Showing papers on "Quantum published in 2005"
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TL;DR: It is proved that the typical quantum precision enhancement is of the order of the square root of the number of times the system is sampled, and it is pointed out the different strategies that permit one to attain this bound.
Abstract: We point out a general framework that encompasses most cases in which quantum effects enable an increase in precision when estimating a parameter (quantum metrology) The typical quantum precision-enhancement is of the order of the square root of the number of times the system is sampled We prove that this is optimal and we point out the different strategies (classical and quantum) that permit to attain this bound
1,858 citations
TL;DR: Optical lattices represent a fast-paced modern and interdisciplinary field of research as discussed by the authors, and they form powerful model systems of quantum many-body systems in periodic potentials for probing nonlinear wave dynamics and strongly correlated quantum phases, building fundamental quantum gates or observing Fermi surfaces.
Abstract: Artificial crystals of light, consisting of hundreds of thousands of optical microtraps, are routinely created by interfering optical laser beams. These so-called optical lattices act as versatile potential landscapes to trap ultracold quantum gases of bosons and fermions. They form powerful model systems of quantum many-body systems in periodic potentials for probing nonlinear wave dynamics and strongly correlated quantum phases, building fundamental quantum gates or observing Fermi surfaces in periodic potentials. Optical lattices represent a fast-paced modern and interdisciplinary field of research.
1,298 citations
TL;DR: Numerical simulations with the ground state of a 1D lattice at criticality show that the resulting coarse-grained sites require a Hilbert space dimension that does not grow with successive RG transformations, and calculations unveil that ground state entanglement in extended quantum systems is organized in layers corresponding to different length scales.
Abstract: In the context of real-space renormalization group methods, we propose a novel scheme for quantum systems defined on a D-dimensional lattice It is based on a coarse-graining transformation that attempts to reduce the amount of entanglement of a block of lattice sites before truncating its Hilbert space Numerical simulations involving the ground state of a 1D system at criticality show that the resulting coarse-grained site requires a Hilbert space dimension that does not grow with successive rescaling transformations As a result we can address, in a quasi-exact way, tens of thousands of quantum spins with a computational effort that scales logarithmically in the system's size The calculations unveil that ground state entanglement in extended quantum systems is organized in layers corresponding to different length scales At a quantum critical point, each rellevant length scale makes an equivalent contribution to the entanglement of a block with the rest of the system
982 citations
TL;DR: In this paper, the authors explore the prospects to control by use of time-dependent fields quantum transport phenomena in nanoscale systems and study for driven conductors the electron current and its noise properties.
605 citations
TL;DR: In this paper, the authors investigate the set of correlations that are constrained only by the no-signaling principle and determine the vertices of such correlations in the case that two observers each choose from two possible measurements with $d$ outcomes.
Abstract: It is well known that measurements performed on spatially separated entangled quantum systems can give rise to correlations that are nonlocal, in the sense that a Bell inequality is violated. They cannot, however, be used for superluminal signaling. It is also known that it is possible to write down sets of ``superquantum'' correlations that are more nonlocal than is allowed by quantum mechanics, yet are still nonsignaling. Viewed as an information-theoretic resource, superquantum correlations are very powerful at reducing the amount of communication needed for distributed computational tasks. An intriguing question is why quantum mechanics does not allow these more powerful correlations. We aim to shed light on the range of quantum possibilities by placing them within a wider context. With this in mind, we investigate the set of correlations that are constrained only by the no-signaling principle. These correlations form a polytope, which contains the quantum correlations as a (proper) subset. We determine the vertices of the no-signaling polytope in the case that two observers each choose from two possible measurements with $d$ outcomes. We then consider how interconversions between different sorts of correlations may be achieved. Finally, we consider some multipartite examples.
585 citations
TL;DR: This work introduces a scheme for linear optics quantum computation, that makes no use of teleported gates, and requires stable interferometry over only the coherence length of the photons, and demonstrates the universality and usefulness of generic parity measurements.
Abstract: We introduce a scheme for linear optics quantum computation, that makes no use of teleported gates, and requires stable interferometry over only the coherence length of the photons. We achieve a much greater degree of efficiency and a simpler implementation than previous proposals. We follow the "cluster state" measurement based quantum computational approach, and show how cluster states may be efficiently generated from pairs of maximally polarization entangled photons using linear optical elements. We demonstrate the universality and usefulness of generic parity measurements, as well as introducing the use of redundant encoding of qubits to enable utilization of destructive measurements--both features of use in a more general context.
