Maximum relative excitation of a specific vibrational mode via optimum laser pulse duration Xiang Zhou, 1, 2 Zhibin Lin, 2, 3 Chenwei Jiang, 4, 2 Meng Gao, 2 and Roland E. Allen 2, ∗ $^{1}$Department of Physics, Wuhan University, Wuhan 430072, China $^{2}$Department of Physics, Texas A&M University, College Station, Texas 77843, USA $^{3}$Renewable Energy Materials Research Science and Engineering Center, and Department of Physics, Colorado School of Mines, Golden, Colorado 80401, USA $^{4}$Department of Applied Physics, Xi’an Jiaotong University, Xi’an 710049, China (Dated: January 22, 2018) For molecules and materials responding to femtosecond-scale optical laser pulses, we predict max- imum relative excitation of a Raman-active vibrational mode with period T when the pulse has an FWHM duration τ ≈ 0 . 42 T . This result follows from a general analytical model, and is precisely confirmed by detailed density-functional-based dynamical simulations for C 60 and a carbon nan- otube, which include anharmonicity, nonlinearity, no assumptions about the polarizability tensor, and no averaging over rapid oscillations within the pulse. The mode specificity is, of course, best at low temperature and for pulses that are electronically off-resonance, and the energy deposited in any mode is proportional to the fourth power of the electric field. For a quarter century there has been considerable in- terest in optimizing the vibrational response of molecules and materials to ultrafast laser pulses [1–15]. This prob- lem is directly relevant to the broader issue of coherent control in physical, chemical [16–19], and biological [20– 23] systems. Here we consider excitation via impulsive stimu- lated Raman scattering and related techniques using femtosecond-scale optical pulses. We find that the opti- mum full-width-at-half-maximum (FWHM) pulse dura- tion for exciting a specific vibrational mode with angular frequency 2 π/T is given by τ ≈ 0 . 42 T . Our prediction results from a general analytical model, and is precisely confirmed by completely independent density-functional- based simulations for C$_{60}$ [24–28] and a small carbon nanotube [29–31]. Unlike the model, these simulations include anharmonic effects in the vibrations, nonlinear effects in the response to the applied field, and no sim- plifying assumptions about the electronic polarizability tensor. Our general model consists of the following: (1) The electric field has the form with ω ≫ π/τ and ℏ ω off resonance. Since the oscilla- tions of sin $^{2}$( ωt + δ ) will average out to 1 / 2 over a period that is short compared to the response time of the vibrat- ing nuclei, we will actually replace the square of Eq. (1) by the envelope function (2) The initial conditions Q$_{k}$ (0) = ˙ Q$_{k}$ (0) = 0 are imposed on the normal-mode coordinates Q$_{k}$ ( t ). This approximation is valid for the expectation value below Eq. (4) at low temperature, or an ensemble average at higher temperature, in the linearized Eq. (3). with ¯ E 2 0 = E 2 $_{0}$/ 2. This form has the following nice fea- tures [32]: (i) The duration is finite and need not be truncated. (ii) The FWHM duration is exactly half the full duration 2 τ . (iii) A plot reveals that it closely resem- bles a Gaussian. (iv) The slope is zero at beginning and end. with Q$_{k}$ ≡ 〈 ˆ Q$_{k}$ 〉 and α ′ k ≡ 〈 ∂ ˆ α /∂ ˆ Q$_{k}$ 〉 , when the term involving ∂ ˆ µ /∂Q$_{k}$ vanishes or is neglected (e.g. in taking rapid oscillations of the electric field E ( t ) to average out to zero, an approximation which also causes the lowest- order nonlinear term to vanish in Eq. (4)). If Eq. (2) is substituted into Eq. (3), the solution after the pulse is found to reduce to where the anharmonic and damping terms (in the normal-mode coordinates Q$_{k}$ ), nonlinear terms (in the electric field E ( t )), and off-diagonal terms (in the polar- izability tensor α ) have been neglected, with the electric dipole moment given by µ$_{tot}$ = µ + µ$_{ind}$ , µ$_{ind}$ = α · E + . . . , α$_{xx}$ = α$_{0}$ + ∑ k α ′ $_{k}$Q$_{k}$ + . . . , and the radiation field taken to be linearly polarized in the x -direction. Equation (3) follows from e.g. the Heisenberg equations of motion for the quantized Hamiltonian (3) The equation of motion is given by the standard (Placzek) model for Raman-active modes [8, 21, 33]: d $^{2}$Q$_{k}$ /dt 2 + ω 2 $_{k}$Q$_{k}$ = α ′ k E ( t ) 2 / 2 (3) ˆ H = ∑ k ( ˆ P 2 $_{k}$/ 2 + ω 2 k ˆ Q 2 $_{k}$/ 2 ) − ˆ µ · E − E · ˆ α · E / 2 (4) E ( t ) = E $_{0}$sin ( πt/ 2 τ ) sin ( ωt + δ ) , 0 < t < 2 τ (1) ¯ E ( t ) 2 = ¯ E 2 0 sin $^{2}$( πt/ 2 τ ) , 0 < t < 2 τ (2) Q$_{k}$ ( t ) = α ′ k E 2 0 4 ω 2 k sin( r$_{k}$π ) 1 − r 2 k sin( ω$_{k}$t − r$_{k}$π ) , t ≥ 2 τ (5) r$_{k}$ = 2 τ/T$_{k}$ . (6) where arXiv:1001.1016v2 [cond-mat.mes-hall] 1 Oct 2010