We study the dynamics of bosonic and fermionic anyons defined on a one-dimensional lattice, under the effect of Hamiltonians quadratic in creation and annihilation operators, also called Gaussian Hamiltonians. These anyonic models are obtained from deformations of the standard bosonic or fermionic commutation relations via the introduction of a non-trivial exchange phase between different lattice sites. We study the effects of the anyonic exchange phase on the usual bosonic and fermionic bunching behaviors. We show how to exploit the inherent Aharonov-Bohm effect exhibited by these particles to build a deterministic, entangling two-qubit gate and prove quantum computational universality in these systems. We define coherent states for bosonic anyons and study their behavior under two-mode passive, Gaussian devices. In particular we prove that, for a particular value of the exchange factor, an anyonic mirror can generate cat states, an important resource in quantum information processing with continuous variables.
We describe a continuous-variable scheme for simulating the Kitaev lattice model and for detecting statistics of abelian anyons. The corresponding quantum optical implementation is solely based upon Gaussian resource states and Gaussian operations, hence allowing for a highly efficient creation, manipulation, and detection of anyons. This approach extends our understanding of the control and application of anyons and it leads to the possibility for experimental proof-of-principle demonstrations of anyonic statistics using continuous-variable systems.
We point out some major technical and conceptual mistakes which invalidate the conclusion drawn in "Anyonic braiding in optical lattices" by C. Zhang, V. W. Scarola, S. Tewari, and S. Das Sarma published in PNAS 104, 18415 (2007).
We propose a scheme to probe the non-Abelian statistics of the collective anyonic excitation in Kitaev's honeycomb model with cold atoms in an optical lattice. The generation of the anyonic excitation can be realized by simple rotating operation acting on an effective spin-1/2 system, which is encoded in the atomic hyperfine energy levels. The non-Abelian nature of the anyonic excitation is manifested by the braiding of four vortices, which leads to different operations on the subspace of degenerate ground states, and thus results in different final states. Here, by introducing an ancilla atom…
Strongly correlated quantum systems can exhibit exotic behavior called topological order which is characterized by non-local correlations that depend on the system topology. Such systems can exhibit remarkable phenomena such as quasi-particles with anyonic statistics and have been proposed as candidates for naturally fault-tolerant quantum computation. Despite these remarkable properties, anyons have never been observed in nature directly. Here we describe how to unambiguously detect and characterize such states in recently proposed spin lattice realizations using ultra-cold atoms or molecules…
Fermionic linear optics corresponds to the dynamics of free fermions, and is known to be efficiently simulable classically. We define fermionic anyon models by deforming the fermionic algebra of creation and annihilation operators, and consider the dynamics of number-preserving, quadratic Hamiltonians on these operators. We show that any such deformation results in an anyonic linear optical model which allows for universal quantum computation.
Anyons are particlelike excitations of strongly correlated phases of matter with fractional statistics, characterized by nontrivial changes in the wave function, generalizing Bose and Fermi statistics, when two of them are interchanged. This can be used to perform quantum computations [A. Yu. Kitaev, Ann. Phys. (N.Y.) 303, 2 (2003)]. We show how to simulate the creation and manipulation of Abelian and non- Abelian anyons in topological lattice models using trapped atoms in optical lattices. Our proposal, feasible with present technology, requires an ancilla particle which can undergo single-pa…
Multicomponent anyons in one-dimensional optical lattices
Rice University（莱斯大学）；The University of Oklahoma（俄克拉荷马大学）；Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences（中国科学院精密测量科学与技术创新研究院）；Research School of Physics, Australian National University（澳大利亚国立大学物理研究学院）
We investigate continuous-time quantum walks of two indistinguishable anyons in one-dimensional lattices with both on-site and nearest-neighbor interactions based on the fractional Jordan-Wigner transformation. It is shown that the two-body correlations in position space are symmetric about the initial sites of two quantum walkers in the Bose limit ($χ=0$ ) and Fermi limit ( $χ=1$), while in momentum space this happens only in the Bose limit. An interesting asymmetry arises in the correlation for most cases with the statistical parameter $χ$ varying in between. It turns out that the origin of …