545 citations
TL;DR: The spectral dimension of universes emerging from nonperturbative quantum gravity is measured, defined through state sums of causal triangulated geometries, and it is concluded that quantum gravity may be "self-renormalizing" at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction.
Abstract: We measure the spectral dimension of universes emerging from nonperturbative quantum gravity, defined through state sums of causal triangulated geometries. While four dimensional on large scales, the quantum universe appears two dimensional at short distances. We conclude that quantum gravity may be ``self-renormalizing'' at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction.
537 citations
TL;DR: In most introductory courses on quantum mechanics one is taught that the Hamiltonian operator must be Hermitian in order that the energy levels be real and that the theory be unitary (probability conserving).
Abstract: In most introductory courses on quantum mechanics one is taught that the Hamiltonian operator must be Hermitian in order that the energy levels be real and that the theory be unitary (probability conserving) To express the Hermiticity of a Hamiltonian, one writes H = H †, where the symbol † denotes the usual Dirac Hermitian conjugation; that is, transpose and complex conjugate In the past few years it has been recognized that the requirement of Hermiticity, which is often stated as an axiom of quantum mechanics, may be replaced by the less mathematical and more physical requirement of space – time reflection symmetry (𝒫𝒯 symmetry) without losing any of the essential physical features of quantum mechanics Theories defined by non-Hermitian 𝒫𝒯-symmetric Hamiltonians exhibit strange and unexpected properties at the classical as well as at the quantum level This paper explains how the requirement of Hermiticity can be evaded and discusses the properties of some non-Hermitian 𝒫𝒯-symmetric quantum theories
501 citations
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TL;DR: In this paper, the same experimental measurement that puzzled Einstein and his contemporaries is forcing us to question our understanding of how quantum matter transforms at ultra-low temperatures, as well as how quantum theory of solids transforms at low temperatures.
Abstract: As we mark the centenary of Albert Einstein's seminal contribution to both quantum mechanics and special relativity, we approach another anniversary--that of Einstein's foundation of the quantum theory of solids But 100 years on, the same experimental measurement that puzzled Einstein and his contemporaries is forcing us to question our understanding of how quantum matter transforms at ultra-low temperatures
478 citations
TL;DR: In this paper, the quantum hydrodynamic model for charged particle systems is extended to the cases of nonzero magnetic fields, and conditions for equilibrium in ideal quantum magnetohydrodynamics are established.
Abstract: The quantum hydrodynamic model for charged particle systems is extended to the cases of nonzero magnetic fields. In this way, quantum corrections to magnetohydrodynamics are obtained starting from the quantum hydrodynamical model with magnetic fields. The importance of the quantum corrections is described by a parameter H which can be significant in dense astrophysical plasmas. The quantum magnetohydrodynamic model is analyzed in the infinite conductivity limit. The conditions for equilibrium in ideal quantum magnetohydrodynamics are established. Translationally invariant exact equilibrium solutions are obtained in the case of the ideal quantum magnetohydrodynamic model.
426 citations
TL;DR: This Letter presents solvable examples of quantum many-body Hamiltonians of systems that are unable to reach their ground states as the environment temperature is lowered to absolute zero.
Abstract: This Letter presents solvable examples of quantum many-body Hamiltonians of systems that are unable to reach their ground states as the environment temperature is lowered to absolute zero. These examples, three-dimensional generalizations of quantum Hamiltonians proposed for topological quantum computing, (1) have no quenched disorder, (2) have solely local interactions, (3) have an exactly solvable spectrum, (4) have topologically ordered ground states, and (5) have slow dynamical relaxation rates akin to those of strong structural glasses.
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TL;DR: A comprehensive review of quantum cloning machines can be found in this paper, where the role of cloning in quantum cryptography, the link between optimal cloning and light amplification via stimulated emission, and the experimental demonstrations of optimal quantum cloning.
