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arXiv:1207.4974 · Physical Review A

Simulating the coupling of angular momenta in distant matter qubits

C. Ammon, Andreas Maser, U. Schilling, Thierry Bastin, Joachim von Zanthier

Published Nov 6, 2012• 2 citations
arXivHTMLPDF

Abstract

We present a mathematical proof of the algorithm allowing one to generate all---symmetric and nonsymmetric---total angular momentum eigenstates in remote matter qubits by projective measurements, proposed in Maser et al. [Phys. Rev. A 79, 033833 (2009)]. By deriving a recursion formula for the algorithm we show that the generated states are equal to the total angular momentum eigenstates obtained via the usual quantum-mechanical coupling of angular momenta. In this way we demonstrate that the algorithm is able to simulate the coupling of $N$ spin-1/2 systems, and to implement the required Clebsch-Gordan coefficients, even though the particles never directly interact with each other.

DOI
10.1103/physreva.86.052308

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We present a mathematical proof of the algorithm allowing one to generate all---symmetric and nonsymmetric---total angular momentum eigenstates in remote matter qubits by projective measurements, proposed in Maser et al. [Phys. Rev. A 79, 033833 (2009)]. By deriving a recursion formula for the algorithm we show that the generated states are equal to the total angular momentum eigenstates obtained via the usual quantum-mechanical coupling of angular momenta. In this way we demonstrate that the algorithm is able to simulate the coupling of NNN spin-1/2 systems, and to implement the required Clebsch-Gordan coefficients, even though the particles never directly interact with each other.
Coupled angular-momentum eigenstates are widely used in atomic and nuclear physics calculations and are building blocks for spin networks and the Schur transform. To combine two angular momenta J1{\mathbf{J}}_{1}J1​ and J2{\mathbf{J}}_{2}J2​, forming eigenstates of their total angular momentum J=J1+J2\mathbf{J}={\mathbf{J}}_{1}+{\mathbf{J}}_{2}J=J1​+J2​, we develop a quantum-walk scheme that does not require inputting O(j3)O({j}^{3})O(j3) nonzero Clebsch--Gordan (CG) coefficients classically. In fact, our scheme may be regarded as a unitary method for computing CG coefficients on quantum computers with a typical complexity of O(j)O(j)O(j) and a worst-case complexity of O(j3)O({j}^{3})O(j3). Equivalently, our scheme provides decompositions of the dense CG unitary into sparser unitary operations. Our scheme prepares angular-momentum eigenstates using a sequence of Hamiltonians to move an initial state deterministically to desired final states, which are usually highly entangled states in the computational basis. In contrast with usual quantum walks, whose Hamiltonians are prescribed, we engineer the Hamiltonians in $\text{su}(2)\ifmmode\times\else\texttimes\fi{}\text{su}(2)$, which are inspired by, but different from, Hamiltonians that govern magnetic resonances and dipole interactions. To achieve a deterministic preparation of both ket and bra states, we use projection and destructive interference to double pinch the quantum walks, such that each step is a unit-probability population transfer within a two-level system. We test our state preparation scheme on classical computers, reproducing tables of CG coefficients. We also implement small test problems on current quantum hardware.
We propose a scheme enabling the universal coupling of angular momentum of NNN remote noninteracting qubits using linear optical tools only. Our system consists of NNN single-photon emitters in a $\ensuremath{\Lambda}$ configuration that are entangled among their long-lived ground-state qubits through suitably designed measurements of the emitted photons. In this manner, we present an experimentally feasible algorithm that is able to generate any of the 2N{2}^{N}2N symmetric and nonsymmetric total angular momentum eigenstates spanning the Hilbert space of the NNN-qubit compound.
The addition theorem of angular momentum states that the angular momentum operator of a coupled state is the vector sum of the angular momentum operators of the uncoupled states. Given that, the eigenfunction of the angular momentum of a system in the coupled state can be represented as a linear combination of eigenfunctions of the system in its uncoupled state and, in turn, the eigenfunctions of the uncoupled state are a direct product of eigenfunctions of the individual particles. Clebsch-Gordan coefficients emerge as an overlap integral in the series expansion of the coupled state in the basis of uncoupled states. This chapter provides a detailed derivation of the addition theorem of angular momentum, including obtaining the lower and upper boundaries of the quantum numbers, deriving the orthonormality conditions for Clebsch-Gordan coefficients, and using shift operators to generate the members of a multiplet. It also derives a general expression for the evaluation of Clebsch-Gordan coefficients, provides a detailed analysis of the addition of angular momenta and electron spin to build the resulting Clebsch-Gordan matrix, and shows that a rotation transformation of the direct product of eigenfunctions of the angular momentum and spin leads to the recurrence relations for the Wigner matrix.
This chapter discusses the addition of angular momenta in the physicist’s way. From the considerations it follows that there are several ways to construct complete systems of commuting angular momentum observables. Belonging to two complete sets (two different ‘couplings’) of commuting angular momentum observables, there must exist a unitary transformation. For more than two angular momenta there will be more sets and more unitary transformations between the different couplings. The matrix elements of these transformations are called ‘recoupling coefficients’. They can be expressed by sums over products of Clebsch-Gordan coefficients (CGCs); this is to be expected from the fact that repeated coupling of two angular momenta generates any desired coupling scheme. It also is to be expected that the transformations between different coupling schemes become rapidly more complicated with growing numbers of subsystems. A powerful and elegant graphical representation for CGCs exists which can be used in the calculations of coupling of angular momenta.
This chapter presents key results on angular momentum in quantum mechanics, beginning with a theorem on the simultaneous diagonalization of commuting operators. The eigenvalues and eigenstates of both the total angular momentum operator and its z-component are derived. For a quantum rotator in the p-state with zero magnetic quantum number, it is shown that the most probable trajectory lies in a plane containing the z-axis, and the angular momentum vector is perpendicular to it. The procedure for angular momentum addition is explained in detail, including the use of Clebsch–Gordan coefficients to perform the explicit decomposition of tensor products of irreducible representations. The Bloch sphere representation of qubit spin states is introduced. The chapter then provides an experimental background to the discovery of spin, introduces the Pauli matrices, and demonstrates how they act on spin states. It concludes with the formulation of the Pauli equation, which incorporates spin into the framework of nonrelativistic quantum mechanics. Metadata/Keywords: angular momentum, simultaneous diagonalization, magnetic quantum number, Clebsch–Gordan coefficients, spin, Pauli matrices, Bloch sphere, spin states, tensor product, Pauli equation