Anyons - particles carrying fractional statistics that interpolate between bosons and fermions - have been conjectured to exist in low dimensional systems. In the context of the fractional quantum Hall effect (FQHE), quasi-particles made of electrons take the role of anyons whose statistical exchange phase is fixed by the filling factor. Here we propose an experimental setup to create anyons in one-dimensional lattices with fully tuneable exchange statistics. In our setup, anyons are created by bosons with occupation-dependent hopping amplitudes, which can be realized by assisted Raman tunneli…
Topological quantum states of matter, both Abelian and non-Abelian, are characterized by excitations whose wavefunctions undergo non-trivial statistical transformations as one excitation is moved (braided) around another. Topological quantum computation proposes to use the topological protection and the braiding statistics of a non-Abelian topological state to perform quantum computation. The enormous technological prospect of topological quantum computation provides new motivation for experimentally observing a topological state. Here we explicitly work out a realistic experimental scheme to …
We consider a two-dimensional spin system that exhibits abelian anyonic excitations. Manipulations of these excitations enable the construction of a quantum computational model. While the one-qubit gates are performed dynamically the model offers the advantage of having a two-qubit gate that is of topological nature. The transport and braiding of anyons on the lattice can be performed adiabatically enjoying the robust characteristics of geometrical evolutions. The same control procedures can be used when dealing with non-abelian anyons. A possible implementation of the manipulations with optic…
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 \textit{dynamic}laser manipulation of cold atoms in an optical lattice. Due to 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…
Generalized Effective Spin-Chain formalism for multicomponent anyons in one-dimensional optical lattices
Rice University（莱斯大学）；The University of Oklahoma（俄克拉荷马大学）；Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences（中国科学院精密测量科学与技术创新研究院）；Research School of Physics, Australian National University（澳大利亚国立大学物理研究学院）
Revealing Pseudo-Fermionization and Chiral Binding of One-Dimensional Anyons using Adiabatic State Preparation
Department of Physics, Harvard University（哈佛大学物理系）；Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-Université PSL, Sorbonne Université（卡斯勒·布罗塞尔实验室，法兰西公学院，法国国家科学研究中心，巴黎文理研究大学，索邦大学）；JILA, NIST and Department of Physics, University of Colorado（JILA，美国国家标准与技术研究院和科罗拉多大学物理系）；Institut für Physik und Astronomie, Technische Universität Berlin（柏林工业大学物理与天文学研究所）；Naquidis Center, Institut d’Optique Graduate School（Naquidis中心，高等光学学院）
Traditional anyons in two dimensions have generalized exchange statistics governed by the braid group. By analyzing the topology of configuration space, we discover that an alternate generalization of the symmetric group governs particle exchanges when there are hard-core three-body interactions in one-dimension. We call this new exchange symmetry the traid group and demonstrate that it has abelian and non-abelian representations that are neither bosonic nor fermionic, and which also transform differently under particle exchanges than braid group anyons. We show that generalized exchange stati…
We proposed an entangled multi-knot lattice model to explore the exotic statistics of anyon. This knot lattice model bears abelian and non-abelian anyons as well as integral and fractional filling states that is similar to quantum Hall system. The fusion rules of anyon are explicitly demonstrated by braiding on crossing states of the multi-knot lattice. The statistical character of anyon is quantified by topological linking number of multi-knot link. Long-range coupling interaction is a fundamental character of this knot lattice model. The short range coupling models, such as Ising model, ferm…
A review of probability representation of quantum states in given for optical and photon number tomography approaches. Explicit connection of photon number tomogram with measurable by homodyne detector optical tomogram is obtained. New integral relations connecting Hermite polynomials of two variables with Laguerre polynomials are found. Examples of generic Gaussian photon states (squeezed and correlated states) are studied in detail.