Abstract: The impossibility of perfectly copying (or cloning) an arbitrary quantum state is one of the basic rules governing the physics of quantum systems. The processes that perform the optimal approximate cloning have been found in many cases. These "quantum cloning machines" are important tools for studying a wide variety of tasks, e.g. state estimation and eavesdropping on quantum cryptography. This paper provides a comprehensive review of quantum cloning machines (both for discrete-dimensional and for continuous-variable quantum systems); in addition, it presents the role of cloning in quantum cryptography, the link between optimal cloning and light amplification via stimulated emission, and the experimental demonstrations of optimal quantum cloning.
TL;DR: It is demonstrated that polarization effects can play a significant role in determining the structures of protein‐ligand complexes, and provide a promising start towards the development of more accurate docking methods for lead optimization applications.
Abstract: The extent to which accuracy of electric charges plays a role in protein-ligand docking is investigated through development of a docking algorithm, which incorporates quantum mechanical/molecular mechanical (QM/MM) calculations. In this algorithm, fixed charges of ligands obtained from force field parameterization are replaced by QM/MM calculations in the protein environment, treating only the ligands as the quantum region. The algorithm is tested on a set of 40 cocrystallized structures taken from the Protein Data Bank (PDB) and provides strong evidence that use of nonfixed charges is important. An algorithm, dubbed "Survival of the Fittest" (SOF) algorithm, is implemented to incorporate QM/MM charge calculations without any prior knowledge of native structures of the complexes. Using an iterative protocol, this algorithm is able in many cases to converge to a nativelike structure in systems where redocking of the ligand using a standard fixed charge force field exhibits nontrivial errors. The results demonstrate that polarization effects can play a significant role in determining the structures of protein-ligand complexes, and provide a promising start towards the development of more accurate docking methods for lead optimization applications.
TL;DR: This work demonstrates the self-assembled formation of concentric quantum double rings with high uniformity and excellent rotational symmetry using the droplet epitaxy technique, and shows that Varying the growth process conditions can control each ring's size.
Abstract: We demonstrate the self-assembled formation of concentric quantum double rings with high uniformity and excellent rotational symmetry using the droplet epitaxy technique. Varying the growth process conditions can control each ring's size. Photoluminescence spectra emitted from an individual quantum ring complex show peculiar quantized levels that are specified by the carriers' orbital trajectories.
TL;DR: A general approach to the nonequilibrium dynamics of quantum-impurity systems for arbitrary coupling strength by using the numerical renormalization group to generate a complete basis set necessary for the correct description of the time evolution.
Abstract: We develop a general approach to the nonequilibrium dynamics of quantum-impurity systems for arbitrary coupling strength. The numerical renormalization group is used to generate a complete basis set necessary for the correct description of the time evolution. We benchmark our method with the exact analytical solution for the resonant-level model. As a first application, we investigate the equilibration of an ultrasmall quantum dot subject to a sudden change of gate voltage and external magnetic field. Two distinct relaxation times are identified for the spin and charge dynamics.
TL;DR: An analysis of the equilibrium limits of the two most widely used approaches for simulating the dynamics of molecular systems that combine both quantum and classical degrees of freedom and shows that the self-consistent-field (Ehrenfest) method deviates substantially from Boltzmann.
Abstract: We present an analysis of the equilibrium limits of the two most widely used approaches for simulating the dynamics of molecular systems that combine both quantum and classical degrees of freedom. For a two-level quantum system connected to an infinite number of classical particles, we derive a simple analytical expression for the equilibrium mean energy attained by the self-consistent-field (Ehrenfest) method and show that it deviates substantially from Boltzmann. By contrast, “fewest switches” surface hopping achieves Boltzmann quantum state populations. We verify these analytical results with simulations.
TL;DR: In this article, a Schnol-type theorem is proven that allows one to detect that a point λ belongs to the spectrum when a generalized eigenfunction with an subexponential growth integral estimate is available.
Abstract: The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics and other areas. A Schnol-type theorem is proven that allows one to detect that a point λ belongs to the spectrum when a generalized eigenfunction with an sub-exponential growth integral estimate is available. A theorem on spectral gap opening for 'decorated' quantum graphs is established (its analogue is known for the combinatorial case). It is also shown that if a periodic combinatorial or quantum graph has a point spectrum, it is generated by compactly supported eigenfunctions ('scars').
