Interferometry of non-Abelian anyons

We develop the general quantum measurement theory of non-Abelian anyons through interference experiments. The paper starts with a terse introduction to the theory of anyon models, focusing on the basic formalism necessary to apply standard quantum measurement theory to such systems. This is then applied to give a detailed analysis of anyonic charge measurements using a Mach-Zehnder interferometer for arbitrary anyon models. We find that, as anyonic probes are sent through the legs of the interferometer, superpositions of the total anyonic charge located in the target region collapse when they are distinguishable via monodromy with the probe anyons, which also determines the rate of collapse. We give estimates on the number of probes needed to obtain a desired confidence level for the measurement outcome distinguishing between charges, and explicitly work out a number of examples for some significant anyon models. We apply the same techniques to describe interferometry measurements in a double point-contact interferometer realized in fractional quantum Hall systems. To lowest order in tunneling, these results essentially match those from the Mach-Zehnder interferometer, but we also provide the corrections due to processes involving multiple tunnelings. Finally, we give explicit predictions describing state measurements for experiments in the Abelian hierarchy states, the non-Abelian Moore-Read state at ν=5/2 and Read-Rezayi state at ν=12/5.     

New implementation of composite fermions in terms of subgroups of a braid group

The new implementation of composite fermions and more generally — of composite anyons is formulated, exploiting one-dimensional unitary representations of appropriately constructed subgroups of the full braid group. The nature of hypothetical fluxes attached to the Jain's composite fermions is explained via additional cyclotron trajectory loops consistently with the braid subgroup structure. It is demonstrated that composite fermions are proper 2D particles (not an auxiliary construction), but associated with braid subgroups instead of the full braid group.     

Pauli Paramagnetic Susceptibility of an Ideal Anyon Gas within Haldane Fractional Exclusion Statistics

The finite-temperature Pauli paramagnetic susceptibility of a three-dimensional ideal anyon gas obeying Haldane fractional exclusion statistics is studied analytically.Different from the result of an ideal Fermi gas,the susceptibility of an ideal anyon gas depends on a statistical factor g in Haldane statistics model.The low-temperature and high-temperature behaviors of the susceptibility are investigated in detail.The Pauli paramagnetic susceptibility of the two-dimensional ideal anyons is also derived.It is found that the reciprocal of the susceptibility has the similar factorizable property which is exhibited in some thermodynamic quantities in two dimensions.     

Anyon Quantum Transport and Noise Away from Equilibrium

The quantum transport of anyons in one space dimension is investigated. After establishing some universal features of non-equilibrium systems in contact with two heat reservoirs in a generalized Gibbs state, the abelian anyon solution of the Tomonaga–Luttinger model possessing axial-vector duality is focused upon. In this context a non-equilibrium representation of the physical observables is constructed, which is the basic tool for a systematic study of the anyon particle and heat transport. The associated Lorenz number is determined and the deviation from the standard Wiedemann–Franz law induced by the interaction and the anyon statistics is explicitly described. The quantum fluctuations generated by the electric and helical currents are investigated and the dependence of the relative noise power on the statistical parameter is established.     

Anyon statistics with single-valued wave functions

The fractional statistics of the anyons proposed by Wilczek are demonstrated in a simple manner using single-valued wave functions. Taking the magnetic flux tube and charge comprising each anyon to be bosons, the wave function for two identical anyons is symmetrical with respect to the interchange, but for ρΦ = π, where ρ is the charge and Φ the magnetic flux in each anyon, the anyons behave as fermions, and for other values of ρΦ, the anyons obey intermediate statistics.     

The relation between properties of Gentile statistics and fractional statistics of anyon

In this paper, we discuss the relationship of two kinds of intermediate-statistics, the Gentile statistics and the fractional statistics of anyons. The anyon winding number representation is introduced. We construct the transformation between anyon winding number representation and the occupation number representation of particles of Gentile statistics. We study intermediate-statistics quantum bracket and coherent states for anyons in the winding number representation. We demonstrate that anyons can be simulated by Gentile statistics with a geometric phase.     

Decoherence-free subspaces of anyon states

In this paper, we propose decoherence-free subspaces (DFSs) of anyon states under two kinds of noises. The forms of collective dephasing and collective rotation noises are introduced. The invariance requirement under these noises give some restrictions of the winding number of anyons. We put forward a scheme to realize the DFS of anyon states in the Kitaev spin-lattice model, and give some discussion of general kinds of anyons.     

Anyon-fermion mapping and applications to ultracold gases in tight waveguides

The Fermi-Bose mapping method for one-dimensional Bose and Fermi gases with zero-range interactions is generalized to an anyon-fermion mapping and applied to exact solution of several models of ultracold gases with anyonic exchange symmetry in tight waveguides: anyonic Calogero-Sutherland model, anyons with point hard-core interaction (anyonic Tonks-Girardeau gas), and spin-aligned anyon gas with infinite zero-range odd-wave attractions (attractive anyonic Tonks-Girardeau, or AATG, gas). It is proved that for even N>or=4 there are states of the AATG gas on a ring, with anyonic phase slips which are odd integral multiples of pi/(N-1), of energy lower than that of the corresponding fermionic ground state. A generalization to a spinor Fermi gas state with anyonic symmetry under purely spatial exchange enables energy lowering by the same mechanism.     

