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Physical Review A

Analytical wave-packet design scheme: Control of dynamics and creation of exotic wave packets

Xin Chen, John A. Yeazell

Published Apr 1, 1998• 9 citations
Publisher

Abstract

A general analytical formalism is described for the design of the excitation pulse (in the weak-field limit) that will excite a specified target wave packet. These analytical formulas (accurate to third order in the phase) provide multiple solutions for a class of target wave packets and provide a basis for choosing between these solutions. In addition, the possibility of independent compensation of the second- and third-order phase terms offers flexibility in the creation of exotic wave packets. A concrete design example is given for the generation of a well-localized Rydberg atomic wave packet at a specific target time.

DOI
10.1103/physreva.57.r2274

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A general analytical formalism is described for the design of the excitation pulse (in the weak-field limit) that will excite a specified target wave packet. These analytical formulas (accurate to third order in the phase) provide multiple solutions for a class of target wave packets and provide a basis for choosing between these solutions. In addition, the possibility of independent compensation of the second- and third-order phase terms offers flexibility in the creation of exotic wave packets. A concrete design example is given for the generation of a well-localized Rydberg atomic wave packet at a specific target time.
Target states are specifically desired states of systems that allow the control of processes or the investigation of unique phenomena. A time-domain scheme for the excitation of such states by the interference of multiple wave packets is developed. The scheme leads to analytical results for the optical field required to excite an arbitrary target state. The scheme suggests a clear physical picture for the design process that is applicable to many systems. The design has strong similarities to the phase-conjugation process in optics. A strong point of the scheme is that it picks a simple set of solutions from the infinite number of possible solutions for a particular target state. This characteristic may be attributed to the choice of a classical wave-packet basis as the appropriate basis for this time-domain scheme. We show that a small number of pulses are sufficient to produce a wide range of target states (both classical and nonclassical target states are designed).
In this article, we describe an analytic approach to the quantum control of wave-packet dynamics. In particular, we show that the field that will produce a specified wave packet in a Rydberg atom can be found approximately from a simple analytic formula. This analytic solution is shown to hold even in the regime of strong excitation, and is found by limiting the duration of the pulsed control field to less than one Kepler period of the system. Comparison with a direct numerical integration of Schr\"odinger's equation shows that our solution yields the target state and population transfer with very high fidelity. The possibility of ionization to the continuum is also included in the model. The solution provides some insight into the multiplicity of solutions to the control problem and the relative complexity of the control-field shapes in the weak- and strong-response regimes.
A simple analytic formula is derived for the impulse that will produce approximately, for arbitrarily large population transfer, a specified wave packet in a Rydberg atom. This solution to the nonlinear control problem is obtained in the limit that the duration of the pulsed control field is less than the Kepler period of the system. The validity of our solution is tested by comparison with a direct numerical integration of Schr\"odinger's equation, and it is found to yield the target state and population transfer with very high fidelity.
Summary form only given. Design and excitation of novel quantum states and the measurement of those states is driven by both fundamental and technological goals. For example, the study of Schrodinger cat states touches upon the basic measurement hypotheses of quantum mechanics while the establishment of particular coherent superposition states is the starting point for the process of coherent control or for quantum information applications. The goals of the design research range from the manipulation of reaction pathways to the creation of novel states for the exploration of aspects of measurement theory. In addition to its fundamental interest the accurate measurement of both amplitude and phase of a wave function plays a key roles in the area of coherent control. The seminal work by Risken and Vogel (1989) reopened the modern discussion of the reconstruction problem that was first broached by Pauli. Since their work, reconstruction techniques have been developed for a variety of systems including the optical field in atomic systems. The measurement scheme used in the paper employs a noninterferometric technique. The subject of the measurement is a designed Rydberg wave packet in an atomic system. However, the noninterferometric character of the technique suggests that this technique may have wide applicability.
A wave-packet interference technique is proposed to measure exotic atomic Rydberg wave functions. For example, the subject wave packet is excited by an arbitrary pulse or pulse sequence. The resulting wave function is probed by a well-characterized pulse that shares a phase relationship with the excitation pulse(s). The observable is the population in each eigenstate after the probe pulse. By making a series of such measurements, varying the phase between excitation and probe pulses, the wave function created by the excitation pulse can be inferred. The method is experimentally feasible for atomic systems, and the extension of the method to molecular systems also appears possible.
We derive a simple approximate analytic solution for the driving field that will generate a desired target vibrational wavepacket in a diatomic molecule. This solution is valid for arbitrary population transfer from the initial to the final electronic states. Analytic solutions to the control problem are possible when the control field is of shorter duration than the characteristic time scale of the system being manipulated. For target states that are localized wavepackets, our technique yields multiple solutions for the control field. We describe a procedure to choose from the set of solutions one that achieves both the target state and population transfer with high accuracy, for a large class of target states. The fidelity of each solution is tested by comparing the state obtained from a direct numerical integration of Schrödinger's equation using the analytic solution with the specified target state.