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Physical Review A

Coherent time evolution on a grid of Landau-Zener anticrossings

David A. Harmin

Published Jul 1, 1997• 38 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.56.232

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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 DDD 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 DDD 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 DDD and a diabatic pattern for D\ensuremath→1D\ensuremath{\rightarrow}1D\ensuremath→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{}n10{\mathit{n}}^{10}n10 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.
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
The Landau--Zener (LZ) model describes a two-level quantum system that undergoes an avoided crossing. In the adiabatic limit, the transition probability vanishes. An auxiliary control field HCDH_\text{CD}HCD​ can be reverse-engineered so that the full Hamiltonian H0+HCDH_0 + H_\text{CD}H0​+HCD​ reproduces adiabaticity for all parameter values. Our aim is to construct a single control field H1H_1H1​ that drives an ensemble of LZ-type Hamiltonians with a distribution of energy gaps. H1H_1H1​ works best statistically, minimizing the average transition probability. We restrict our attention to a special class of H1H_1H1​ controls, motivated by HCDH_\text{CD}HCD​. We found a systematic trade-off between instantaneous adiabaticity and the final transition probability. Certain limiting cases with a linear sweep can be treated analytically; one of them being the LZ system with Dirac δ(t)δ(t)δ(t) function. Comprehensive and systematic numerical simulations support and extend the analytic results.
In a model of interacting Rydberg manifolds, two sets of parallel energy levels, E{sub m}(t) = pt - m{epsilon} and E{sub m{prime}}(t)=p{prime}t + m{prime}{epsilon}{prime} (m,m{prime} = 0, {plus_minus}1,...;p{prime} {ne} p), are linearly shifted by an electric field F(t) = {dot F}t. Atomic core coupling V occurs only at the point of each pair of levels intersection, with a Landau-Zener diabatic transition probability D = d{sup 2} = e{sup {minus}2}{pi}V{sup 2}/{vert_bar}p{minus}p{prime}{vert_bar}. Adiabatic phases depend on the unit of action, {var_phi} = {contour_integral} dt E(t) = {epsilon}{epsilon}{prime}/{vert_bar}p{minus}p{prime}{vert_bar}. The amplitudes for approaching the intersection of levels m and m{prime} from an initial m = 0 at m{prime} = 0 are found from recursion relations, similar to an earlier model but including paths` interference via operators {xi} and {eta}, where {eta}{xi} = {xi}{eta}e{sup i{var_phi}}. Coefficients of ordered power {xi}{sup m}{eta}{sup m {prime}} in the generating functions 1 + {eta}[1 + d({eta}{minus}{xi}) {minus} {xi}{eta}]{sup {minus}1}({xi} {minus} d) and {alpha}{xi}[1 + d({eta} {minus} {xi}) {minus} {eta}{xi}]{sup {minus}1} yield the amplitudes for m- and m{prime}-levels. Another exact form for the amplitudes is obtained from integral representations. In the adiabatic limit (d {r_arrow} 0) the one for m-levels reduces to J{sub m{minus}m{prime}} (2d sin [1/2m{prime}{var_phi}]/sin[1/2{var_phi}]); in the diabaticmore » limit (d {r_arrow} 1) it reduces to a Whittaker function. These results help explain overall patterns in the population distributions during coherent evolution under field ramping.« less
The mixing of substates of an isolated Rydberg manifold (constant n,m) is studied using time-evolution matrices U(t,t0). Coherent mixing is effected by a linearly ramped electric field, F(t)=F\ifmmode \dot{}\else \.{}\fi{}t. The spherical l basis, diabatic (hydrogenic) p basis, and adiabatic (nonhydrogenic) q basis are considered. In the linear Stark regime, the diabatic levels are coupled by F(t) through the atomic core, parametrized by quantum defects. The resulting adiabatic levels EqF in this model group into a submanifold degenerate at F=0 plus shifted levels split off from the adiabatic manifold. Analytical expressions for the q levels Eq\ensuremath{\tau} and their coupling \ensuremath{\Gamma}qq\ensuremath{'}\ensuremath{\tau} are derived as a function of rescaled field or time \ensuremath{\tau}. The time evolution of the levels' populations is studied first for the two-level case---the usual Landau-Zener effect (LZE)---which is generalized to a multilevel Landau-Zener effect (MLZE) for any n manifold. The matrices U(t,t0) are constructed in the Riemann product representation, including Magnus corrections up to fourth order. The probability Pq\ensuremath{\rightarrow}q\ensuremath{'}(Vl) of making a transition from state q to q\ensuremath{'} upon a one-way pass across the manifold depends on a single parameter Vl=[2\ensuremath{\mu}l2/(3F\ifmmode \dot{}\else \.{}\fi{}n9)]1/2 (when one \ensuremath{\mu}l\ensuremath{\ne}0). Numerical results for l=m=0 indicate that either diabatic (p\ensuremath{\rightarrow}p\ensuremath{'}=p) or adiabatic (q\ensuremath{\rightarrow}q\ensuremath{'}=q) transitions always predominate, no matter what the ramp rate F\ifmmode \dot{}\else \.{}\fi{}. Strong evidence supports the conjecture that outer diabatic transitions between edge states of a manifold obey Pndiab(V)=exp(-\ensuremath{\beta}n\ensuremath{\pi}V2) for l=m=0, with \ensuremath{\beta}n=1 for n=2 (the LZE) and \ensuremath{\beta}n\ensuremath{\approx}lnn for n\ensuremath{\gg}2 (the MLZE), and that a similar analytical result holds for arbitrary {\ensuremath{\mu}l} and m.
Two aspects of the classic two-level Landau-Zener (LZ) problem are considered. First, we address the LZ problem when one or both levels decay, i.e., ${\ensuremath{\varepsilon}}_{j}(t)\ensuremath{\rightarrow}{\ensuremath{\varepsilon}}_{j}(t)\ensuremath{-}i{\ensuremath{\Gamma}}_{j}/2$. We find that if the system evolves from an initial time $\ensuremath{-}T$ to a final time +T+T+T such that $|{\ensuremath{\varepsilon}}_{1}(\ifmmode\pm\else\textpm\fi{}T)\ensuremath{-}{\ensuremath{\varepsilon}}_{2}(\ifmmode\pm\else\textpm\fi{}T)|$ is not too large, the LZ survival probability of a state $|j\ensuremath{\rangle}$ can increase with increasing decay rate of the other state $|i\ensuremath{\ne}j\ensuremath{\rangle}$. This surprising result occurs because the decay results in crossing of the two eigenvalues of the instantaneous non-Hermitian Hamiltonian. On the other hand, if $|{\ensuremath{\varepsilon}}_{1}(\ifmmode\pm\else\textpm\fi{}T)\ensuremath{-}{\ensuremath{\varepsilon}}_{2}(\ifmmode\pm\else\textpm\fi{}T)|\ensuremath{\rightarrow}\ensuremath{\infty}$ as $T\ensuremath{\rightarrow}\ensuremath{\infty}$, the probability is independent of the decay rate. These results are based on analytic solutions of the time-dependent Schr\"odinger equations for two cases: (a) the energy levels depend linearly on time, and (b) the energy levels are bounded and of the form ${\ensuremath{\varepsilon}}_{1,2}(t)=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varepsilon}tanh(t/\mathcal{T})$. Second, we study LZ transitions affected by dephasing by formulating the Landau-Zener problem with noise in terms of a Schr\"odinger-Langevin stochastic coupled set of differential equations. The LZ survival probability then becomes a random variable whose probability distribution is shown to behave very differently for long and short dephasing times. We also discuss the combined effects of decay and dephasing on the LZ probability.