We establish the fundamental limit of communication capacity within Gaussian schemes under phase-insensitive Gaussian channels, which employ multimode Gaussian states for encoding and collective Gaussian operations and measurements for decoding. We prove that this Gaussian capacity is additive, i.e., its upper bound occurs with separable encoding and separable receivers so that a single-mode communication suffices to achieve the largest capacity under Gaussian schemes. This rigorously characterizes the gap between the ultimate Holevo capacity and the capacity within Gaussian communication, sho…
It has recently been shown that optical parametric oscillator (OPO) Ising machines, consisting of coupled optical pulses circulating in a cavity with parametric gain, can be used to probabilistically find low-energy states of Ising spin systems. In this work, we study optical Ising machines that operate under simplified Gaussian dynamics. We show that these dynamics are sufficient for reaching probabilities of success comparable to previous work. Based on this result, we propose modified optical Ising machines with simpler designs that do not use parametric gain yet achieve similar performance…
We prove that a classical computer can efficiently sample from the photon-number probability distribution of a Gaussian state prepared by using an optical circuit that is shallow and local. Our work generalizes previous known results for qubits to the continuous-variable domain. The key to our proof is the observation that the adjacency matrices characterizing the Gaussian states generated by shallow and local circuits have small bandwidth. To exploit this structure, we devise fast algorithms to calculate loop hafnians of banded matrices. Since sampling from deep optical circuits with exponent…
We present an implementation of quantum logic gates via internal state dependent displacements of ions in a linear Paul trap caused by optical dipole forces. Based on a general quantum analysis of the system dynamics we consider specific implementations with alkaline earth ions. For experimentally realistic parameters gate infidelities as low as $10^{-4}$ can be obtained.
We study fast expansions of cold atoms in a three-dimensional Gaussian-beam optical trap. Three different methods to avoid final motional excitation are compared: inverse engineering using Lewis-Riesenfeld invariants, which provides the best overall performance, a bang-bang approach, and a fast adiabatic approach. We analyze the excitation effect of anharmonic terms, radial-longitudinal coupling, and radial-frequency mismatch. In the inverse engineering approach these perturbations can be suppressed or mitigated by increasing the laser beam waist.
We construct a class of variational methods for the study of open quantum systems based on Gaussian ansatzes for the quantum trajectory formalism. Gaussianity in the conjugate position and momentum quadratures is distinguished from Gaussianity in density and phase. We apply these methods to a driven-dissipative Kerr cavity where we study dephasing and the stationary states throughout the bistability regime. Computational cost proves to be similar to the truncated Wigner (TWA) method, with at most quadratic scaling in system size. Meanwhile, strong correspondence with the numerically exact traj…
Laser-light (coherent-state) modulation is sufficient to achieve the ultimate (Holevo) capacity of classical communication over a lossy and noisy optical channel, but requires a receiver that jointly detects long modulated codewords with highly nonlinear quantum operations, which are near-impossible to realize using current technology. We analyze the capacity of the lossy-noisy optical channel when the transmitter uses coherent state modulation but the receiver is restricted to a general quantum-limited Gaussian receiver, i.e., one that may involve arbitrary combinations of Gaussian operations…
In all practical communication channels, the code word consist of Gaussian states and the measurement strategy is often a Gaussian detector such as homodyning or heterodyning. In this paper we investigate the communication performance using a phase-conjugated alphabet and joint Gaussian detection in an phase-insensitive amplifying channel. We find that a communication scheme consisting of a phase-conjugated alphabet of coherent states and a joint detection strategy significantly outperforms a standard coherent-state strategy based in individual detection. Moreover, we show that the performance…
We experimentally investigate the mode characteristics of multimode radiation fields propagating through frequency dependent Gaussian channels. After manipulating the twin beams emitted from a conventional parametric down-conversion source via spectral filtering, we study the changes in their mode characteristics, utilizing the joint normalized correlation functions. While filtering reduces the number of spectral modes, it also leads to an apparent mode mismatch, which destroys the perfect photon-number correlation between the twin beams, and influences the mode properties of heralded states.
