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arXiv:1309.3758

Unitary manipulation of a single atom in time and space --- The spatially-selective and internal-state-selective triggering pulses

Xijia Miao

Published Sep 15, 2013• 0 citations
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Abstract

A spatially-selective and internal-state-selective triggering pulse is an important component to realize the unitary dynamical state-locking process which may be used to realize the reversible and unitary halting protocol and construct the unstructured quantum search process based on the tensor-product Hilbert-space symmetric structure and the unitary quantum dynamics. In this paper it is shown how a spatially-selective and internal-state-selective triggering pulse is constructed generally in the quantum system of a single atom. Rigorous theoretical calculation and error estimation, which are based on the Trotter-Suzuki decomposition method and the multiple-Gaussian-wave-packet expansion method, are carried out for the time evolution process of a single atom in the double-well potential field and in the presence of the spatially-selective and internal-state-selective triggering pulse. It proves that the spatially-selective and internal-state-selective triggering pulse is feasible, that is, the possible errors generated by the triggering pulse are shown to be controllable. This rigorous theoretical calculation also shows that the spatially-selective and internal-state-selective triggering pulse may be different from its corresponding ideal internal-state-selective triggering pulse that is not spatially selective, but the difference between their final states of the time evolution process is controllable. These computational results are achieved with the help of the Gaussian wave-packet motional states and the space-dependent quadratic Hamiltonians of a single atom. The methods and techniques developed in the paper are useful for studying the quantum-computing speedup mechanism of the unitary quantum dynamics in the quantum system of a single atom.

