Molecular spin aqc and ensemble spin manipulation technology

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The series Lecture Notes in Physics (LNP), founded in 1969, reports new developments in physics research and teaching-quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research and to serve three purposes:

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Kouichi Semba
National Institute of Information and Communications Technology Koganei, Tokyo, Japan

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Part II: Quantum Metrology and Sensing is composed of four chapters. Chapter introduces an optical lattice clock operated at the so-called magic wavelength, in which an atomic ensemble trapped in an optical lattice provides a precisely

v

the coherent injection of feedback optical pulses back into the main cavity. The numerical benchmark study for the NP-hard MAX-CUT problems in sparse G-set graphs and dense complete graphs features substantial speedup against the conventional approximation algorithms based on semi-definite programming (SDP) and simulated annealing (SA).

Part IV: Quantum Simulation is composed of five chapters. Chapter reviews the basics of Bose-Einstein condensation (BEC) and introduces two different experimental systems, which provide unique platforms for quantum simulation experiments. A dilute atomic BEC is suitable for implementing an equilibrium many-body system which is well isolated from reservoirs. A dense exciton-polariton BEC is suitable for studying a nonequilibrium many-body system in open-dissipative environments. The chapter covers such topics as the fundamental concepts of BEC, the Bogoliubov theory of interacting particles, superfluidity, and various lattice implementing techniques for atomic and polaritonic BEC. Chapter discusses the recent progress of quantum simulation experiments with ultracold ytterbium (Yb) atoms in optical lattices at Kyoto University. They describe experimental results pertaining to a strongly interacting Bose-Fermi Yb mixture, an SU(N) Mott insulator, an artificial impurity system with Yb-Li (lithium) atomic mixture, flat bands and Dirac cones in nonstandard optical lattices, and optical Feshbach resonance. Chapter reports the recent quantum simulation experiments with trapped CaCions at Osaka University. They implement the Jaynes-Cummings-Hubbard (JCH) model, which describes an array of coupled cavities with a single two-level atom and thus features strongly correlated phenomena such as quantum phase transition from Mott insulator to superfluid. Chapter discusses the quantum simulation experiments with exciton-polaritons in various two-dimensional lattice structures at Stanford University. In particular, they report the observation of the spontaneously formed high-orbital (p-wave, d-wave, and f-wave) condensation due to the open-dissipative nature of exciton-polaritons. Finally, Chap. presents the theoretical framework for describing the equilibrium BEC, a high-density BCS to nonequilibrium lasing crossover in driven-dissipative semiconductor systems. The gap equation in the BCS theory (GE-BCS) and the Maxwell semiconductor Bloch equation (MSBE) are formulated for the coupled electron-hole-photon system, and their mutual connection is established.

Part VII: Semiconductor and Molecular Spin Qubits is composed of four chap-ters. Chapter reports the recent progress in semiconductor quantum dot spin qubit technologies. The chapter discusses the experimental schemes of initializing and detecting single qubit gates on the basis of electric dipole-induced spin resonance and implementing two qubit gates on the basis of the exchange coupling between nearby quantum dots. Chapter describes the quantum dynamics of nuclear spins and electron spins associated with an impurity in silicon crystal. In particular, they review the recent nuclear magnetic resonance (NMR)/electron spin resonance (ESR) double resonance experiments using a phosphorus (31P) donor impurity in isotope-purified silicon crystal. Chapter introduces nuclear spins and electron spins in molecules, which have relatively long decoherence time and can be controlled by magnetic resonance techniques. The chapter reports on a newly developed hyperpolarization technique, spin amplification, and an arbitrary waveform pulsed ESR. Finally, Chap. discusses the synthetic approaches to scalable molecular spin-based quantum information processing. The chapter proposes using nuclear spins in the topological network of molecular frames as client qubits and delocalized electron spins as bus qubits, which are simultaneously controlled by RF and microwave pulse techniques.

We hope the research results presented in this book will be a useful source of ideas and knowledge for the future development of quantum information technologies, which are actively being investigated in numerous countries around the world.

