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arXiv:1909.07769 · Physical Review A

Generation of arbitrary qubit states by adiabatic evolution split by a phase jump

Kaloyan N. Zlatanov, Nikolay V. Vitanov

Published Jan 17, 2020• 15 citations
arXivHTMLPDF

Abstract

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 $A$ 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 $N={2}^{n}$ 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.

DOI
10.1103/physreva.101.013426

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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-\ensuremathπ\ensuremath{\pi}\ensuremathπ 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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ of each individual pulse down to 4\ensuremathϵ24{\ensuremath{\epsilon}}^{2}4\ensuremathϵ2 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 \ensuremathϕ\ensuremath{\phi}\ensuremathϕ. 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{A}^{\ensuremath{-}1}A\ensuremath−1 for a single pulse and A\ensuremath−2{A}^{\ensuremath{-}2}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-\ensuremathπ\ensuremath{\pi}\ensuremathπ sequences and splitting them by a phase jump, thereby further reducing the nonadiabatic error \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ to (2\ensuremathϵ)N{(2\ensuremath{\epsilon})}^{N}(2\ensuremathϵ)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.
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.
We show how one can perform arbitrary rotation of any qubit, using delayed laser pulses through nonadiabatic evolution, i.e., via transitions among the adiabatic states. We use a double-Lambda scheme and use a set of control parameters such as detuning, ratio of pulse amplitudes, time-separation of two pulses for realizing different rotations of the qubit. We also investigate the effect of different kinds of chirping, namely linear chirping and hyperbolic tangent chirping. Our work using nonadiabatic evolution adds to the flexibility in the implementation of logic gate operations and show how to achieve control of quantum systems by using different types of pulses.
We demonstrate a method of adiabatic population transfer from a single quantum state into a coherent superposition of states. The transfer is executed with femtosecond pulses, spectrally shaped in a simple and intuitive manner, which does not require iterative feedback-controlled loops. In contrast to nonadiabatic methods of excitation, our approach is not sensitive to the exact value of laser intensity. We show that the population transfer is complete, and analyze the possibility of controlling the relative phases and amplitudes of the excited eigenstates. We discuss the limitations of the proposed control methods due to the dynamic level shifts and suggest ways of reducing their influence.
We propose a controlled-Z gate scheme for Rydberg atoms based on level-crossing-free echoing rapid adiabatic population transfer. We design antisymmetric Rabi frequency pulses and symmetric detuning pulses, enabling the system to completely avoid level-crossing points throughout the evolution, and the dynamical phase is naturally eliminated by the time-reversal symmetry of the double-pulse sequence. We incorporate dissipative effects through the Lindblad master equation. The numerical simulation yields a two-qubit CZ gate fidelity of 0.9999. When the Rabi-frequency fluctuation is within ±2%\pm 2\%±2%, and the detuning offset is within ±1%\pm 1\%±1%, the fidelity can still remain above 0.999. Under the same dissipative model, the three-qubit CCZ gate achieves a fidelity of 0.999. When a single-parameter fluctuation does not exceed ±3%\pm 3\%±3%, the fidelity is always higher than 0.997. Our scheme requires no laser phase jumps or fast switching operations. The zero-area pulse structure suppresses first-order intensity noise, and the symmetric double-pulse sequence avoids spatially resolved laser switching, making it suitable for parallel gate operations in large-scale neutral-atom arrays.
We describe a technique for creating superpositions of degenerate quantum states, such as are needed for beam splitters used in matter-wave optics, by manipulating the timing of three orthogonally polarized laser beams through which moving atoms (or molecules) pass; motion across the laser beams produces pulses in the atomic rest frame. As illustrated with representative simulations for transitions in metastable neon, a single pass through three overlapping laser beams can produce superpositions (with preselected phase) of atomic beams differing by transverse momentum corresponding to the momentum of four photons. Like the two-photon momentum transfer of the tripod linkage pattern which it extends, the method relies on controlled adiabatic time evolution in the Hilbert subspace of two degenerate dark states. It is thus a generalization to multiple dark states (and larger transfers of linear momentum to the atomic beam) of the single dark state occurring with the stimulated Raman adiabatic passage (STIRAP) technique, and therefore it is potentially insensitive to decoherence due to spontaneous emission. By extending the tripod-linkage system to more numerous degenerate states, the technique not only increases the atomic beam deflections but, as we demonstrate, allows control over the superposition phase and amplitudes. LIke other techniques based on adiabatic time evolution, the technique is robust with respect to variations of the intensity, timing, and other characteristics of the laser fields. Unlike STIRAP, the same robust partial population transfer occurs for opposite timings of the pulse sequence, as is needed for such procedures as Hadamard gates.