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Physical Review A

Adiabatic generation of arbitrary coherent superpositions of two quantum states: Exact and approximate solutions

Kaloyan N. Zlatanov, Nikolay V. Vitanov

Published Jul 17, 2017• 13 citations
Publisher

Abstract

The common objective of the application of adiabatic techniques in the field of quantum control is to transfer a quantum system from one discrete energy state to another. These techniques feature both high efficiency and insensitivity to variations in the experimental parameters, e.g., variations in the driving field amplitude, duration, frequency, and shape, as well as fluctuations in the environment. Here we explore the potential of adiabatic techniques for creating arbitrary predefined coherent superpositions of two quantum states. We show that an equally weighted coherent superposition can be created by temporal variation of the ratio between the Rabi frequency $\mathrm{\ensuremath{\Omega}}(t)$ and the detuning $\mathrm{\ensuremath{\Delta}}(t)$ from 0 to $\ensuremath{\infty}$ (case 1) or vice versa (case 2), as it is readily deduced from the explicit adiabatic solution for the Bloch vector. We infer important differences between cases 1 and 2 in the composition of the created coherent superposition: The latter depends on the dynamical phase of the process in case 2, while it does not depend on this phase in case 1. Furthermore, an arbitrary coherent superposition of unequal weights can be created by using asymptotic ratios of $\mathrm{\ensuremath{\Omega}}(t)/\mathrm{\ensuremath{\Delta}}(t)$ different from 0 and $\ensuremath{\infty}$. We supplement the general adiabatic solution with analytic solutions for three exactly soluble models: two trigonometric models and the hyperbolic Demkov-Kunike model. They allow us not only to demonstrate the general predictions in specific cases but also to derive the nonadiabatic corrections to the adiabatic solutions.

DOI
10.1103/physreva.96.013415

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Phase shifts in nonresonant coherent excitation

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Far-off-resonant pulsed laser fields produce negligible excitation between two atomic states but may induce considerable phase shifts. The acquired phases are usually calculated by using the adiabatic-elimination approximation. We analyze the accuracy of this approximation and derive the conditions for its applicability to the calculation of the phases. We account for various sources of imperfections, ranging from higher terms in the adiabatic-elimination expansion and irreversible population loss to couplings to additional states. We find that as far as the phase shifts are concerned, the adiabatic elimination is accurate only for a very large detuning. We show that the adiabatic approximation is a far more accurate method for evaluating the phase shifts, with a vast domain of validity; the accuracy is further enhanced by superadiabatic corrections, which reduce the error well below ${10}^{\ensuremath{-}4}$. Moreover, owing to the effect of adiabatic population return, the adiabatic and superadiabatic approximations allow one to calculate the phase shifts even for a moderately large detuning and even when the peak Rabi frequency is larger than the detuning; in these regimes the adiabatic elimination is completely inapplicable. We also derive several exact expressions for the phases using exactly soluble two-state and three-state analytical models.