Masato Koashi

Masahide Sasaki, Mikio Fujiwara, and Masahiro Takeoka

5 Optical Lattice Clocks for Precision Time and Frequency
Metrology.................................................................... 93 Masao Takamoto and Hidetoshi Katori

6 Cold Atom Magnetometers ................................................ 111 Yujiro Eto, Mark Sadrove, and Takuya Hirano

ix

x Contents

Part IV Quantum Simulation

13 Bose-Einstein Condensation: A Platform for Quantum
Simulation Experiments ................................................... 265 Yoshihisa Yamamoto and Yoshiro Takahashi

Part V Quantum Computing

18 Layered Architectures for Quantum Computers and
Quantum Repeaters ........................................................ 387 Nathan C. Jones

21 Microwave Photonics on a Chip: Superconducting
Circuits as Artificial Atoms for Quantum Information Processing ... 461 Franco Nori and J.Q. You

22 Achievements and Outlook of Research on Quantum
Information Systems Using Superconducting Quantum Circuits ..... 477 Jaw-Shen Tsai

26 Silicon Quantum Information Processing................................ 569 Takeharu Sekiguchi and Kohei M. Itoh

27 Quantum Information Processing Experiments Using
Nuclear and Electron Spins in Molecules ................................ 587 Masahiro Kitagawa, Yasushi Morita, Akinori Kagawa,
and Makoto Negoro

Quantum Information Theory for Quantum

Communication

Rule 1 A physical system is associated with a Hilbert space H . Every pure state of this system is represented by a normalized vector ji 2 H . For any normalized vector ji 2 H , it is possible to prepare the system in the state represented by ji.

dim H of the Hilbert space is finite.2A physical system with a Hilbert space of To avoid complications, we assume in this section that the dimension d D

© Springer Japan 2016 3 Y. Yamamoto, K. Semba (eds.), Principles and Methods of Quantum Information
Technologies, Lecture Notes in Physics 911, DOI 10.1007/978-4-431-55756-2_1

4 M. Koashi

In this rule, we are not interested in the state of the measured system after the measurement is performed. The two rules above only refer to the feasibility of the limited sets of transformations and measurements. In general, a much wider variety of operations should be available on a physical system, and we will see the whole landscape of these operations in Sect. 1.4.

1 Quantum Information Theory for Quantum Communication 5

jiA ˝ jiB 2 HAB. The state that can be written in this form is called a product state, and is often abbreviated as jiAjiB or even jiAB. The unitary transformationsOUA on system A andOVB on B result in the unitary transformation OUA ˝ OVB on the composite system AB. Performing an orthogonal measurement with basis fjuiiAgiD1;:::;d on system A and another with basis fjvjiBgjD1;:::;d0 on system B can be regarded as the performance of a single orthogonal measurement, where the outcome is represented by two numbers .i; j/, carried out on the composite system AB with the orthonormal basis fjuiiA ˝ jvjiBgiD1;:::;d According to Rule 1, we should be able to prepare a state represented by any jD1;:::;d0 of HAB.

1.2 Density Operators

In classical mechanics, a mixed state is simply regarded as a way to formulate an observer’s lack of knowledge of the true state of a system. In principle, it is always possible to assume that there is an omnipotent observer who knows the exact state (the pure state) of every system. In quantum mechanics, however, this simple picture does not hold. When a composite system is in a pure state jiAB, we cannot associate the state of the subsystem A with a single vector jiA 2 HA unless jiAB is a product state. Therefore, it is not always possible to assume that every system is in a pure state at the same time. In this subsection, we determine how we can represent the state of a subsystem when it is a part of a composite

Our strategy is to observe what happens if we perform a measurement with arbitrary basis fjuiiAgiD1;:::;d on system A. Regardless of the temporal order of the measurements on A and B, Rules 3 and 4 dictate that the joint probability of the two outcomes .i; j/ is given by pi;j D j.Ahuij ˝ Bhvjj/jiABj2. Let us introduce the unnormalized vector j QjiA WD BhvjjjiAB 2 HA. We then have pi;j D jAhuij QjiAj2 and pj DPd jjiA WD j QjiA=ppj, we obtain an expression for the conditional probability, pijj WD pi;j=pj D jAhuijjiAj2. Because the choice of the basis fjuiiAgiD1;:::;d was arbitrary, comparison of this relationship to Rule 3 shows that the state of the subsystem A iD1pi;j D Pd iD1jAhuij QjiAj2 D jAh Qjj QjiAj2. Using a normalized vector

conditioned on the outcome j must be a pure state, which is represented by the vector jjiA. Noting that the measurement on A can be performed immediately after the preparation of jiAB, we arrive at the following theorem.

