Bilha Segev

Phase-space derivation of propensity rules for energy transfer processes between Born–Oppenheimer surfaces

We consider a simple method for calculating weak Franck–Condon factors. The Wigner transform of the wave function on an initial Born–Oppenheimer state is calculated for the donor potential surface and projected onto the acceptor energy surface energy shell. The integrated projection yields an approximation for the relevant Franck–Condon factors, while phase-space integrand shows where in phase space the leakage occurs between the donor and acceptor states. This in turn determines the initial conditions on the acceptor surface for subsequent IVR and energy flow. Propensity rules are obtained by recognizing phase-space points of closest approach of the initial-state Wigner function and the final-state energy surface. The example of two coupled harmonic oscillators is explicitly solved to demonstrate the power of this phase-space approach.

Semiclassical estimation of Franck-Condon factors and transition rates for vertical and nonvertical transitions

We develop a systematic way for estimating multidimensional Franck–Condon factors and transition rates for vertical and nonvertical transitions. By analyzing the phase-space overlap integral, we find the most probable positions and momenta of the nuclei immediately after the electronic transition. We find the transition rate by treating the dominant region in phase space as a funnel for the transition and by calculating the flow of probability through this funnel. We use the Wigner representation and its semiclassical limit and find that the transition occurs through a point(s) on the final surface of constant energy where the initial Wigner function is maximal. This dominant contribution is estimated analytically. Results are illustrated for Harmonic, Morse and Poeschl–Teller oscillators.

A phase-space approach to the T1⇝S0 radiationless decay in benzene: The effect of deuteration

The influence of full deuteration on the T1 right arrow-wavy S0 intersystem crossing in benzene is studied by a phase space approach. A full treatment of all the vibrational modes in the molecule leads to a ratio of the rate between the two isotopomers which is very close to the experimental value. Several aspects of the results are compared to previous estimates, and the effects of anharmonicity on the rates and accepting modes are examined. This first successful application of the method to a real physical system encourages the possibility of establishing a routine procedure for simple calculations of transition rates even for relatively large molecules.

Most probable path in phase space for a radiationless transition in a molecule

We study a radiationless transition in a polyatomic molecule when the electronic energy of an excited electronic state is transferred to the vibrational degrees of freedom of the nuclei, and when some nuclear coordinates change abruptly. This jump between the donor energy surface and the acceptor one gives the initial conditions for the subsequent dynamics on the acceptor surface, and the partition of energy between competing accepting modes. In the Wigner representation, the physical problem of recognizing the accepting modes for a radiationless vibronic relaxation reduces to the mathematical problem of finding the maximum of a function of many variables under a constraint. The function is the initial Wigner function of the nuclei and the constraint is energy conservation. In a harmonic approximation for the potential surfaces, the problem is equivalent to finding the distance from a given point to a multidimensional ellipsoid. This geometrical problem is solved in closed form. For nonharmonic potentials, the optimization problem is solved perturbatively.

Engineering entanglement: The fast-approach phase gate

Optimal-control techniques and a fast-approach scheme are used to implement a collisional controlled-phase gate in a model of cold atoms in an optical lattice, significantly reducing the gate time as compared to adiabatic evolution while maintaining high fidelity. Objective functionals are given for which optimal paths are obtained for evolution that yields a controlled-phase gate up to single-atom Rabi shifts. Furthermore, the fast-approach procedure is used to design a path to significantly increase the fidelity of nonadiabatic transport in a recent experiment. In addition, the entanglement power of phase gates is quantified.

Dominant channels of vibronic transitions in molecules with several identical modes

Abstract Weak-coupling radiationless transitions (internal conversion or inter system crossing) are studied assuming separability and symmetry over N identical modes. Franck–Condon factors control the branching ratios between exciting just one of the equivalent modes, or equally distributing the available energy. The dominant process can be predicted by an exact quantum mechanical solution if the wavefunctions are known (Gaussian initial distributions and accepting Morse or Poeschl-Teller oscillators, for example); or more generally by a Wigner phase space surface-jumping analysis based on a classical limit of the Wigner function, using only the donor distribution and the acceptor potential surface.

Surface jumping in a harmonic model of trans‐octatetraene: Franck—condon factors and accepting vibrational modes in s1→S0 non‐vertical radiationless transition

A phase-space method for finding the accepting modes in a non-vertical radiationless vibronic transition and for recognizing the final state with the largest Franck-Condon factor is applied to a harmonic model of the S 1 → S 0 relaxation in trans-octatetraene. Input required for the analysis includes the energy gap between S 1 and S 0 , normal mode frequencies, reduced masses, and eigenvectors (including the Duschinsky rotation matrix), and the molecule equilibrium configurations (bond lengths and angles) in S 1 and S 0 . Some of these data are taken from published experimental results and some are calculated in this work. The energy gap of 0.132 au is much larger than the energy of a vertical transition, which is only 0.047 au. The phase-space method gives a closed-form analytic solution for how to divide the excess energy between the accepting modes. The final distribution includes a large excitation of the two CH 2 end groups, where the motion of the two hydrogen atoms within each quasilocal CH 2 group is antisymmetric; a symmetric stretch of the two central C-H bonds of the molecule; and small totally symmetric bending of the whole molecule. Comparison of Franck-Condon factors (exact within the harmonic model) of the final state obtained by the phase-space analysis and of other similar isoenergetic states shows that the phase-space method indeed chooses the most probable final energy distribution. Possible modifications of these results due to anharmonic effects are discussed.

Fermi’s golden rule in the Wigner representation

When Fermis golden rule (FGR) is studied in the Wigner representation, the transition rate from an initial pure state or from an initial thermal distribution into a quasicontinuum manifold of degenerate states is given by an overlap integral of Wigner functions in phase space. In the semiclassical limit the transition rate is obtained by integrating over the regions in phase space where the energy difference between the initial and final potential surfaces is equal to the available energy. The integral is weighted by the initial probability density to be at that phase-space region. The classical limit of FGR is thus both simple and intuitive. In one dimension a relation to the Landau–Zener–Stuckelberg formula is established. The multi-dimensional case is considered by induction, proving that for separable multi-dimensional systems deviations of the logarithm of the transition rate from its classical limit scale at worst linearly with the dimension.

Quantum and QED Effects on Reflection from an Atomic Mirror

I review recent work on the evanescent wave atomic mirror [B. Segev, R. Cote, and M.G. Raizen, PRA 56, R3350 (1997); R. Cote, B. Segev, and M.G. Raizen, PRA 58, 3999 (1998)] in which we demonstrated that reflection of cold sodium atoms from such a mirror is expected to exhibit both quantum and quantum- electrodynamic (QED) effects. I then present a new phase-space wave-packet analysis of the time dependent reflection process. The time dependent distribution of cold atoms reflected from an atomic mirror can be used to probe the effective potential of the ‘mirror’.

Irreversible transitions in the Wigner representation

Irreversible transitions are studied in the Wigner representation. When Fermi Golden Rule (FGR) applies, the transition rate is given by the phase-space overlap integral between the initial Wigner function and the final quasi-distribution which is the Wigner transform of the microcanonical density matrix. Classical approximations and various applications are discussed.