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The common objective of the application of adiabatic techniques in the field of quantum control is to transfer a quantum system from one discrete energy state to another. These techniques feature both high efficiency and insensitivity to variations in the experimental parameters, e.g., variations in the driving field amplitude, duration, frequency, and shape, as well as fluctuations in the environment. Here we explore the potential of adiabatic techniques for creating arbitrary predefined coherent superpositions of two quantum states. We show that an equally weighted coherent superposition can be created by temporal variation of the ratio between the Rabi frequency \ensuremathΩ(t)\mathrm{\ensuremath{\Omega}}(t)\ensuremathΩ(t) and the detuning \ensuremathΔ(t)\mathrm{\ensuremath{\Delta}}(t)\ensuremathΔ(t) from 0 to \ensuremath∞\ensuremath{\infty}\ensuremath∞ (case 1) or vice versa (case 2), as it is readily deduced from the explicit adiabatic solution for the Bloch vector. We infer important differences between cases 1 and 2 in the composition of the created coherent superposition: The latter depends on the dynamical phase of the process in case 2, while it does not depend on this phase in case 1. Furthermore, an arbitrary coherent superposition of unequal weights can be created by using asymptotic ratios of \ensuremathΩ(t)/\ensuremathΔ(t)\mathrm{\ensuremath{\Omega}}(t)/\mathrm{\ensuremath{\Delta}}(t)\ensuremathΩ(t)/\ensuremathΔ(t) different from 0 and \ensuremath∞\ensuremath{\infty}\ensuremath∞. We supplement the general adiabatic solution with analytic solutions for three exactly soluble models: two trigonometric models and the hyperbolic Demkov-Kunike model. They allow us not only to demonstrate the general predictions in specific cases but also to derive the nonadiabatic corrections to the adiabatic solutions.
We propose a technique for the accurate, flexible, and robust generation of arbitrary preselected coherent superpositions of two quantum states. It uses a sequence of two adiabatic pulses split by a phase jump serving as a control parameter. Each pulse has a chirped detuning, which induces a half crossing, and acts approximately as a half-$\ensuremath{\pi}$ pulse in the adiabatic regime. The phase jump is imprinted onto the population ratio of the created superposition state. Of the various possible relations between the two pulses, we select the case when the Rabi frequency and the detuning of the second pulse are mirror images of those of the first pulse, and the two detunings have opposite signs. Then the mixing angle of the created superposition state depends on the phase jump only. For other arrangements, the superposition mixing angle is shifted by the dynamic phases of the propagators, which makes these cases suitable for state tomography. This sandwich setup comes along with the advantage that it reduces the error $\ensuremath{\epsilon}$ of each individual pulse down to $4{\ensuremath{\epsilon}}^{2}$ overall. Therefore, the proposed technique combines the benefits of robustness stemming from adiabatic evolution with accuracy generated by the twin-pulse error suppression, and flexibility of the created superposition state controlled by the value of the phase jump $\ensuremath{\phi}$. In addition to the general analysis, we present a simple exactly soluble trigonometric model in order to illustrate the proposed technique. In this model, when the pulse area AAA increases, the nonadiabatic oscillations are damped as ${A}^{\ensuremath{-}1}$ for a single pulse and ${A}^{\ensuremath{-}2}$ for the two-pulse sequence. Finally, the proposed technique is iteratively extended to sequences of N=2nN={2}^{n}N=2n pulses by concatenating half-$\ensuremath{\pi}$ sequences and splitting them by a phase jump, thereby further reducing the nonadiabatic error $\ensuremath{\epsilon}$ to ${(2\ensuremath{\epsilon})}^{N}$. This makes the proposed technique suitable for generating high-fidelity quantum rotation gates even when starting with errant pulses.
Abstract The generation of arbitrary preselected coherent superposition of states in multi-state systems using adiabatic pulses split by a phase jump has been investigated. We consider a linkage pattern that includes a ground state and N excited states so that the ground state is coupled to the excited states using N pulses with a chirped detuning. In this scheme, time-dependent Rabi frequencies and detuning are used in two steps, and in each step, it is assumed that the Rabi frequencies have the same time dependence. Also, the Rabi frequencies in the second step have a phase jump compared to the first step, and the amount of this phase jump is the same for all pulses. It is shown that by properly adjusting the amount of phase jump in the second step and also the maximum value of Rabi frequencies in both steps, an arbitrary coherent superposition of all states can be created. In this study, the maximum values of Rabi frequencies and detuning are considered to be large enough to establish the adiabatic passage condition, and it is numerically shown that this scheme is insensitive to the change of the maximum value of Rabi frequencies and detuning.