In a recent paper [I.I. Arkhipov, Phys. Rev. A 98, 021803 (2018)], it was shown that one can completely identify the nonclassicality of single- and two-mode Gaussian states by means of certain nonclassicality witnesses which are based on intensity moments up to the third order of optical fields, provided that an appropriate coherent displacement is applied to a given Gaussian state. Here, we utilize a mathematical equivalence between the description of the coherent displaced Gaussian states generated in the spontaneous parametric processes and the Gaussian states generated in the corresponding…
We show how the entanglement in a wide range of continuous variable non-Gaussian states can be preserved against decoherence for long-range quantum communication through an optical fiber. We apply protection via decoherence-free subspaces and quantum dynamical decoupling to this end. The latter is implemented by inserting phase shifters at regular intervals $Δ$ inside the fiber, where $Δ$ is roughly the ratio of the speed of light in the fiber to the bath high-frequency cutoff. Detailed estimates of relevant parameters are provided using the boson-boson model of system-bath interaction for sil…
We describe a general technique that allows to induce and control strong interaction between spin states of neighboring atoms in an optical lattice. We show that the properties of spin exchange interactions, such as magnitude, sign, and anisotropy can be designed by adjusting the optical potentials. We illustrate how this technique can be used to efficiently ``engineer'' quantum spin systems with desired properties, for specific examples ranging from scalable quantum computation to probing a model with non-trivial topological orders that supports exotic non-abelian anyonic excitations.
Quantum reading aims at retrieving classical information stored in an optical memory with low energy and high accuracy by exploiting the inherently quantum properties of light. We provide an optimal Gaussian strategy for quantum reading with phase-shift keying encoding that makes use of squeezed coherent light and homodyne detectors to largely outperform the Standard Quantum Limit, even in the presence of loss. This strategy, being feasible with current quantum optical technology, represents a viable prototype for a highly efficient and reliable quantum-enhanced optical reader.
We demonstrate quantum correlations in the transverse plane of continuous wave light beams by producing -4.0 dB, -2.6 dB and -1.5 dB of squeezing in the TEM00, TEM10 and TEM20 Hermite- Gauss modes with an optical parametric amplifier, respectively. This has potential applications in quantum information networking, enabling parallel quantum information processing. We describe the setup for the generation of squeezing and analyze the effects of various experimental issues such as mode overlap between pump and seed and nonlinear losses.
An arbitrary quantum-optical process (channel) can be completely characterized by probing it with coherent states using the recently developed coherent-state quantum process tomography (QPT) [Lobino et al., Science 322, 563 (2008)]. In general, precise QPT is possible if an infinite set of probes is available. Thus, realistic QPT of infinite-dimensional systems is approximate due to a finite experimentally-feasible set of coherent states and its related energy-cut-off approximation. We show with explicit formulas that one can completely identify a quantum-optical Gaussian process just with a f…
Entanglement is an essential ingredient for building a quantum network that can have many applications. Understanding how entanglement is distributed in a network is a crucial step to move forward. Here we study the conservation and distribution of Gaussian entanglement in a linear network using a new quantifier for bipartite entanglement. We show that the entanglement can be distributed through a beam-splitter in the same way as the transmittance and the reflectance. The requirements on the entangled states and the type of networks to satisfy this relation are presented explicitly. Our result…
Quantum states with non-Gaussian statistics generated by optical parametric oscillators (OPO) with fluctuating parameters are studied by means of the Kurtosis excess of the external field quadratures. The field generated is viewed as the response of a nonlinear device to the fluctuations of laser pump amplitude and frequency, crystal temperature and cavity detuning, in addition to quantum noise sources. The Kurtosis excess has been evaluated perturbatively up to the third order in the strength of the crystal nonlinear coupling factor and the second order in the classical fluctuating parameters…
Non-Gaussian states of an optical field are important as a proposed resource in quantum information applications. While conditional preparation is a highly successful approach to preparing such states, their quality is limited by detector non-idealities such as dead time, narrow dynamic range, limited quantum efficiency and dark noise. Mesoscopic photon counters, with peak performance at higher photon number, offer many practical advantages over single-photon level conditioning detectors. Here we propose a novel approach involving displacement of the ancilla field into the regime where mesosco…