The argument in the previous subsection immediately provides a description of the marginal state of the subsystem A when the composite system AB is prepared in the pure state jiAB. If the value of the outcome j of the measurement on subsystem B

a mixed state by the ensemble is by no means unique. If we change the basis

fjvjiBgjD1;:::;d0 of the measurement to another basis, then the description of the state f.pj; jjiA/gjD1;:::;d0 also changes through Eq. (1.1). This new ensemble should also be a valid representation of the same state.

Consider a physical system that is associated with a Hilbert space H , and let us

operator, i.e.,

When an orthogonal measurement with a basis fjujigj is performed on a mixed state f.qi; jii/gi, the probability of the outcome j is calculated using Rule 3 to be pj DP outcome depend only on the density operator. This also shows that each physical state is associated with a single density operator. Consider two mixed states with iqijhujjiij2 D hujj Ojuji. This shows that the statistics of the measurement

different density operators Oand O0.¤ O/. Because jui 2 H exists with huj. O

First, we consider how a general bipartite pure state jiAB can be written in terms of the orthonormal bases fjuiiAgi and fjvjiBgj for the subsystems A and B. Because fjuiiAjvjiBgi;j is a basis of HAB, it is always possible to decompose jiAB as jiAB DP simpler form of decomposition, jiAB DP fjuiiAgi and fjvjiBgj appropriately for the given vector jiAB. This decomposition is called Schmidt decomposition, and it will be convenient to describe Schmidt i;jci;jjuiiAjvjiB. The special aspect of bipartite states is that a much icijuiiAjviiB, is available if we select

decomposition in the form of the following theorem.

1 Quantum Information Theory for Quantum Communication 9

smaller than dim HA or dim HB, we can always augment the orthonormal sets to The number s is often called the Schmidt number of the state jiAB. If s is

j˚0iAB D .O1A ˝ OVB/j˚iAB: (1.6)

Proof. When we write down a diagonal form OA DPs ensures that the purifications are decomposed as j˚iAB DPs j˚0iAB DPs unitary operatorOVB exists such that jv0 iiB D OVBjviiB for all i. iiBgi are orthonormal sets, a iD1pijuiiAAhuij, Theorem 2

the system AB is in state j˚iAB or in state j˚0iAB. There are then only two possible situations: (i) The marginal density operators of subsystem A are different for j˚iAB and j˚0iAB, and thus Alice can locally distinguish state j˚iAB from state j˚0iAB to some extent. (ii) The marginal density operators of subsystem A are the same

and according to Theorem 3, Bob can switch locally between state j˚iAB and state j˚0iAB. As a result, we see that there is no situation whatsoever in which Alice is unable to distinguish between the two states locally and Bob is unable

the physical states and the density operators. Consider two states represented by the

ensembles f.pj; jjiA/gjD1;:::;d and f.p0 j; j0 jiA/gjD1;:::;d0, which are associated with the same density operator OA. We will show that these two states are in fact the same state [1, 2].

another orthonormal basis fjv0 confirm that the requisite of Lemma 1, Eq. (1.2), holds, and thus the two states are jiBgjD1;:::;d0 using jv0 ji WD OV Bjvji. It is then simple to

the same state. When combined with the previous observation in Sect. 1.2.3, we can conclude that:

pj O.j/
Oout D OU OinOU

OA ˝ OB
ODP OA D TrB OAB AD Bhvjj OABjvjiB iqi O.i/

1 Quantum Information Theory for Quantum Communication 11

1.3 Qubits

They satisfy the following commutation and anti-commutation relations:

ŒOi; OjD 2iijk Ok and fOi; Ojg D 2ıi;jO1; (1.9)

OA D .P0O1 C P O/=2 (1.11)

with P0 D Tr.OA/ and P D Tr. OOA/.