TL;DR: This work presents a new route for distributed optical QIP, based on generalized quantum non-demolition measurements, providing a unified approach for quantum communication and computing.
Abstract: Quantum information processing (QIP) offers the promise of being able to do things that we cannot do with conventional technology. Here we present a new route for distributed optical QIP, based on generalized quantum non-demolition measurements, providing a unified approach for quantum communication and computing. Interactions between photons are generated using weak non-linearities and intense laser fields--the use of such fields provides for robust distribution of quantum information. Our approach requires only a practical set of resources, and it uses these very efficiently. Thus it promises to be extremely useful for the first quantum technologies, based on scarce resources. Furthermore, in the longer term this approach provides both options and scalability for efficient many-qubit QIP.
TL;DR: In this article, entangled trinary quantum systems (qutrits) were used for quantum key distribution and two identical keys were obtained with a qutrit error rate of approximately 10 %.
Abstract: We produce two identical keys using, for the first time, entangled trinary quantum systems (qutrits) for quantum key distribution. The advantage of qutrits over the normally used binary quantum systems is an increased coding density and a higher security margin. The qutrits are encoded into the orbital angular momentum of photons, namely Laguerre-Gaussian modes with azimuthal index l +1, 0 and -1, respectively. The orbital angular momentum is controlled with phase holograms. In an Ekert-type protocol the violation of a three-dimensional Bell inequality verifies the security of the generated keys. A key is obtained with a qutrit error rate of approximately 10 %.
TL;DR: In this article, a new derivation of filtering equations is presented, in the cases of counting processes and of measurement processes of diffusive type, and it is also shown that the equation for the a posteriori dynamics in the diffusive case can be obtained, by a suitable limit, from that one in the counting case.
Abstract: Measurements continuous in time were consistently introduced in quantum mechanics and applications worked out, mainly in quantum optics. In this context a quantum filtering theory has been developed giving the reduced state after the measurement when a certain trajectory of the measured observables is registered (the a posteriori states). In this paper a new derivation of filtering equations is presented, in the cases of counting processes and of measurement processes of diffusive type. It is also shown that the equation for the a posteriori dynamics in the diffusive case can be obtained, by a suitable limit, from that one in the counting case. Moreover, the paper is intended to clarify the meaning of the various concepts involved and to discuss the connections among them. As an illustration of the theory, simple models are worked out.
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TL;DR: In this article, a *-algebraic indefinite structure of quantum stochastic calculus is introduced and a continuity property of generalized nonadapted QS integrals is proved under the natural integrability conditions in an infinitely dimensional nuclear space.
Abstract: A *-algebraic indefinite structure of quantum stochastic (QS) calculus is introduced and a continuity property of generalized nonadapted QS integrals is proved under the natural integrability conditions in an infinitely dimensional nuclear space. The class of nondemolition output QS processes in quantum open systems is characterized in terms of the QS calculus, and the problem of QS nonlinear filtering with respect to nondemolition continuous measurments is investigated. The stochastic calculus of a posteriori conditional expectations in quantum observed systems is developed and a general quantum filtering stochastic equation for a QS process is derived. An application to the description of the spontaneous collapse of the quantum spin under continuous observation is given.
TL;DR: In this article, the authors proposed a method that uses fixed, minimal physical resources to achieve generation and nested purification of quantum entanglement for quantum communication over arbitrarily long distances and discuss its implementation using realistic photon emitters and photonic channels.
Abstract: We analyze a method that uses fixed, minimal physical resources to achieve generation and nested purification of quantum entanglement for quantum communication over arbitrarily long distances and discuss its implementation using realistic photon emitters and photonic channels. In this method, we use single-photon emitters with two internal degrees of freedom formed by an electron spin and a nuclear spin to build intermediate nodes in a quantum channel. State-selective fluorescence is used for probabilistic entanglement generation between electron spins in adjacent nodes. We analyze in detail several approaches which are applicable to realistic, homogeneously broadened single-photon emitters. Furthermore, the coupled electron and nuclear spins can be used to efficiently implement entanglement swapping and purification. We show that these techniques can be combined to generate high-fidelity entanglement over arbitrarily long distances. We present a specific protocol that functions in polynomial time and tolerates percent-level errors in entanglement fidelity and local operations. The scheme has the lowest requirements on physical resources of any current scheme for fully fault-tolerant quantum repeaters.
