Photon echo signature of vibrational superposition states created by femtosecond excitation of molecules

A pair of coherent femtosecond pulse excitations applied to a molecule with strong electron-phonon coupling creates a coherent superposition of a low momentum and a high momentum wavepacket in the vibrational states of both the excited state and the ground state of the coherent transition. As the excited state is accelerated further, interference between the high momentum ground state contribution and the low momentum excited state contribution causes a photon echo. This photon echo is a direct consequence of quantum interference between separate vibrational trajectories and can therefore provide experimental evidence of the non-classical properties of molecular vibrations. Typeset using REVTEX 1 “The feature of quantum mechanics which most distinguishes it from classical mechanics is the coherent superposition of distinct physical states” [1]. In the case of molecular vibrations, the motion of a single Gaussian wavepacket usually corresponds well with the classical motion for the potential considered. However, quantum mechanics also allows coherent superpositions between two wavepackets with distinct positions and momenta. The properties of such superpositions depend on the phase between the two Gaussian wavepackets, a property that has no analog in the classical theory. The study of such superpositions should therefore provide insights into the non-classical features of quantum mechanics. The creation and measurement of vibrational wavepackets by optical excitations has been studied both experimentally and theoretically [2–10]. In particular, the creation of a coherent superposition of two distinct Gaussian wavepackets (also referred to as a “cat state”) in a molecular vibration has been predicted if the molecule is excited by a sequence of two femtosecond pulses [4]. However, it is difficult to obtain experimental evidence indicating the successful creation of the coherent superposition. In the following, it will be shown that the nonlinear optical contributions in the formation of the superposition state automatically produce a photon echo effect that corresponds to the “which path” interference between two distinct trajectories of the molecular vibration. Thus, the vibrational photon echo may provide direct experimental evidence of quantum coherence between two vibrational wavepackets. The Hamiltonian describing the electron-phonon interaction of an electronic two level system and a single vibrational degree of freedom may be written as Ĥ0 = p̂ 2m + VG(x̂)⊗|G〉〈G |+VE(x̂)⊗|E〉〈E |, (1) where m is the effective mass of the vibration, and VG(x̂) and and VE(x̂) describe the vibrational potentials associated with the electronic ground state | G〉 and the electronic excited state | E〉, respectively. The position operator x̂ and the conjugate momentum operator p̂ represent the dynamical variables of the vibrational mode under consideration. Initially, the molecular system is in the electronic ground state |G〉 and the vibrational state | ψ0〉 is localized near the minimum of the ground state potential VG(x̂). It is therefore convenient to define this minimum as x = 0 and its potential as VG(0) = 0. If the timescales considered are much shorter than the period of a molecular vibration, the vibrational wavefunction will always remain close to x = 0. Moreover, the momentum is also close to zero initially, and its changes can be considered small enough to neglect the quadratic term p̂/m. The total Hamiltonian can then be linearized in x̂ and p̂. The approximate Hamiltonian reads Ĥ0 ≈ (h̄ω0 − FEx̂)⊗|E〉〈E |, with h̄ω0 = VE(x = 0) and FE = − ∂ ∂x VE(x)|x=0. (2) This Hamiltonian describes the linear acceleration of the excited state component of the vibrational state by the force FE. In terms of the momentum eigenstate components ψG(p; t) = 〈G; p |ψ(t)〉 and ψE(p; t) = 〈E; p |ψ(t)〉, this acceleration can be written as 2 ψG(p; t) = ψG(p; 0) ψE(p; t) = exp(−iω0t)ψE(p−FE t; 0). (3) Note that this evolution of the vibrational wavefunction preserves the quantum coherence between the excited state and the ground state contributions. It is therefore not possible to assign separate “realities” to excited state and ground state molecules. Instead, any coherent overlap between the vibrational states corresponds to a coherent electronic dipole. This coherent dipole is given by the operator d̂ =|G〉〈E |. Its expectation value reads 〈d̂〉(t) = ∫ dpψ∗ G(p; t)ψE(p; t). (4) The electronic dipole of the molecular transition thus represents an interference between the accelerated excited state and the non-accelerated ground state. The acceleration process separates the vibrational state like a beam splitter separates the incoming fields. In a photon echo experiment, the first pulse at t0 − τ splits the vibrational dynamics, the second pulse at t0 “reflects” the excited state into the ground state and vice versa, and the photon echo indicates interference between two indistinguishable paths of acceleration. The ground state component of the photon echo dipole corresponds to acceleration during t0 − τ < t < t0 followed by a