” actually overlapped). Fig. 2 shows that, although the EGL reproduces thegeneral decrease of the broadening coefficients with increasing rotation, it exhibits additionaloscillations. It is worth to point out that the EGL cannot adequately reflect the effects ofthe energy differences as all parameters are the same except for the proportionality weightsof 10 and 90 % imposed to the contributions of processes R1 and R3 (for Γ ) and R2 and14 η β EGL Γ Γ Γ Γ K are in cm − atm − ; the other parameters are unitless. Γ and Γ identifythe eigenstates of the upper level of the transition (appearing as an exponent in Eqs. 13 and 14).The numbers between parentheses are the standard deviations in the units of the last digit quoted. R4 (for Γ ) (Eqs. 13 and 14). As the EGL parameters of Frost and Henton et al. yieldbroadening coefficients smaller than observed, it is not surprising that the present fit resultsin larger values of the K and η parameters. The fitted exponential parameters ( η ) howeverlie within the uncertainties of most reported ones (see Table II).As shown in Fig. 2, the introduction of an additional parameter β in the EPGL (Eq. 9)results in a better description of the rotational dependence of the broadening coefficients.The resulting best-fit values of the parameters of the EPGL are also presented in Table III.Fig. 3 shows a simulation of population dynamics for selected states in the polyad ofinterest, similar to those presented in Fig. 1. The Lindblad parameters are calculated usingboth the EPGL approach and its parameters listed in Table III and the EGL with its pa-rameters listed in Table II. The state basis used for the simulation is the same as for Fig.1. Note that our fitting procedure within the impact and binary collision approximationsincludes level energies up to J =100 and broadening coefficients up to | m | = 34 while IRU-VDR measurements were fitted to the EGL model relying on transitions from a singlypopulated state (see Table II) and including transitions with − ≤ ∆J ≤ +10 at most.The population transfers based on the present EPGL fit compare well with those pre-dicted using the EGL experimental model. As expected from the broadening coefficientspredicted from the measurements of Frost and Henton et al. being smaller than the fit-15 J=4
J=6
J=8
J=10
J=12 R e l a t i v e popu l a t i on J=14
J=16
J=18
J=20
Time (ps)
FIG. 3. Collision induced dynamics in Γ ( continuous ) and in Γ ( dashed ) from unitary pop-ulation in the | Γ , J = 12 (cid:105) state under P = 1 atm. The state-to-state transition frequenciesare calculated using the EPGL parameters listed in Table III (black) and the EGL experimentalparameters reproduced in Table II (green). ted broadening coefficients presented in Fig. 2, the transition frequencies calculated usingthe EPGL model are larger than the measured transitions frequencies. However, thedynamics happen on a similar timescale and the trends captured by the experiment arereproduced by the present model, i.e. the | ∆J | = 2 propensity shown by the J = 10 and J = 14 curves that reflects the vicinity of levels energies and the faster fall of upwardstransfers ( ∆J >
0) rather than the downwards transfers ( ∆J <
VI. CONCLUSIONS AND PERSPECTIVES
Two different methods allowing to extract the phenomenological parameters of the Lind-blad equations from experimental data have been presented. In the first approach, thetransitions frequencies are taken directly from a pump-probe experiment. In the secondcase, they are extracted from measured broadening coefficients. Both approaches rely onthe use of the very simple Energy Gap laws and provide similar results with an unbeat-able, inexpensive computational effort. Alternatively, the parameters could for example becalculated ab-initio . However, this would require the computation of an interaction poten-tial, which can be very expensive due to the number of degrees of freedom, followed by thecalculation of the S matrix.The parameters obtained in this way allow to include rotational relaxation in dynamicssimulations and in particular for laser control optimisations. We point out that the modelbeing as good as its underlying approximations, it still depends strongly on assumptions.Main limitations follow that transient effects of collisions (non-Markovian effects ) are notincluded limiting the validity to a finite range of frequencies and Doppler broadening isneglected so that the model does not hold at low pressure.In the case of the selected polyad of acetylene, we show that the rotational relaxationwill induce a reorganisation of the populations initially confined in one J level in about200 ps, which is very short to engage in a specific process. As stated before, the “dark” | (cid:105) mode is not affected by the collisions on the same timescale due to selection rules.Indeed, parity-violation transitions should happen at the second to kilosecond timescale. The | (cid:105) mode can be seen as a “decoherence free” subspace, which makes it an inter-esting target for possible state-control studies. ACKNOWLEDGMENTS
The authors warmly thank Michel Herman for very fruitful discussions. The IISN (In-stitut Interuniversitaire des Sciences Nucl´eaires) is acknowledged for its financial support.Computational resources have been provided by the Shared ICT Services Centre, Universit´e17ibre de Bruxelles.
DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the correspondingauthor upon reasonable request.
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S1. LINDBLAD MASTER EQUATION IMPLEMENTATION
We develop here the second term of the Lindblad master equation: d ˆ ρ ( t ) dt = − i (cid:126) (cid:104) ˆ H ( t ) , ˆ ρ ( t ) (cid:105) + ˆˆ L D ρ ( t ) (S1)= − i (cid:126) (cid:104) ˆ H ( t ) , ˆ ρ ( t ) (cid:105) + (cid:88) jk (cid:18) ˆ L jk ˆ ρ ( t ) ˆ L † jk − (cid:104) ˆ ρ ( t ) , ˆ L † jk ˆ L jk (cid:105) + (cid:19) with ˆ H ( t ) the total hamiltonian in absence of interaction with the environment, ˆ ρ the densityoperator, ˆ L jk are transitions operators between states j and k and [ A, B ] + is the anti-commutator of arguments A and B . A phenomenological representation of the transitionoperator is used, with θ jk the transmission frequencies between the system hamiltonianeigenstates {| j (cid:105)} : ˆ L jk = (cid:112) θ jk | j (cid:105)(cid:104) k | . (S2)Development of Eq. S1 is given below:ˆˆ L D ρ ( t ) = (cid:88) jk (cid:18) ˆ L jk ˆ ρ ( t ) ˆ L † jk − (cid:104) ˆ ρ ( t ) , ˆ L † jk ˆ L jk (cid:105) + (cid:19) = (cid:88) jk (cid:112) θ jk | j (cid:105)(cid:104) k | ˆ ρ ( t ) (cid:112) θ kj | k (cid:105)(cid:104) j |− (cid:104) ˆ ρ ( t ) (cid:112) θ kj | k (cid:105)(cid:104) j | (cid:112) θ jk | j (cid:105)(cid:104) k | + (cid:112) θ kj | k (cid:105)(cid:104) j | (cid:112) θ jk | j (cid:105)(cid:104) k | ˆ ρ ( t ) (cid:105) θ jk et θ kj are scalars and not necessarily equal;ˆˆ L D ˆ ρ ( t ) = (cid:88) jk (cid:112) θ jk (cid:112) θ kj | j (cid:105)(cid:104) k | ˆ ρ ( t ) | k (cid:105)(cid:104) j | − (cid:112) θ jk (cid:112) θ kj [ ˆ ρ ( t ) | k (cid:105)(cid:104) j | j (cid:105)(cid:104) k | + | k (cid:105)(cid:104) j | j (cid:105)(cid:104) k | ˆ ρ ( t )]= (cid:88) jk (cid:112) θ jk (cid:112) θ kj | j (cid:105)(cid:104) k | ˆ ρ ( t ) | k (cid:105)(cid:104) j | − (cid:112) θ jk (cid:112) θ kj [ ˆ ρ ( t ) | k (cid:105)(cid:104) k | + | k (cid:105)(cid:104) k | ˆ ρ ( t )]The diagonal elements are given by: (cid:104) n | ˆˆ L D ˆ ρ ( t ) | n (cid:105) = (cid:88) jk (cid:16)(cid:112) θ jk (cid:112) θ kj (cid:104) n | j (cid:105)(cid:104) k | ˆ ρ ( t ) | k (cid:105)(cid:104) j | n (cid:105)− (cid:112) θ jk (cid:112) θ kj [ (cid:104) n | ˆ ρ ( t ) | k (cid:105)(cid:104) k | n (cid:105) + (cid:104) n | k (cid:105)(cid:104) k | ˆ ρ ( t ) | n (cid:105) ] (cid:19) The use of θ is a deliberate choice, literature generally uses γ . S1he first terms is non-zero if and only if (cid:104) n | j (cid:105) (cid:54) = 0, i.e. , if j = n . In the second term, (cid:104) n | k (cid:105) (cid:54) = 0 only if n = k . The sum over j disappears in the first term and over k in thesecond term. (cid:104) n | ˆˆ L D ˆ ρ ( t ) | n (cid:105) = (cid:88) k (cid:112) θ nk (cid:112) θ kn ρ kk ( t ) − (cid:88) j (cid:104)(cid:112) θ jn (cid:112) θ nj ρ nn ( t ) + (cid:112) θ jn (cid:112) θ nj ρ nn ( t ) (cid:105) = (cid:88) k (cid:112) θ nk (cid:112) θ kn ρ kk ( t ) − (cid:88) j (cid:112) θ jn (cid:112) θ nj ρ nn ( t ) (S3)The non-diagonal elements are given by: (cid:104) n | ˆˆ L D ˆ ρ ( t ) | m (cid:105) = (cid:88) jk (cid:16)(cid:112) θ jk (cid:112) θ kj (cid:104) n | j (cid:105)(cid:104) k | ˆ ρ ( t ) | k (cid:105)(cid:104) j | m (cid:105)− (cid:112) θ jk (cid:112) θ kj [ (cid:104) n | ˆ ρ ( t ) | k (cid:105)(cid:104) k | m (cid:105) + (cid:104) n | k (cid:105)(cid:104) k | ˆ ρ ( t ) | m (cid:105) ] (cid:19) The first term is always zero ( n (cid:54) = m ) and the second term is non-zero if k = m , (cid:104) n | ˆˆ L D ˆ ρ ( t ) | m (cid:105) = − (cid:34)(cid:88) j (cid:112) θ jm (cid:112) θ mj ρ nm ( t ) + (cid:112) θ jn (cid:112) θ nj ρ nm ( t ) (cid:35) = − ρ nm ( t ) (cid:88) j (cid:16)(cid:112) θ jm (cid:112) θ mj + (cid:112) θ jn (cid:112) θ nj (cid:17) (S4)S2
2. SIMULATION OF SPECTRA USING THE RELAXATION MATRIX
Absorption spectra, i.e. the absorption coefficient as a function of the wavenumber, canbe calculated using Eq. 5 of the main text. All the parameters involved are self-explanatoryor identified there. However, the calculation of the equilibrium relative population ρ(cid:96) in theground vibrational state of acetylene associated with line (cid:96) is worth an explanation. Separaterelative equilibrium populations are calculated for ortho ( J odd) and para ( J even) C H . They are given by ρ J ortho = 3 exp [ − ( E J ortho − E J =1 ) /k B T ] (cid:80) J ortho exp [ − ( E J ortho − E J =1 ) /k B T ] (S5) ρ J para = exp (cid:2) − (cid:0) E J para − E J =0 (cid:1) /k B T (cid:3)(cid:80) J para exp (cid:2) − (cid:0) E J para − E J =0 (cid:1) /k B T (cid:3) (S6)with k B the Boltzmann constant, and T the temperature. The distributions of ortho- andpara-acetylene, centered at J = 9 are illustrated in Fig. S1. FIG. S1. Boltzmann population distribution of ortho- and para-acetylene in its ground vibrationalstate at T = 296 K, normalized so that the sum of all the values is equal to 4. As an argument supporting the validity of the broadening coefficients calculated usingthe EGL parameters determined in this work (listed in Table III), the absorption spectrumof the ν band of C H at 1 atm and 296 K was calculated two ways. The black tracein Fig. S2 was obtained using Eq. 5 with only the real part of the diagonal elementsS3f the relaxation matrix W being non zero, therefore ignoring line shifts and line mixing.The broadening coefficients of R and P branch lines were calculated as described in themain text using the EGL parameters of Table III. The red trace displays the differencesbetween the same spectrum calculated using a sum of Lorentzian lines with the parametersavailable in the HITRAN database and the black trace. Note that the black spectrumwas actually calculated using line positions from the effective Hamiltonian of Amyay et al. and transition dipole moments from Vander Auwera et al. However, the differences withthe line positions and intensities available in HITRAN are only marginal, with a root meansquare deviation of 0.0007 cm − . Fig. S2 shows that, although the EGL model does not fitperfectly the experimental broadening coefficients (see Fig. 2), the differences are at most10%. FIG. S2. Calculated spectrum of the ν band of C H at P = 1 atm and T = 296 K.= 296 K.