TL;DR: By using inelastic neutron scattering, a complete data set of the magnetic correlations of KCuF3 are collected as a function of momentum, energy and temperature and the LL description is found to be valid over an extensive range of these parameters.
Abstract: Quantum effects dominate the behaviour of many diverse materials. Of particular current interest are those systems in the vicinity of a quantum critical point (QCP). Their physical properties are predicted to reflect those of the nearby QCP with universal features independent of the microscopic details. The prototypical QCP is the Luttinger liquid (LL), which is of relevance to many quasi-one-dimensional materials. The magnetic material KCuF3 realizes an array of weakly coupled spin chains (or LLs) and thus lies close to but not exactly at the LL quantum critical point. By using inelastic neutron scattering we have collected a complete data set of the magnetic correlations of KCuF3 as a function of momentum, energy and temperature. The LL description is found to be valid over an extensive range of these parameters, and departures from this behaviour at high and low energies and temperatures are identified and explained.
TL;DR: In this paper, the authors describe a scalable stochastic method for the experimental measurement of generalized fidelities characterizing the accuracy of the implementation of a coherent quantum transformation, based on the motion reversal of random unitary operators.
Abstract: We describe a scalable stochastic method for the experimental measurement of generalized fidelities characterizing the accuracy of the implementation of a coherent quantum transformation. The method is based on the motion reversal of random unitary operators. In the simplest case our method enables direct estimation of the average gate fidelity. The more general fidelities are characterized by a universal exponential rate of fidelity loss. In all cases the measurable fidelity decrease is directly related to the strength of the noise affecting the implementation -- quantified by the trace of the superoperator describing the non--unitary dynamics. While the scalability of our stochastic protocol makes it most relevant in large Hilbert spaces (when quantum process tomography is infeasible), our method should be immediately useful for evaluating the degree of control that is achievable in any prototype quantum processing device. By varying over different experimental arrangements and error-correction strategies additional information about the noise can be determined.
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TL;DR: Wang et al. as mentioned in this paper proposed a time-shift attack that exploits the efficiency mismatch of two single photon detectors (SPD) in a quantum key distribution (QKD) system, which shifts the arrival time of either the signal pulse or the synchronization pulse or both between Alice and Bob.
Abstract: Recently, a new type of attack, which exploits the efficiency mismatch of two single photon detectors (SPD) in a quantum key distribution (QKD) system, has been proposed. In this paper, we propose another "time-shift" attack that exploits the same imperfection. In our attack, Eve shifts the arrival time of either the signal pulse or the synchronization pulse or both between Alice and Bob. In particular, in a QKD system where Bob employs time-multiplexing technique to detect both bit "0" and bit "1" with the same SPD, Eve, in principle, could acquire full information on the final key without introducing any error. Finally, we discuss some counter measures against our and earlier attacks.
TL;DR: In this article, the Curie temperature of half-metallic Heusler alloys was investigated using first-principles calculations in conjunction with the frozen-magnon approximation.
Abstract: We study the exchange interactions in half-metallic Heusler alloys using first-principles calculations in conjunction with the frozen-magnon approximation. The Curie temperature is estimated within both mean-field (MF) and random-phase-approximation (RPA) approaches. For the half-Heusler alloys NiMnSb and CoMnSb, the dominant interaction is between the nearest Mn atoms. In this case, the MF and RPA estimations differ strongly. The RPA approach provides better agreement with the experiment. The exchange interactions are more complex in the case of full-Heusler alloys ${\mathrm{Co}}_{2}\mathrm{Mn}\mathrm{Si}$ and ${\mathrm{Co}}_{2}\mathrm{Cr}\mathrm{Al}$ where the dominant effects are the intersublattice interactions between the Mn(Cr) and Co atoms and between Co atoms at different sublattices. For these compounds, we find that both MF and RPA give very close values of the Curie temperature slightly underestimating experimental quantities. We study the influence of the lattice compression on the magnetic properties. The temperature dependence of the magnetization is calculated using the RPA method within both quantum mechanical and classical approaches.
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TL;DR: A new formalism for determining energy eigenstates of spherical quantum dots and cylindrical quantum wires in the multiple-band envelope-function approximation is described and Conduction-band–valence-band coupling is shown to be critical in a "type-II" InAs/GaSb quantum dot.
