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arXiv:quant-ph/0208044

Arbitrary qubit rotation through nonadiabatic evolution

Asoka Biswas, G. S. Agarwal

Published Aug 7, 2002• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0208044

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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.
Abstract Quantum gates can be implemented adiabatically and nonadiabatically. Many schemes used at least two sequentially implemented gates to obtain an arbitrary one-qubit gate. Recently, it has been shown that nonadiabatic gates can be realized by single-shot implementation. It has also been shown that quantum gates can be implemented with controlled adiabatic evolutions. In this paper, we combine the advantage of single-shot implementation with controlled adiabatic evolutions to obtain controlled nonadiabatic evolutions. We also investigate the robustness to different types of errors. We find that the fidelity is close to unity for realistic decoherence rates.
Arbitrary rotation of a qubit can be performed with a three-pulse sequence, for example, ZYZZYZZYZ rotations. However, this requires precise control of the relative phase and timing between the pulses, making it technically challenging in optical implementation in a short time scale. Here we show any ZYZZYZZYZ rotations can be implemented with a single laser pulse, that is, a chirped pulse with a temporal hole. The hole of this shaped pulse induces a nonadiabatic interaction in the middle of the adiabatic evolution of the chirped pulse, converting the central part of an otherwise simple ZZZ rotation to a YYY rotation, constructing ZYZZYZZYZ rotations. The result of our experiment performed with shaped femtosecond laser pulses and cold rubidium atoms shows strong agreement with the theory.
The most common method to implement quantum gates is based on the Rabi solution regime of a two-level system excited by external field. There is a very simple reason for this choice. To construct a quantum gate one needs to know the exact form of evolution operator of the qubit under the external excitation. In the case of Rabi solution the qubit evolution has very simple and easily interpreted form. In essence, the whole dynamics of the qubit wave function is governed by the pulse area and it exhibits Rabi oscillations for the time less than the decoherence time. Various implementations of the quantum gates employ relatively weak pulses that results in slow gate operations of order microseconds. To make quantum gates faster one may use strong femtosecond pulses. However, direct implementation schemes are not so obvious because of strong field effects and large bandwidth of the pulses might give rise to unwanted excitations. Here we propose a simple scheme which provides a possibility to utilize strong pulses while keep all advantages of Rabi solution regime. We design fast gates (picosecond time scale) by choosing proper parameters of the chirped pulses as a way to control nonadiabatic terms and to satisfy the adiabaticity conditions. Proposed Hadamard and phase-shift gate allow us to construct universal set of single qubit gates by controlling the effective pulse area and two-photon detuning. The proposed excitation scheme can be used to implement CNOT gate as well.
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.
Nonadiabatic holonomic quantum computation has received increasing attention due to its robustness against control errors. However, all the previous schemes have to use at least two sequentially implemented gates to realize a general one-qubit gate. In this paper we put forward a scheme by which one can directly realize an arbitrary holonomic one-qubit gate with a single-shot implementation, avoiding the extra work of combining two gates into one. Based on a three-level model driven by laser pulses, we show that any single-qubit holonomic gate can be realized by varying the detuning, amplitude, and phase of lasers. Our scheme is compatible with previously proposed nonadiabatic holonomic two-qubit gates, combining with which the arbitrary holonomic one-qubit gates can play universal nonadiabatic holonomic quantum computation. We also investigate the effects of some unavoidable realistic errors on our scheme.
A new scheme of controlling molecular processes by a sequence of linearly chirped pulses is proposed and is applied to selective excitation of an energy level among closely lying ones and to complete electronic excitation of a diatomic molecule. The basic idea is quite different from the conventional ones utilizing chirped pulses in the sense that the present one does not rely on the idea of adiabatic rapid passage at all, but tries to control basic nonadiabatic transitions explicitly. Control of molecular processes can be achieved by controlling nonadiabatic transitions among Floquet (or dressed) states with use of the interference effects. The scheme can be formulated with use of the analytical theories of nonadiabatic transitions, and the proper control parameters can be estimated theoretically. Numerical demonstrations are provided to confirm the robustness of the method in comparison with the other conventional ones. Namely, the present scheme is shown to be stable against the variation of pulse area, complete and selective in population transfer, and fast to accomplish the transition. It is expected that the method can be applied to general multilevel systems and various types of wave packet dynamics. Its experimental realizability can also be expected, since linear chirping can now be relatively easily realized and manipulated.