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Bulletin of the American Physical Society

Coherent time evolution of Rydberg level populations in a Landau-Zener grid

David A. Harmin

Published May 1, 1993• 0 citations

Abstract

The time evolution groups of parallel energy levels (m, m{prime} = 0, 1, 2,...) is considered as a model of dynamic state mixing of interacting Rydberg manifolds by ramped electric fields used in selective field ionization (SFI). Each intersection [m, m{prime}] of the triangular grid of level crossings is treated as an isolated 2-level Landau-Zener anticrossing with amplitudes d and a for diabatic and adiabatic transitions, respectively ({vert_bar}d{vert_bar}{sup 2} + {vert_bar}a{vert_bar}{sup 2} = 1). Beginning on the uppermost upward-going level (m = 0) and following successive time steps {tau}{sup (N)} (N = 0,1,2,...), a path analysis leads to complex interference patterns among the many paths arriving at an intersection [m, m{prime}] after passing through N = m + m{prime} previous ones. The relative phase between any two paths is a multiple of the unit of action {var_phi} defined by the two pairs of adiabatic levels connecting four neighboring anticrossings. Compared to results from an analysis neglecting interference effects, there occurs here a general {open_quotes}emptying out{close_quotes} of population at the center of the interaction region (where the majority of paths lead). Most striking are resonances at high ramp rates (d {yields} 1), which with high probability enhance upward-going levels whose indexmore » m is an integral multiple of 27{pi}/{var_phi}. Though {var_phi} is sensitive to ramp rate, it is nevertheless expected that observed diabatic SFI signals should exhibit sequences of secondary peaks traceable to such resonances, which should not be assigned to extraneous initial states.« less

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Coherent time evolution on a grid of Landau-Zener anticrossings

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A model of the time evolution of two interacting Rydberg manifolds of energy levels subject to a linearly ramped electric field is solved exactly in the Landau-Zener (LZ) approximation. Each manifold's levels are treated as linear in time, parallel, equally spaced, and infinite in number. Their pairwise interactions produce a regular two-dimensional grid of isolated anticrossings. The time development of an initially populated state is then governed by two-level LZ transitions at avoided crossings and adiabatic evolution between them, parametrized by the LZ transition probability $D$ and a dynamical phase unit \ensuremath{\varphi}. The resulting probability distributions of levels are given analytically in the form of recursion relations, generating functions, integral representations involving $D$ and \ensuremath{\varphi}, and in certain limits by Bessel or Whittaker functions. Level populations are mapped out versus location on the grid for a range of cases. Interference effects lead to two principal types of probability distributions: a braiding adiabatic pattern with revivals for small $D$ and a diabatic pattern for $D\ensuremath{\rightarrow}1$ in which only certain levels parallel to the initial one are appreciably populated. The sensitivity of the coherent evolution to \ensuremath{\varphi} is discussed, along with the relation of this model to others and to selective-field ionization.

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The time evolution groups of parallel energy levels (m, m{prime} = 0, 1, 2,...) is considered as a model of dynamic state mixing of interacting Rydberg manifolds by ramped electric fields used in selective field ionization (SFI). Each intersection [m, m{prime}] of the triangular grid of level crossings is treated as an isolated 2-level Landau-Zener anticrossing with amplitudes d and a for diabatic and adiabatic transitions, respectively ({vert_bar}d{vert_bar}{sup 2} + {vert_bar}a{vert_bar}{sup 2} = 1). Beginning on the uppermost upward-going level (m = 0) and following successive time steps {tau}{sup (N)} (N = 0,1,2,...), a path analysis leads to complex interference patterns among the many paths arriving at an intersection [m, m{prime}] after passing through N = m + m{prime} previous ones. The relative phase between any two paths is a multiple of the unit of action {var_phi} defined by the two pairs of adiabatic levels connecting four neighboring anticrossings. Compared to results from an analysis neglecting interference effects, there occurs here a general {open_quotes}emptying out{close_quotes} of population at the center of the interaction region (where the majority of paths lead). Most striking are resonances at high ramp rates (d {yields} 1), which with high probability enhance upward-going levels whose indexmore » m is an integral multiple of 27{pi}/{var_phi}. Though {var_phi} is sensitive to ramp rate, it is nevertheless expected that observed diabatic SFI signals should exhibit sequences of secondary peaks traceable to such resonances, which should not be assigned to extraneous initial states.« less

A model of the time evolution of two interacting Rydberg manifolds of energy levels subject to a linearly ramped electric field is solved exactly in the Landau-Zener (LZ) approximation. Each manifold's levels are treated as linear in time, parallel, equally spaced, and infinite in number. Their pairwise interactions produce a regular two-dimensional grid of isolated anticrossings. The time development of an initially populated state is then governed by two-level LZ transitions at avoided crossings and adiabatic evolution between them, parametrized by the LZ transition probability