In this thesis, several new procedures for the generation and manipulation of entangled states in quantum optical systems are introduced. Each is evaluated in terms of realistic models of the imperfect apparatus of any real laboratory implementation. The first half of the thesis considers entanglement generation in Cavity QED, introducing several proposals for the creation of maximally entangled two-qubit states. The second half of the thesis focuses on the manipulation of entangled states of light pulses. In particular, an entanglement distillation procedure for Gaussian states is introduced.…
Since a general Gaussian process is phase-sensitive, a stable phase reference is required to take advantage of this feature. When the reference is missing, either due to the volatile nature of the measured sample or the measurement's technical limitations, the resulting process appears as random in phase. Under this condition, we consider two single-mode Gaussian processes, displacement and squeezing. We show that these two can be efficiently estimated using photon number states and photon number resolving detectors. For separate estimation of displacement and squeezing, the practical estimati…
Non-Gaussian states are essential for many optical quantum technologies. The so-called optical quantum state synthesizer (OQSS), consisting of Gaussian input states, linear optics, and photon-number resolving detectors, is a promising method for non-Gaussian state preparation. However, an inevitable and crucial problem is the complexity of the numerical simulation of the state preparation on a classical computer. This problem makes it very challenging to generate important non-Gaussian states required for advanced quantum information processing. Thus, an efficient method to design OQSS circuit…
The behavior of general nonclassical two-mode Gaussian states at a beam splitter is investigated. Single-mode nonclassicality as well as two-mode entanglement of both input and output states are analyzed suggesting their suitable quantifiers. These quantifiers are derived from local and global invariants of linear unitary two-mode transformations such that the sum of input (or output) local nonclassicality measures and entanglement measure gives a global invariant. This invariant quantifies the global nonclassicality resource. Mutual transformations of local nonclassicalities and entanglement …
Quantum bit commitment has long been known to be impossible. Nevertheless, just as in the classical case, imposing certain constraints on the power of the parties may enable the construction of asymptotically secure protocols. Here, we introduce a quantum bit commitment protocol and prove that it is asymptotically secure if cheating is restricted to Gaussian operations. This protocol exploits continuous-variable quantum optical carriers, for which such a Gaussian constraint is experimentally relevant as the high optical nonlinearity needed to effect deterministic non-Gaussian cheating is inacc…
We demonstrate how to build a simulation of two dimensional physical theories describing topologically ordered systems whose excitations are in one to one correspondence with irreducible representations of a Hopf algebra, D(G), the quantum double of a finite group G. Our simulation uses a digital sequence of operations on a spin lattice to prepare a ground "vacuum" state and to create, braid and fuse anyonic excitations. The simulation works with or without the presence of a background Hamiltonian though only in the latter case is the system topologically protected. We describe a physical real…
The connection between Poincaré spheres for polariz-ation and Gaussian beams is explored, focusing on the interpretation of elliptic polarization in terms of the isotropic 2-dimensional harmonic oscillator in Hamiltonian mechanics, its canonical quantization and semiclassical interpretation. This leads to the interpretation of structured Gaussian modes, the Hermite-Gaussian, Laguerre-Gaussian and Generalized Hermite-Laguerre Gaussian modes as eigenfunctions of operators corresponding to the classical constants of motion of the 2-dimensional oscillator, which acquire an extra significance as fa…
Many nonlinear optical technologies require the two-mode spectral amplitude function that describes them---the \emph{joint spectral amplitude} (JSA)---to be separable. We prove that the JSA factorizes \emph{only} when the incident pump field and phase-matching function are Gaussian functions. We show this by mapping our problem to a known result, in continuous variable quantum information, that only squeezed states remain unentangled when combined on a beam splitter. We then conjecture that only a squeezed state minimizes entanglement when sent through a beam splitter with another pre-specifie…
We study ground states and elementary excitations of a system of bosonic atoms and diatomic Feshbach molecules trapped in a one-dimensional optical lattice using exact diagonalization and variational Monte Carlo methods. We primarily study the case of an average filling of one boson per site. In agreement with bosonization theory, we show that the ground state of the system in the thermodynamic limit corresponds to the Pfaffian-like state when the system is tuned towards the superfluid-to-Mott insulator quantum phase transition. Our study clarifies the possibility of the creation of exotic Pfa…