Abstract: We describe a new formalism for determining energy eigenstates of spherical quantum dots and cylindrical quantum wires in the multiple-band envelope-function approximation. The technique is based upon a reformulation of the K·P theory in a basis of eigenstates of total angular momentum. Stationary stales are formed by mixing bulk energy eigenvectors and imposing matching conditions across the heterostructure interface, yielding dispersion relations for eigenenergies in quantum wires and quantum dots. The bound states are studied for the conduction band and the coupled light and heavy holes as a function of radius for the GaAs/Al 2 Ga 1-x As quantum dot. Conduction-band-valence-band coupling is shown to be critical in a "type-II" InAs/GaSb quantum dot, which is studied here for the first time. Quantum-wire valence-subband dispersion and effective masses are determined for GaAs/Al x Ga 1-x As wires of several radii. The masses are found to be independent of wire radius in an infinite-well model, but strongly dependent on wire radius for a finite well, in which the effective mass of the highest-energy valence subband is as low as 0.16m o . Implications of the band-coupling effects on optical matrix elements in quantum wires and dots are discussed.
TL;DR: In this article, the authors derived an adiabatic approximation in the limit that the oscillator frequency is much larger than the characteristic frequency of the two-level system, together with a discussion of its applicability in a system consisting of a Cooper-pair box coupled to a nanomechanical resonator.
Abstract: Recent experiments on quantum behavior in microfabricated solid-state systems suggest tantalizing connections to quantum optics. Several of these experiments address the prototypical problem of cavity quantum electrodynamics: a two-level system coupled to a quantum harmonic oscillator. Such devices may allow the exploration of parameter regimes outside the near-resonance and weak-coupling assumptions of the ubiquitous rotating-wave approximation (RWA), necessitating other theoretical approaches. One such approach is an adiabatic approximation in the limit that the oscillator frequency is much larger than the characteristic frequency of the two-level system. A derivation of the approximation is presented, together with a discussion of its applicability in a system consisting of a Cooper-pair box coupled to a nanomechanical resonator. Within this approximation the time evolution of the two-level-system occupation probability is calculated using both thermal- and coherent-state initial conditions for the oscillator, focusing particularly on collapse and revival phenomena. For thermal-state initial conditions parameter regimes are found in which collapse and revival regions may be clearly distinguished, unlike the erratic evolution of the thermal-state RWA model. Coherent-state initial conditions lead to complex behavior, which exhibits sensitive dependence on the coupling strength and the initial amplitude of the oscillator state. One feature of the regime considered here is that closed-form evaluation of the time evolution may be carried out in the weak-coupling limit, which provides insight into the differences between the thermal- and coherent-state models. Finally, potential experimental observations in solid-state systems, particularly the Cooper-pair box—nanomechanical resonator system, are discussed and found to be promising.
TL;DR: In this paper, the quantum Zakharov equations are applied to two model cases, namely, the four-wave interaction and the decay instability, to describe the nonlinear interaction between quantum Langmuir waves and quantum ion-acoustic waves.
Abstract: Quantum Zakharov equations are obtained to describe the nonlinear interaction between quantum Langmuir waves and quantum ion-acoustic waves. These quantum Zakharov equations are applied to two model cases, namely, the four-wave interaction and the decay instability. In the case of the four-wave instability, sufficiently large quantum effects tend to suppress the instability. For the decay instability, the quantum Zakharov equations lead to results similar to those of the classical decay instability except for quantum correction terms in the dispersion relations. Some considerations regarding the nonlinear aspects of the quantum Zakharov equations are also offered.
TL;DR: In the present work, it is demonstrated how to realize a 1D closed optical lattice experimentally, including a tunable boundary phase twist, which may induce "persistent currents" visible by studying the atoms' momentum distribution.
Abstract: In the present work we demonstrate how to realize a 1D closed optical lattice experimentally, including a tunable boundary phase twist. The latter may induce ''persistent currents,'' visible by studying the atoms' momentum distribution. We show how important phenomena in 1D physics can be studied by physical realization of systems of trapped atoms in ring-shaped optical lattices. A mixture of bosonic and/or fermionic atoms can be loaded into the lattice, realizing a generic quantum system of many interacting particles.