D

and a dynamical phase unit \ensuremath{\varphi}. The resulting probability distributions of levels are given analytically in the form of recursion relations, generating functions, integral representations involving

D

and \ensuremath{\varphi}, and in certain limits by Bessel or Whittaker functions. Level populations are mapped out versus location on the grid for a range of cases. Interference effects lead to two principal types of probability distributions: a braiding adiabatic pattern with revivals for small

D

and a diabatic pattern for $D\ensuremath{\rightarrow}1$ in which only certain levels parallel to the initial one are appreciably populated. The sensitivity of the coherent evolution to \ensuremath{\varphi} is discussed, along with the relation of this model to others and to selective-field ionization.

The mixing of Rydberg manifolds, induced by a ramped electric field F(t)=F\ifmmode \dot{}\else \.{}\fi{}t, is modeled by interactions among two intersecting groups of parallel energy levels. The levels form a grid, each node of which is treated as an isolated two-level Landau-Zener anticrossing, characterized by probabilities D=exp(-2\ensuremath{\pi}\ensuremath{\gamma}) and A=1-D for diabatic and adiabatic transitions, respectively. The model assumes a core with one nonvanishing quantum defect ${\mathrm{\ensuremath{\mu}}}_{0}$ and with extremal Stark-level slopes, so that \ensuremath{\gamma}=\ensuremath{\Vert}${\mathrm{\ensuremath{\mu}}}_{0}$${\mathrm{\ensuremath{\Vert}}}^{2}$/3F\ifmmode \dot{}\else \.{}\fi{}

{\mathit{n}}^{10}

for all anticrossings, where the manifolds have principal quantum numbers n and n+1. Interference effects are ignored, which allows analytical treatment of the system's evolution using path statistics or recursion relations. An initially populated edge state leads to a single-humped distribution of level populations. Analytical expressions are found for the probability distribution as a function of time, as well as its average and standard deviation. Limiting forms of the distribution at large times and in the diabatic (D\ensuremath{\rightarrow}1) and adiabatic (D\ensuremath{\rightarrow}0) limits are also given. These features are contrasted with those of a random walk. The relevance of the model to selective-field ionization is discussed.

We present a direct comparison between calculated and measured ionization in Li and Rb when an initially bound Rydberg electron is stripped from the atom by ramping an electric field [0.415 V/(cm ns)]. We describe the method used to calculate the field dependence of the ionization; the method can be used to evolve the population from field strengths where three or more n manifolds cross to the fields where the electron is stripped from the atom. The essential feature of the method is the repeated application of the Landau-Zener approximation for all of the level crossings as the field is ramped. We also give a description of the dynamics of Li and Rb in the ramping field. We observe that the Li selective field ionization is sensitive to the Stark levels within an n manifold, while for Rb only an n dependence is observed.

We report the observation of a controlled Landau-Zener transition (LZT) in Rydberg atoms by breaking the symmetry of the underlying Hamiltonian. For a nonhydrogenic Rydberg atom inside a changing electric (F) field, a LZT occurs between the avoided crossing energy levels of neighboring Rydberg states only for a sufficiently high changing rate. If a transverse magnetic (B) field is applied as we implement, the atomic level symmetry is broken, which causes the Stark manifolds denoted by a different |m| (m is the magnetic quantum number) to interact with each other. The mixed state levels end up pushing the adiabatically repelled target states closer and additionally they serve as stepping stones for the sequential LZTs between the neighboring sublevels. Such a feature significantly decreases the changing rate required for an efficient LZT inside a pure electric field. We report experimental observations that support the above scenario. It opens a versatile approach for engineering a controlled LZT in more general systems.

We propose a universal approach to the Landau-Zener problem in a three-level system. The problem is formulated in terms of Gell-Mann operators which generate SU (3) algebra and map the Hamiltonian on the effective anisotropic pseudospin 1 model. The vector Bloch equation for the density matrix describing the temporal evolution of the three-level crossing problem is also derived and solved analytically for the case where the diabatic states of the SU (3) Hamiltonian form a triangle. This analytic solution is in excellent quantitative agreement with the numerical solution of the Schrödinger equation for a 3-level crossing problem. The model demonstrates oscillation patterns which radically differ from the standard patterns for the two-level Landau-Zener problem. The triangle works as an interferometer and the interplay between two paths results in formation of "beats" and "steps" pattern in the time-dependent transition probability. The characteristic time scales describing the "beats" and "steps" depend on a dwell time in the triangle. These scales are related to the geometric size of the interferometer. The possibilities of the experimental realization of this effect in triple quantum dots and in two-well traps for cold gases are discussed.