Wavepacket Reconstruction Research Articles

Communication: Systematic elimination of Stokes divergences emanating from complex phase space caustics.

Stokes phenomenon refers to the fact that an asymptotic expansion of complex functions can differ in different regions of the complex plane, and that beyond the so-called Stokes lines the expansion has an unphysical divergence. An important special case is when the Stokes lines emanate from phase space caustics of a complex trajectory manifold. In this case, symmetry determines that to second order there is a double coverage of the space, one portion of which is unphysical. Building on the seminal but laconic findings of Adachi, we show that the deviation from second order can be used to rigorously determine the Stokes lines and therefore the region of the space that should be removed. The method has applications to wavepacket reconstruction from complex valued classical trajectories. With a rigorous method in hand for removing unphysical divergences, we demonstrate excellent wavepacket reconstruction for the Morse, Quartic, Coulomb, and Eckart systems.

Multivalued classical mechanics arising from singularity loops in complex time.

Complex-valued classical trajectories in complex time encounter singular times at which the momentum diverges. A closed time contour around such a singular time may result in final values for q and p that differ from their initial values. In this work, we develop a calculus for determining the exponent and prefactor of the asymptotic time dependence of p from the singularities of the potential as the singularity time is approached. We identify this exponent with the number of singularity loops giving distinct solutions to Hamilton's equations of motion. The theory is illustrated for the Eckart, Coulomb, Morse, and quartic potentials. Collectively, these potentials illustrate a wide variety of situations: poles and essential singularities at finite and infinite coordinate values. We demonstrate quantitative agreement between analytical and numerical exponents and prefactors, as well as the connection between the exponent and the time circuit count. This work provides the theoretical underpinnings for the choice of time contours described in the studies of Doll et al. [J. Chem. Phys. 58(4), 1343-1351 (1973)] and Petersen and Kay [J. Chem. Phys. 141(5), 054114 (2014)]. It also has implications for wavepacket reconstruction from complex classical trajectories when multiple branches of trajectories are involved.

Attosecond correlation dynamics

Photoemission of an electron is commonly treated as a one-particle phenomenon. With attosecond streaking spectroscopy we observe the breakdown of this single active-electron approximation by recording up to six attoseconds retardation of the dislodged photoelectron due to electronic correlations. We recorded the photon-energy-dependent emission timing of electrons, released from the helium ground state by an extreme-ultraviolet photon, either leaving the ion in its ground state or exciting it into a shake-up state. We identify an optical field-driven d.c. Stark shift of charge-asymmetric ionic states formed after the entangled photoemission as a key contribution to the observed correlation time shift. These findings enable a complete wavepacket reconstruction and are universal for all polarized initial and final states. Sub-attosecond agreement with quantum mechanical ab initio modelling allows us to determine the absolute zero of time in the photoelectric effect to a precision better than 1/25th of the atomic unit of time. Photoemission is not a simple process and it is not instantaneous. Delays of a few attoseconds have now been measured in helium and it seems that they are partly due to electronic correlations.

Perfect wave-packet splitting and reconstruction in a one-dimensional lattice

Particle delocalization is a common feature of quantum random walks in arbitrary lattices. However, in the typical scenario a particle spreads over multiple sites and its evolution is not directly useful for controlled quantum interferometry, as may be required for technological applications. In this paper we devise a strategy to perfectly split the wave-packet of an incoming particle into two components, each propagating in opposite directions, which reconstruct the shape of the initial wavefunction after a particular time $t^*$. Therefore, a particle in a delta-like initial state becomes exactly delocalized between two distant sites after $t^*$. We find the mathematical conditions to achieve the perfect splitting which are satisfied by viable example Hamiltonians with static site-dependent interaction strengths. Our results pave the way for the generation of peculiar many-body interference patterns in a many-site atomic chain (like the Hanbury Brown and Twiss and quantum Talbot effects) as well as for the distribution of entanglement between remote sites. Thus, as for the case of perfect state transfer, the perfect wave-packet splitting can be a new tool for varied applications.

