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arXiv:1712.06998

Optimal control theory for rapid-adiabatic passage techniques in inhomogeneous external fields

E. J. Zak

Published Dec 19, 2017• 2 citations
arXivHTMLPDF

Abstract

The present paper reports on results of quantum dynamics calculations for Stark-chirp rapid-adiabatic passage (SCRAP) in two-level systems with electric fields computed with the optimal control theory. The Pontryagin maximum principle is used to determine the robust optimal control fields in the presence of time-varying and spatially-inhomogeneous perturbing electric fields. The concept of a non-adiabatic correction to the Bloch vector dynamics is introduced and discussed. The existence of a non-zero geometric phase is proved for certain adiabatic paths, which correspond to the complete population return in the rapid-adiabatic passage. A connection is shown between the geometric phase and a measure of the non-adiabatic effects in the time evolution of the state vector during SCRAP. Different cost functionals used in the optimal control scheme are shown to correlate with different topologies of the paths followed by the parameters of the Hamiltonian, which tightly relates to the values of the geometric phase acquired by the adiabatic wavefunction.

DOI
10.48550/arxiv.1712.06998

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The present paper reports on results of quantum dynamics calculations for Stark-chirp rapid-adiabatic passage (SCRAP) in two-level systems with electric fields computed with the optimal control theory. The Pontryagin maximum principle is used to determine the robust optimal control fields in the presence of time-varying and spatially-inhomogeneous perturbing electric fields. The concept of a non-adiabatic correction to the Bloch vector dynamics is introduced and discussed. The existence of a non-zero geometric phase is proved for certain adiabatic paths, which correspond to the complete population return in the rapid-adiabatic passage. A connection is shown between the geometric phase and a measure of the non-adiabatic effects in the time evolution of the state vector during SCRAP. Different cost functionals used in the optimal control scheme are shown to correlate with different topologies of the paths followed by the parameters of the Hamiltonian, which tightly relates to the values of the geometric phase acquired by the adiabatic wavefunction.
Inhomogeneous broadening of energy levels is one of the principal limiting factors for achieving ``slow'' or ``stationary'' light in solid-state media by means of electromagnetically induced transparency, a quantum version of stimulated Raman adiabatic passage (STIRAP). Stark-shift-chirped rapid adiabatic-passage (SCRAP) has been shown to be far less sensitive to inhomogeneous broadening than STIRAP, a population transfer technique to which it is closely related. We further optimize the pulses used in SCRAP to be even less sensitive to inhomogeneous broadening in a $\ensuremath{\Lambda}$-type three-level system. The optimized pulses perform at a higher fidelity than the standard Gaussian pulses for a wide range of detunings (i.e., large inhomogeneous broadening).
Stark-chirped rapid adiabatic passage (SCRAP) is an important technique used for coherent quantum controls. In this paper we investigate how the practically-existing dissipation of the system influences on the efficiency of the passage, and thus the fidelities of the SCRAP-based quantum gates. With flux-biased Josephson qubits as a specifical example, our results show clearly that the efficiency of the logic gates implemented by SCRAP are robust against the weak dissipation. The influence due to the non-adiabtic transitions between the adiabatic passages is comparatively significantly small. Therefore, the SCRAP-based logic gates should be feasible for the realistic physical systems with noises.
We show that the technique of Stark-chirped rapid adiabatic passage (SCRAP), hitherto used for complete population transfer between two quantum states, offers a simple and robust method for complete population transfer amongst three states in atoms and molecules. In this case SCRAP uses three laser pulses: a strong far-off-resonant pulse modifies the transition frequencies by inducing dynamic Stark shifts and thereby creating time-dependent level crossings amongst the three diabatic states, while near-resonant and moderately strong pump and Stokes pulses, appropriately offset in time, drive the population between the initial and final states via adiabatic passage. The population transfer efficiency is robust to variations in the intensities of the lasers, as long as these intensities are sufficiently large to enforce adiabatic evolution. With suitable pulse timings the population in the (possibly decaying) intermediate state can be minimized, as with stimulated Raman adiabatic passage (STIRAP). This technique applies to one-photon as well as multiphoton transitions and it is also applicable to media exhibiting inhomogeneous broadening; these features represent clear advantages over STIRAP by overcoming the inevitable dynamical Stark shifts that accompany multiphoton transitions as well as unwanted detunings, e.g., induced by Doppler shifts.
This work explores the relationship between optimal control theory and adiabatic passage techniques in quantum systems. The study is based on a geometric analysis of the Hamiltonian dynamics constructed from Pontryagin's maximum principle. In a three-level quantum system, we show that the stimulated Raman adiabatic passage technique can be associated to a peculiar Hamiltonian singularity. One deduces that the adiabatic pulse is solution of the optimal control problem only for a specific cost functional. This analysis is extended to the case of a four-level quantum system.
We propose a modified Stark-chirped rapid adiabatic passage technique for a robust rovibrational population transfer in the gas phase molecules in the presence of certain inhomogeneous electric fields. As an example application, the new state switching scheme is shown to greatly enhance the process of slowing polar ammonia molecules in the Stark decelerator. High-level quantum mechanical simulations show that a virtually complete population inversion between a selected pair of weak-field and strong-field seeking states of NH3_33​ can be achieved. Strong dc electric fields within the Stark decelerator are used as part of the rovibrational population transfer protocol. Classical-dynamics simulations for ammonia demonstrate notable improvements in the longitudinal phase space acceptance of the Stark decelerator upgraded with the state switching and an increased deceleration efficiency with respect to the standard Stark deceleration technique.
We generalize Stark-chirped rapid adiabatic passage (SCRAP) to the case of two and three degenerate-level manifolds. The analysis of a degenerate-level system is facilitated by its subdivision into a set of smaller independently evolving subsystems corresponding to the minimal-sized invariant subspaces of the Hamiltonian. Population transfer from the starting to the final level is examined for different types of the invariant subspaces depending on the presence of dark states. It is shown that the complete transfer is feasible if the initial state is prepared into specific coherent superpositions. Our formalism is applicable to the general case of arbitrary numbers of degenerate states within each level and arbitrary couplings of the appropriate transitions. It represents a generalization of well-known Morris-Shore transformation to the case when the removed degeneracy of the sublevels leads to detuning from two-photon resonance. We give the application of the SCRAP formalism to the 87Rb^{87}\text{R}\text{b}87Rb atom.