We discuss a simple scheme for preparing atoms and molecules in an arbitrary preselected coherent superposition of quantum states. The technique, which we call fractional stimulated Raman adiabatic passage ({\it f-STIRAP}), is based upon (incomplete) adiabatic population transfer between an initial state ψ1ψ_1ψ1​ and state ψ3ψ_3ψ3​ through an intermediate state ψ2ψ_2ψ2​. As in STIRAP, the Stokes pulse arrives before the pump pulse, but unlike STIRAP, the two pulses terminate simultaneously while maintaining a constant ratio of amplitudes. The independence of f-STIRAP from details of pulse shape and pulse area makes it the analog of conventional STIRAP in the creation of coherent superpositions of states. We suggest a smooth realization of f-STIRAP which requires only two laser pulses (which can be derived from a single laser) and at the same time ensures the automatic fulfillment of the asymptotic conditions at early and late times. Furthermore, we provide simple analytic estimates of the robustness of f-STIRAP against variations in the pulse intensity, the pulse delay, and the intermediate-state detuning, and discuss its possible extension to multistate systems.
A technique for creation of well-defined preselected coherent superpositions of multiple quantum states is proposed. It is based on an extension of the technique of stimulated Raman adiabatic passage (STIRAP) to degenerate levels. As an example, the nine-state system composed of the magnetic sublevels of three levels with angular momenta Jg=0{J}_{g}=0Jg​=0, Je=1{J}_{e}=1Je​=1, and Jf=2{J}_{f}=2Jf​=2 is studied in detail. Starting from the $\ensuremath{\mid}{J}_{g}=0,{M}_{g}=0⟩$ state, STIRAP can create an arbitrary preselected coherent superposition between the five Mf{M}_{f}Mf​ sublevels $({M}_{f}=\ensuremath{-}2,\ensuremath{-}1,0,+1,+2)$ of the Jf=2{J}_{f}=2Jf​=2 level with 100% efficiency in the adiabatic limit. The populations and the phases of the Mf{M}_{f}Mf​ states in this superposition are determined entirely by the polarizations of the two laser fields. It is shown that this technique allows one to create any superposition of the five Mf{M}_{f}Mf​ states, that is, to reach any point in the Hilbert space of the Jf=2{J}_{f}=2Jf​=2 manifold, and the corresponding recipes for choosing the laser polarizations are presented.
Coherent manipulations of atoms using laser light The internal structure of a particle - an atom or other quantum system in which the excitation energies are discrete - undergoes change when exposed to pulses of near-resonant laser light. This tutorial review presents basic concepts of quantum states, of laser radiation and of the Hilbert-space statevector that provides the theoretical portrait of probability amplitudes - the tools for quantifying quantum properties not only of individual atoms and molecules but also of artificial atoms and other quantum systems. It discusses the equations of motion that describe the laser-induced changes (coherent excitation), and gives examples of laser-pulse effects, with particular emphasis on two-state and three-state adiabatic time evolution within the rotating-wave approximation. It provides pictorial descriptions of excitation based on the Bloch equations that allow visualization of two-state excitation as motion of a three-dimensional vector (the Bloch vector). Other visualization techniques allow portrayal of more elaborate systems, particularly the Hilbert-space motion of adiabatic states subject to various pulse sequences. Various more general multilevel systems receive treatment that includes degeneracies, chains and loop linkages. The concluding sections discuss techniques for creating arbitrary pre-assigned quantum states, for manipulating them into alternative coherent superpositions and for analyzing an unknown superposition. Appendices review some basic mathematical concepts and provide further details of the theoretical formalism, including photons, pulse propagation, statistical averages, analytic solutions to the equations of motion, exact solutions of periodic Hamiltonians, and population-trapping "dark" states.
Far-off-resonant pulsed laser fields produce negligible excitation between two atomic states but may induce considerable phase shifts. The acquired phases are usually calculated by using the adiabatic-elimination approximation. We analyze the accuracy of this approximation and derive the conditions for its applicability to the calculation of the phases. We account for various sources of imperfections, ranging from higher terms in the adiabatic-elimination expansion and irreversible population loss to couplings to additional states. We find that as far as the phase shifts are concerned, the adiabatic elimination is accurate only for a very large detuning. We show that the adiabatic approximation is a far more accurate method for evaluating the phase shifts, with a vast domain of validity; the accuracy is further enhanced by superadiabatic corrections, which reduce the error well below ${10}^{\ensuremath{-}4}$. Moreover, owing to the effect of adiabatic population return, the adiabatic and superadiabatic approximations allow one to calculate the phase shifts even for a moderately large detuning and even when the peak Rabi frequency is larger than the detuning; in these regimes the adiabatic elimination is completely inapplicable. We also derive several exact expressions for the phases using exactly soluble two-state and three-state analytical models.