Scheme for demonstration of fractional statistics of anyons in an exactly solvable model

We propose a scheme to demonstrate fractional statistics of anyons in an exactly solvable lattice model proposed by Kitaev that involves four-body interactions. The required many-body ground state, as well as the anyon excitations and their braiding operations, can be conveniently realized through dynamic laser manipulation of cold atoms in an optical lattice. Because of the perfect localization of anyons in this model, we show that a quantum circuit with only six qubits is enough for demonstration of the basic braiding statistics of anyons. This opens up the immediate possibility of proof-of-principle experiments with trapped ions, photons, or nuclear magnetic resonance systems.     

N-anyon problem: cyclic corrections

The cyclic corrections (i.e. the contribution of cyclic particle exchanges) to the anyon partition functions are computed at all perturbative orders of the statistical coupling in the thermodynamic limit. A possible generalization for anyons in a harmonic well is revisited. One shows that this proposal correctly encompasses the contribution of the linear energies both in the symmetric, antisymmetric and mixed representations, but it is not satisfying as regards the contribution of the non-linear energies. Finally, one builds a few-body spectrum which interpolates linearly between the Bose one and the Fermi one in a harmonic well, it reproduces the previous proposal, and it is consistent with a finite virial expansion where only the second coefficient is affected by the statistics in the thermodynamic limit.     

Quantum computation with Turaev-Viro codes

For a 3-manifold with triangulated boundary, the Turaev-Viro topological invariant can be interpreted as a quantum error-correcting code. The code has local stabilizers, identified by Levin and Wen, on a qudit lattice. Kitaev’s toric code arises as a special case. The toric code corresponds to an abelian anyon model, and therefore requires out-of-code operations to obtain universal quantum computation. In contrast, for many categories, such as the Fibonacci category, the Turaev-Viro code realizes a non-abelian anyon model. A universal set of fault-tolerant operations can be implemented by deforming the code with local gates, in order to implement anyon braiding. We identify the anyons in the code space, and present schemes for initialization, computation and measurement. This provides a family of constructions for fault-tolerant quantum computation that are closely related to topological quantum computation, but for which the fault tolerance is implemented in software rather than coming from a physical medium.     

Second Quantization of the Elliptic Calogero-Sutherland Model

We construct a quantum field theory model of anyons on a circle and at finite temperature. We find an anyon Hamiltonian providing a second quantization of the elliptic Calogero-Sutherland model. This allows us to prove a remarkable identity which is a starting point for an algorithm to construct eigenfunctions and eigenvalues of the elliptic Calogero-Sutherland Hamiltonian.Work supported by the Swedish Natural Science Research Council (NFR).     

从一维光晶格中释放的任意子噪声关联函数研究

首先求解具有delta函数型相互作用的任意子气体的含时薛定谔方程,给出了其多体波函数的解析解,并在此基础上详细分析了无相互作用情形和有相互作用情形下任意子噪声关联函数的特性.对于有相互作用的任意子气体,其噪声关联呈现出与无相互作用情形下不同的特性:散射相位具有一定的空间分布,一系列线性而不是尖峰出现在噪声关联函数中;线性的宽度、取向以及位置与任意子的统计参数和粒子间相互作用强度的关系都非常密切.特别地,在TG极限下,也就是相互作用趋于无限大的情形下,粒子间散射相位变为,任意子的噪声关联函数图样与无相互作用情形下的图样完全相反.     

从一维光晶格中释放的任意子噪声关联函数研究

首先求解具有delta函数型相互作用的任意子气体的含时薛定谔方程,给出了其多体波函数的解析解,并在此基础上详细分析了无相互作用情形和有相互作用情形下任意子噪声关联函数的特性。对于有相互作用的任意子气体,其噪声关联呈现出与无相互作用情形下不同的特性：散射相位具有一定的空间分布,一系列线性而不是尖峰出现在噪声关联函数中;线性的宽度、取向以及位置与任意子的统计参数和粒子间相互作用强度的关系都非常密切。特别地,在TG极限下,也就是相互作用趋于无限大的情形下,任意子的噪声关联函数图样与无相互作用情形下的图样完全相反。     

Fermionic behavior of ideal anyons

We prove upper and lower bounds on the ground-state energy of the ideal two-dimensional anyon gas. Our bounds are extensive in the particle number, as for fermions, and linear in the statistics parameter \(\alpha \). The lower bounds extend to Lieb–Thirring inequalities for all anyons except bosons.     

On the connetion between Euclidean and Hamiltonian lattice field theories of vortices and anyons

In the (2+1)-dimensional non-compact Abelian lattice Higgs model Euclidean correlation functions of vortices and, after adding the Chern-Simons term to the action, correlation functions of transmuted matter fields (anyons) are set up. These correlation functions satisfy Osterwalder-Schrader positivity. Via the transition to continuous Euclidean time, vortex and anyon operators within a Hamiltonian lattice formulation are obtained, respectively, and their respective dual algebras are displayed.     

Simulating the fourth-order interference phenomenon of anyons with photon pairs

The generalized Hong-Ou-Mandel interferometer with anyons is studied. Novel interference results different from bosons or fermions are found. An experimental scheme based on linear optics is proposed and realized to simulate the fourth-order interference phenomenon of anyons.