Excited-state wavepacket and potential reconstruction by coherent anti-Stokes Raman scattering

Among the major challenges in the chemical sciences is controlling chemical reactions and deciphering their mechanisms. Since much of chemistry occurs in excited electronic states, in the last three decades scientists have employed a wide variety of experimental techniques and theoretical methods to recover excited-state potential energy surfaces and the wavepackets that evolve on them. These methods have been partially successful but generally do not provide a complete reconstruction of either the excited state wavepacket or potential. We have recently proposed a methodology for reconstructing excited-state molecular wavepackets and the corresponding potential energy surface [Avisar and Tannor, Phys. Rev. Lett., 2011, 106, 170405]. In our approach, the wavepacket is represented as a superposition of the set of vibrational eigenfunctions of the molecular ground-state Hamiltonian. We assume that the multidimensional ground-state potential surface is known, and therefore these vibrational eigenfunctions are known as well. The time-dependent coefficients of the basis functions are obtained by experimental measurement of the resonant coherent anti-Stokes Raman scattering (CARS) signal. Our reconstruction strategy has several significant advantages: (1) the methodology requires no a priori knowledge of any excited-state potential. (2) It applies to dissociative as well as to bound excited-state potentials. (3) It is general for polyatomics. (4) The excited-state potential surface is reconstructed simultaneously with the wavepacket. Apart from making a general contribution to the field of excited-state spectroscopy, our method provides the information on the excited-state wavepacket and potential necessary to design laser pulse sequences to control photochemical reactions.

Low-frequency anomalies in dynamic localization

Quantum mechanical spreading of a particle hopping on tight binding lattices can be suppressed by the application of an external ac force, leading to periodic wave packet reconstruction. Such a phenomenon, referred to as dynamic localization (DL), occurs for certain ‘magic’ values of the ratio Γ = F0/ω between the amplitude F0 and frequency ω of the ac force. It is generally believed that in the low-frequency limit (ω → 0) DL can be achieved for an infinitesimally small value of the force F0, i.e. at finite values of Γ. Such normal behavior is found in homogeneous lattices as well as in inhomogeneous lattices of the Glauber–Fock (GF) type. Here we introduce a tight-binding lattice model with inhomogeneous hopping rates, referred to as a pseudo-GF lattice, which shows DL but fails to reproduce the normal low-frequency behavior of homogeneous and GF lattices. In pseudo-GF lattices, DL can be exactly realized; however, at the DL condition the force amplitude F0 remains finite as ω → 0. Such an anomalous behavior is explained in terms of a symmetry-breaking transition of an associated two-level non-Hermitian Hamiltonian that effectively describes the dynamics of the Hermitian lattice model.

Quantum recurrence and fractional dynamic localization in ac-driven perfect state transfer Hamiltonians

Quantum recurrence and dynamic localization are investigated in a class of ac-driven tight-binding Hamiltonians, the Krawtchouk quantum chain, which in the undriven case provides a paradigmatic Hamiltonian model that realizes perfect quantum state transfer and mirror inversion. The equivalence between the ac-driven single-particle Krawtchouk Hamiltonian Hˆ(t) and the non-interacting ac-driven bosonic junction Hamiltonian enables to determine in a closed form the quasi energy spectrum of Hˆ(t) and the conditions for exact wave packet reconstruction (dynamic localization). In particular, we show that quantum recurrence, which is predicted by the general quantum recurrence theorem, is exact for the Krawtchouk quantum chain in a dense range of the driving amplitude. Exact quantum recurrence provides perfect wave packet reconstruction at a frequency which is fractional than the driving frequency, a phenomenon that can be referred to as fractional dynamic localization.