DOI
10.48550/arxiv.1309.3758

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A spatially-selective and internal-state-selective triggering pulse is an important component to realize the unitary dynamical state-locking process which may be used to realize the reversible and unitary halting protocol and construct the unstructured quantum search process based on the tensor-product Hilbert-space symmetric structure and the unitary quantum dynamics. In this paper it is shown how a spatially-selective and internal-state-selective triggering pulse is constructed generally in the quantum system of a single atom. Rigorous theoretical calculation and error estimation, which are based on the Trotter-Suzuki decomposition method and the multiple-Gaussian-wave-packet expansion method, are carried out for the time evolution process of a single atom in the double-well potential field and in the presence of the spatially-selective and internal-state-selective triggering pulse. It proves that the spatially-selective and internal-state-selective triggering pulse is feasible, that is, the possible errors generated by the triggering pulse are shown to be controllable. This rigorous theoretical calculation also shows that the spatially-selective and internal-state-selective triggering pulse may be different from its corresponding ideal internal-state-selective triggering pulse that is not spatially selective, but the difference between their final states of the time evolution process is controllable. These computational results are achieved with the help of the Gaussian wave-packet motional states and the space-dependent quadratic Hamiltonians of a single atom. The methods and techniques developed in the paper are useful for studying the quantum-computing speedup mechanism of the unitary quantum dynamics in the quantum system of a single atom.
The paper first discusses theoretically the off-resonance selective excitation method that is dependent on the atomic internal states and used to generate approximately a standard coherent state of harmonic oscillator. The coherent average method then is proposed to construct the state-selective trigger pulse. A state-selective trigger pulse can keep Gaussian shape unchanged but change in an internal-state-dependent form the center-of-mass position and/or momentum of an atomic Gaussian wave-packet motional state. A Gaussian wave-packet state is one of the simplest wave-packet states that can be easily manipulated and controlled in time and space. The paper also investigates how to manipulate in time and space an atomic Gaussian wave-packet motional state by a generalized quadratic potential field. A general quadratic Hamiltonian can affect not only the center-of-mass position and momentum but also the complex linewidth of a Gaussian wave-packet motional state while keep Gaussian shape of the motional state unchanged. It is shown that generally quadratic terms of a quadratic Hamiltonian can control directly the complex linewidth, while linear terms of a quadratic Hamiltonian can affect only the center-of-mass position and momentum of a Gaussian wave-packet motional state.
It has been shown (Arxiv: quant-ph/0507236) that a universal quantum computer could be powerful enough to solve efficiently the quantum search problem, and the reversible and unitary halting protocol based on the state-locking pulse field is the key component to construct the efficient quantum search processes, while the state-locking pulse field is the key component to generate the reversible and unitary halting protocol. In this paper the reversible and unitary halting protocol and the generalized state-locking pulse field have been extensively investigated theoretically. The basic principles to construct the state-locking pulse field and design the reversible and unitary halting protocol are studied in detail. A generalized state-locking pulse field is generally dependent upon the time and space variables. It could be a sequence of time- and space-dependent electromagnetic pulse fields and could also contain the time- and space-dependent potential fields. Thus, the reversible and unitary halting protocol built up out of the state-locking pulse field generally consists of a sequence of time- and space-dependent unitary evolution processes. It is shown how the quantum control process is constructed to simulate efficiently the reversible and unitary halting protocol. An improved subspace-reduction quantum program and circuit based on the reversible and unitary halting protocol is proposed as the key component to construct further an efficient quantum search process. A simple atomic physical system that is an atomic ion or a neutral atom in the double-well potential field is proposed to show how the state-locking pulse field is generated and how to implement the reversible and unitary halting protocol.
The quantum-computing speedup theory considers the symmetric structures and properties of quantum systems as the fundamental Quantum-Computing-Speedup (QCS) resources which are responsible for exponentially speeding up quantum computing and simulating. At present a large and important problem is how to make use of the fundamental QCS resources to speed up essentially quantum computing and simulating. Here the author makes a great effort toward solving this important problem. The theoretical research work in this paper is mainly divided into the two Parts I and II. The Part I investigates mainly the multiple-quantum operator algebra spaces. And the relationships are analyzed among the multiple-quantum operator algebra spaces, quantum simulating for the unitary time-evolutional processes, and the fundamental QCS resources which exist in the different kinds of basic quantum spaces: the multiple-quantum operator algebra spaces, the density operator spaces, and the Hilbert spaces. It concludes that the multiple-quantum operator algebra space must be positioned as the central place where the fundamental QCS resources are exploited to speed up quantum computing and simulating. The Part II investigates mainly the subspace-selective unitary manipulation based on the Hilbert-space symmetric structures. Recognize that the multiple-quantum operator algebra space is the central place. Then those fundamental QCS resources original from the Hilbert space (a quantum-state space) must be explicitly taken into account in the multiple-quantum operator algebra space (a linear operator space). This is an important problem. The subspace-selective unitary manipulation is able to solve this problem. It aims to harness the fundamental QCS resources original from the Hilbert space to speed up quantum computing and simulating in the multiple-quantum operator algebra space.
A STIRAP-based unitary decelerating (accelerating) process consists of a train of the standard three-state STIRAP pulse sequences which may act as the basic unitary decelerating (accelerating) sequences. The present work is focused on investigating analytically and quantitatively how the momentum distribution of a momentum superposition state such as a momentum Gaussian wave-packet state of a single freely moving atom affects the STIRAP state transfer in these decelerating and accelerating processes. The complete STIRAP state transfer and the unitarity of these processes are stressed highly in the investigation. It has been shown that the momentum distribution has an important influence upon the STIRAP state-transfer efficiency. In the ideal adiabatic condition these unitary decelerating and accelerating processes for a freely moving atom are studied in detail, and it is shown that they can be used to manipulate and control in time and space the center-of-mass position and momentum of a Gaussian wave-packet motional state of a free atom. Two general (strict and accurate) adiabatic conditions for the basic STIRAP decelerating and accelerating processes are derived analytically. With the help of the STIRAP theory and the unitary quantum dynamics it confirms theoretically that the time- and space-compressing processes of the quantum control process (quant-ph/0607144) can be realized almost perfectly by the STIRAP-based unitary decelerating and accelerating processes in the ideal or nearly ideal adiabatic condition.
Realizing coherent quantum computation requires precise and robust manipulation of quantum systems through quantum control protocols. Most quantum control techniques rely on heuristic methods for designing the driving pulses that steer the system towards a target state. Such methods are often based on brute-force optimization and offer limited understanding of the solution landscape. In contrast, quantum algorithms offer a rich body of analytical methods with rigorous error guarantees for implementing unitary and non-unitary transformations, which suggests a promising direction for developing new approaches to quantum control. Among various such algorithms, quantum signal processing (QSP) has emerged as a powerful framework for quantum algorithm design, implementation, and optimization. However, its potential for quantum control remains largely unexplored. In this work, we establish QSP-Control, an analytical framework for quantum control of qubit-oscillator dynamics. We focus on dispersively coupled qubit-oscillator systems and employ the QSP formalism to mitigate unwanted nonlinear effects arising from cross-Kerr interactions. In addition, we develop constructions for precise manipulation of Fock states by designing Fock-state-selective operators, based on structural parallels between the Jaynes-Cummings interaction and QSP. These findings demonstrate how several practically relevant problems in quantum control can be mapped to forms amenable to QSP, offering both a systematic design framework and an interpretable perspective on quantum control.
Experimental and theoretical work is presented on the control of wave packet dynamics in atomic and molecular systems. Using sequences of phase-locked optical pulses, the link between optical phase and quantum mechanical phase is explored in the Na atom and Naj dimer, representing a step towards the logical engineering of quantum states in more complicated systems. A novel apparatus, constructed to study and control the vibrational dynamics of vibrational wave packets on the Na ionic potential surface, is described in chapter 3. Theoretical simulations for proposed experiment are presented in chapter 4. Control is achieved by exploiting the phase-evolution of the constituent vibrational quantum states within the wave packet superposition. The phase relationship and the accumulated phase difference between the various components of the wave packet is determined, and a sequence of phase-locked optical pulses is employed to selectively enhance or depopulate specific vibrational states, or sets of vibrational states. The quantum state composition of the resulting wave packet, and the efficiency of the control scheme, is determined by calculating the multi-pulse response of the time-dependent vibrational state populations. In chapter 5, the quantum interference between Rydberg electron wave packets in the Na atom is investigated using pairs of phase-locked wave packets, allowing manipulation of the total orbital angular momentum of Na Rydberg atoms. Initially the wave packet is composed of a superposition of s and d Rydberg series. Exploitation of the difference between the quantum defects of the two series allows specific angular momentum compositions within the resultant wave packet to be engineered. Experimentally, this final quantum state distribution is analysed in the frequency domain using state selective field ionisation, and in the time domain using the optical Ramsey method. The theoretical calculations show how the phase difference between pairs of optical pulses is linked to the corresponding Rydberg frequency spectrum, therefore enabling the control of the quantum state composition of the wave packets.