Dynamics of polar polarizable rotors acted upon by unipolar electromagnetic pulses: From the sudden to the adiabatic regime

Abstract

We study, analytically as well as numerically, the dynamics that arises from the interaction of a polar polarizable rigid rotor with single unipolar electromagnetic pulses of varying length, ∆τ , with respect to the rotational period of the rotor, τ r . In the sudden, non-adiabatic limit, ∆τ τ r , we derive analytic expressions for the rotor's wavefunctions, kinetic energies, and field-free evolution of orientation and alignment. We verify the analytic results by solving the corresponding timedependent Schrödinger equation numerically and extend the temporal range of the interactions considered all the way to the adiabatic limit, ∆τ τ r , where general analytic solutions beyond the field-free case are no longer available. The effects of the orienting and aligning interactions as well as of their combination on the post-pulse populations of the rotational states are visualized as functions of the orienting and aligning kick strengths in terms of populations quilts. Quantum carpets that encapsulate the evolution of the rotational wavepackets provide the space-time portraits of the resulting dynamics. The population quilts and quantum carpets reveal that purely orienting, purely aligning, or even-break combined interactions each exhibit a sui generis dynamics. In the intermediate temporal regime, we find that the wavepackets as functions of the orienting and aligning kick strengths show resonances that correspond to diminished kinetic energies at particular values of the pulse duration. * burkhard.schmidt@fu-berlin.de † bretislav.friedrich@fhi-berlin.mpg.de arXiv:1806.11329v3 [quant-ph] 9 Aug 2018 basis set for expanding the time-dependent wavefunction, with the resultwhere E J = J(J + 1). Note that the expansion coefficients, C J J 0 ,M 0 , are time-independent, in consequence of the fact that the time dependence in Eq. (10) only arises from the e −iJ 2 τ term.The final form of the wavefunction in the sudden limit is given by (for a detailed derivationare the Clebsch-Gordan coefficients and only c J is a function of the kick strengths, cf. Eq. (A5).Throughout the remainder of this paper, we restrict ourselves to the case when the free rotor is initially in its ground state, J 0 = M 0 = 0. As a result, the C J J 0 ,M 0 coefficients in Eq. (11) reduce toThe coefficients C J J 0 ,M 0 arising in Eq. (11) can be found by expanding the time-independent term in Eq. (10) in terms of spherical harmonics,with Γ the gamma function; we applied the Legendre duplication formula [49] to achieve the final form of Eq. (A4).By making use of Eq. (A2) and Eq. (A5) we can evaluateC J J 0 ,M 0 in Eq. (A1) as follows,where J 1 M 1 , J 2 M 2 |J 3 M 3 are the Clebsch-Gordan coefficients, which vanish unless |J −J 0 | ≤ J ≤ J + J 0 and J + J + J 0 is an even integer [49]. Finally, we obtainIn all the analytic results presented in this paper, Eqs. (A5) and (A7) were found to converge satisfactorily for κ and J up to 80 and 50, respectively (e.g. for P η = P ζ = 8 the 80th term of the sum over k in Eq. (A5) is close to 10 −30 for C 50 0,0 ≈ 1...

Quantum dynamics of a polar rotor acted upon by an electric rectangular pulse of variable duration

“…The range of σ was varied between 0.005 and 10 in steps of 0.005, i.e. from the impulsive, non-adiabatic regime (the results for σ = 0.005 are in agreement with the theory of δ-kicks [56]) to the adiabatic limit (the results for σ = 10 approximate well the stationary solutions of the TDSE, Equation (3)). Owing to the rectangular pulse-shape, the present problem may be well treated in a numerically exact way by the diagonalisation of a time-independent Hamiltonian in the (numerically finite) basis of J-states (since quantum number M (= 0) is conserved).…”

“…However, for longer pulses of small to moderate strength, we observed sudden quasi-periodic drops in the kinetic energy imparted to the rotor by the pulse as a function of the pulse duration, see Figure 4 of Ref. [56].…”

“…Whereas either adiabatic interactions [26][27][28][29][30][31][32][33][34][35][36][37][38] or their impulsive, non-adiabatic counterparts [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53] have received much attention, interactions with finiteduration pulses have been scarce [54][55][56]. It is the last that are the focus of the present study.…”

“…The impulsive interaction of a rotor with a δ-pulse is analytically solvable and represents a benchmark for the behaviour of the expansion (or hybridisation) coefficients of the rotational states, J, that make up the wavepacket created by the pulse. For a purely orienting interaction that arises for a polar rotor acted upon by an electric field, the hybridisation coefficents were found [11,12,56] to be proportional to the spherical Bessel functions of the first kind, J J (P), where P is the pulse strength, see below. The zeroes of the hybridisation coefficients, see Figure 1(b) of Ref.…”

“…The zeroes of the hybridisation coefficients, see Figure 1(b) of Ref. [56], determine the minima of the post-pulse populations of the various rotational states and are key to optimising the post-pulse value of a given observable. The figure also illustrates that, typically, many states are hybridised by a δ-pulse.…”