Multi-dimensional wavepacket and potential reconstruction by resonant coherent anti-Stokes Raman scattering: Application to H2O and HOD

We have recently proposed a methodology for reconstructing excited-state (ExS) molecular wavepackets, and the corresponding potential energy surface, from three-pulse resonant coherent anti-Stokes Raman scattering and knowledge of the ground-state potential [Avisar and Tannor, Phys. Rev. Lett. 106, 170405 (2011)]. The methodology is general for polyatomics and applies to any form of ExS potential--bound or dissociative. In our previous work we demonstrated the method on diatomics. Here, we demonstrate the method on the triatomics H(2)O and HOD, reconstructing the ExS wavepacket and potential in the two bond-stretching coordinates.

Wavepacket and potential reconstruction by four-wave mixing spectroscopy: preliminary application to polyatomic molecules

We have recently shown how the excited-state wavepacket of a polyatomic molecule can be completely reconstructed from resonant coherent anti-Stokes Raman spectroscopy [Avisar and Tannor, Phys. Rev. Lett., 2011, 106, 170405]. The method assumes knowledge of the ground-state potential but not of any excited-state potential, however the latter can be computed once the excited-state wavepacket is known. The formulation applies to dissociative as well as bound excited potentials. We demonstrate the method on the Li2 molecule with its bound first excited-state as well as with a model dissociative excited state potential. Preliminary results are shown for a model two-dimensional molecular system. The calculations assume constant transition dipole moment (Condon approximation), delta-pulse excitation and a single excited-state potential, but we discuss the implications of removing these assumptions.

Periodic wave packet reconstruction in truncated tight-binding lattices

A class of truncated tight-binding Hermitian and non-Hermitian lattices with commensurate energy levels, showing periodic reconstruction of the wave packet, is presented. Examples include exact Bloch oscillations on a finite lattice, periodic wave packet dynamics in non-Hermtian lattices with a complex linear site-energy gradient, and self-imaging in lattices with commensurate energy levels quadratically-varying with the quantum number.

Quantum State Reconstruction of a Rotational Wave Packet Created by a Nonresonant Intense Femtosecond Laser Field

We have experimentally determined the amplitudes and phases of a rotational wave packet in an adiabatically cooled benzene molecule, created by a nonresonant intense femtosecond laser field. In this wave-packet reconstruction, the initial wave packet is further interfered by a replica of the first laser pulse, and the resultant modulation in population is observed in a state-resolved manner. Though several states with different nuclear-spin modifications are populated in the initial condition, a single wave packet created from one of them (with J=0) is specifically reconstructed. Phase shifts characteristic of stepwise Raman excitation beyond the perturbative regime are experimentally identified.

Wave-packet reconstruction via local dynamics in a parabolic lattice

We study the dynamics of a wave packet in a potential formed by the sum of a periodic lattice and a parabolic potential. The dynamics of the wave packet is essentially a superposition of oscillations with frequencies proportional to the local slope of the parabolic potential. The amplitude and the phase of the Fourier transform of a signal characterizing this dynamics then contain information about the amplitude and the phase of the wave packet at a given lattice site. Hence, complete reconstruction of the wave packet in real space can be performed from a study of the dynamics of the system.

Wave-packet reconstruction in a two-electron atom via impulsive isolated core excitation

In a two-electron atom, autoionization from an impulsive isolated core excitation is a sensitive probe of a Rydberg electronic wave packet. This probe via the electron-electron correlation allows a quantum measurement of the wave function to be made. In addition to this capability, the measurement process has the advantage that it is independent of the excitation process. An analytical formula describes this probe. This formula is used in conjunction with a nonlinear least-squares algorithm to reconstruct the wave function of this wave packet. The algorithm is relatively insensitive to noise. With 20% added Gaussian noise, the algorithm can still reproduce the amplitudes and phases of the wave function.

State reconstruction and irregular wavefunctions for the hydrogen atom

Inspired by a recently proposed procedure by Leonhardt and Raymer for wavepacket reconstruction, we calculate the irregular wavefunctions for the bound states of the Coulomb potential. We select the irregular solutions which have the simplest semiclassical limit.