constant momentum of FEτ during t0 < t < t0 + τ , and the excited state component corresponds to zero momentum during t0 − τ < t < t0 followed by acceleration to a momentum of FEτ during t0 < t < t0 + τ . These trajectories are illustrated in figure 1. In the following, we apply this description of the molecular dynamics to a pair of ultrafast excitations at times t0 − τ and t0. The pulses are considered to be much shorter than τ . Before the first pulse, the molecule is in its ground state, given by ψG(p) = ψ0(p) and ψE(p) = 0. Between the two pulses (t0 − τ < t < t0), the coherent evolution of the partially excited state is given by ψG(p; t) = cos(φ/2)ψ0(p) ψE(p; t) = e −iω0(t−t0+τ) sin(φ/2)ψ0 (p−FE(t−t0 + τ)) , (5) where φ is a measure of the pulse area exciting the molecule. The expectation value of the coherent dipole evolves according to 〈d̂〉(t) = e00 1 2 sin(φ) ∫ dpψ∗ 0(p)ψ0(p−FE(t−t0 + τ)), (6) which corresponds to the autocorrelation of the vibrational wavefunction in momentum space, ψ0(p). The dipole dephasing time tφ is therefore given by tφ = δp FE , (7) where δp is the momentum uncertainty of the initial wavepacket. If δp is much smaller than FEτ , then the coherent dipole will be close to zero at t = t0. The second pulse at t = t0 then restores dipole coherence by transferring part of the ground state component to the excited state and vice versa. The evolution of the total molecular state for t > t0 reads 3 ψG(p; t) = 1 2 (1 + cos(φ))ψ0(p)− e −iω0τ 1 2 (1− cos(φ))ψ0 (p−FEτ) ψE(p; t) = e −iω0(t−t0) 1 2 sin(φ)ψ0 (p−FE(t−t0)) +e00 1 2 sin(φ)ψ0 (p−FEτ−FE(t−t0)) . (8) The total vibrational state now consists of four separate contributions. Initially (t = t0), there is dipole coherence between two pairs, the one around p = 0 and the one around p = FEτ . This coherence is lost as the excited state is accelerated. The dipole dynamics of the decoherence process reads 〈d̂〉(t) = e00 1 2 sin(φ) cos(φ) ∫ dpψ∗ 0(p)ψ0(p−FE(t−t0)). (9) This corresponds to the linear response of the partially excited two level system to the second pulse. However, there is a revival of the dipole coherence in the form of a photon echo as the excited state from p = 0 is accelerated to p = FEτ and interferes with the ground state component there. Figure 2 illustrates the coherent wavefunction at t = t0 + τ . The dipole dynamics close to t = t0 + τ are given by 〈d̂〉(t) = e00 1 4 sin(φ) (1− cos(φ)) ∫ dpψ∗ 0(p−FEτ)ψ0(p−FE(t−t0)). (10) This result is again equal to the autocorrelation of ψ0(p), but it is centered around t = t0+τ . Figure 3 shows the sequence of pulses and the dipole response. Since the first two dipole signals are immediate responses to the exciting pulses, they suddenly appear at t0−τ and at t0, followed by a gradual decay given by the autocorrelation of ψ0(p). The echo pulse arises from the hidden coherence in the dynamics following the second pulse at t = t0. It therefore appears gradually and is symmetric around t = t0. Since dipole coherence always indicates an interference between ground state and excited state components, the echo indicates equal momentum of the accelerated excited state and the non-accelerated ground state. In terms of the average momentum of the four separate contributions, figure 1 shows the trajectories involved in the quantum interferences indicated by the dipole expectation value 〈d̂〉. By comparing the dipole evolution given in figure 3 with the trajectories in figure 1, the optical signals can be related to the quantum dynamics of the vibration. The analogy between photon echoes in inhomogeneously broadened transitions and the molecular photon echoes discussed here arises from the assumption that the position coordinate x̂ remains nearly constant during the experiment. The inhomogeneity is then a consequence of the randomness of the position coordinate x̂ given by the spatial width of the vibrational wavepacket. The main difference between the photon echo in an inhomogeneously broadened medium and the vibrational photon echo discussed here is that the coherence of the contributions from different positions corresponds to a well defined momentum. It is therefore impossible to identify each precise position x with a different molecule, since this would imply an infinite momentum uncertainty. For most practical purposes, however, the experimental setup corresponds to a conventional photon echo experiment. In order to satisfy the assumption that x̂ does not change much during the experiment, the spatial shifts induced by the velocity p(t)/m during the delay time τ must be much 4 smaller than the spatial width of the vibrational wavepacket. The shift in position can be determined by integrating the velocity p(t)/m over time. The total shift of position thus contains a “memory” of the momentum path taken by the molecular vibration, destroying the quantum interference. At the interference point, the difference in position between the ground state component and the excited state component with p = FEτ is ∆x = FEτ 2 m . (11) In momentum representation, this shift appears as a phase factor of exp(−ip∆x/h̄). The maximal coherent dipole of the echo pulse is then reduced by a factor of