We investigate the entanglement properties of thermal states of the harmonic lattice in one, two and three dimensions. We establish the value of the critical temperature for entanglement between neighbouring sites and give physical reasons. Further sites are shown to be entangled only due to boundary effects. Other forms of entanglement are addressed in the second part of the paper by using the energy as witness of entanglement. We close with a comprehensive diagram showing the different phases of entanglement versus complete separability and propose techniques to swap and tune entanglement experimentally.
We investigate the second-order non-linear interaction as a means to generate entanglement between fields of differing wavelengths. And show that perfect entanglement can, in principle, be produced between the fundamental and second harmonic fields in these processes. Neither pure second harmonic generation, nor parametric oscillation optimally produce entanglement, such optimal entanglement is rather produced by an intermediate process. An experimental demonstration of these predictions should be imminently feasible.
We study the boson-parafermion entanglement of the parasupersymmetric coherent states of the harmonic oscillator and derive the degree of entanglement in terms of the concurrence. The conditions for obtaining the maximal entanglement is also examined, and it is shown that in the usual supersymmetry situation we can obtain maximally entangled Bell states.
We study the harmonic entanglement and squeezing in a two-mode radiation produced in a degenerate parametric down conversion process coupled to a two-mode vacuum reservoir employing the linearization procedure. It is found that there is a quadrature entanglement between the harmonically related fundamental and second-harmonic modes and the superimposed radiation exhibits a significant two-mode squeezing. The entanglement exits even when there is no two-mode squeezing, since the correlation leading to these phenomena are essentially different. In addition, the more the external coherent light i…
We analyze the quantum entanglement between two interacting atoms trapped in a spherical harmonic potential. At ultra-cold temperature, ground state entanglement is generated by the dominated s-wave interaction. Based on a regularized pseudo-potential Hamiltonian, we examine the quantum entanglement by performing the Schmidt decomposition of low-energy eigenfunctions. We indicate how the atoms are paired and quantify the entanglement as a function of a modified s-wave scattering length inside the trap.
The limitations on the production and profitability of entanglement in a harmonic chain under strong driving are considered. We report on the limits of attainable entanglement for a given set of squeezings of the eigenmodes, showing that the higher the entanglement the more oscillatory and thus less easy to profit from. We also comment on propagation properties of entanglement, discussing the role of fast rotating terms and illustrating several issues with the example of a sudden switch of the coupling.
The entanglement between a Pauli-like two-level system and a quantum harmonic oscillator enhanced by an interaction between them and a $δ$-pulse sequence is studied, with the decoherence due to their coupling with a Markovian bath. Without the Markovian bath, the entanglement is enhanced to maximum possible values. With the Markovian bath, the entanglement is enhanced up to some time and then dissipated, with the system in thermal equilibrium with the Markovian bath after a very long time. The time for achieving the maximum entanglement shows discontinuous jumps over the parameters of decohere…
We investigate quantum entanglement between two (spin-1/2) fermions inside a cylindrical harmonic trap, making use of the von Neumann entropy for the reduced single particle density matrix as the pure state entanglement measure. We explore the dependence of pair entanglement on the geometry and strength of the trap and on the strength of the pairing interaction over the complete range of the effective BCS to BEC crossover. Our result elucidates an interesting connection between our model system of two fermions and that of two interacting bosons.
We analyse a nonlinear optical system which uses cascaded nonlinearities to produce both second and fourth harmonic outputs from an input field at the fundamental frequency. Using fully quantum equations of motion, we show that the system produces quadrature squeezed outputs which exhibit bipartite entanglement, EPR-steering, and asymmetric steering across a two octave frequency range.
We compare the bipartite entanglement and EPR-steering properties of the two different schemes which produce third harmonic optical fields from an input field at the fundamental frequency. The first scheme uses second harmonic cascaded with sum-frequency generation while the second uses triply degenerate four-wave mixing, also known as direct third harmonic generation. We examine and compare the two schemes in both the travelling wave and intra-cavity configurations. We find that both schemes produce continuous-variable bipartite entanglement and EPR-steering. The direct scheme produces a grea…
We apply the Walborn [Walborn S P, Taketani B G, Salles A, Toscano F and de Matos Filho R L 2009 Entropic Entanglement Criteria for Continuous Variables Phys. Rev. Lett. 103 160505] Shannon Entropic Entanglement Criterion (SEEC), to simple harmonic oscillators, a system typically discussed in undergraduate quantum mechanics courses. In particular, we investigate the entanglement of a system of coupled harmonic oscillators. A simple form of entanglement criterion in terms of their interaction is found. It is shown that a pair of interacting ground state coupled oscillators are more likely to be…
We address the presence of non-distillable (bound) entanglement in natural many-body systems. In particular, we consider standard harmonic and spin-1/2 chains, at thermal equilibrium and characterized by few interaction parameters. The existence of bound entanglement is addressed by calculating explicitly the negativity of entanglement for different partitions. This allows to individuate a range of temperatures for which no entanglement can be distilled by means of local operations, despite the system being globally entangled. We discuss how the appearance of bound entanglement can be linked t…
The entanglement properties of the phase transition in a two dimensional harmonic lattice, similar to the one observed in recent ion trap experiments, are discussed both, for finite number of particles and thermodynamical limit. We show that for the ground state at the critical value of the trapping potential two entanglement measures, the negativity between two neighbouring sites and the block entropy for blocks of size 1, 2 and 3, change abruptly. Entanglement thus indicates quantum phase transitions in general; not only in the finite dimensional case considered in [Phys. Rev. Lett. {\bf 93}…
We study the entanglement evolution between two harmonic oscillators having different free frequencies each leaking into an independent bath. We use an exact solution valid in the weak coupling limit and in the short time non-Markovian regime. The reservoirs are identical and characterized by an Ohmic spectral distribution with Lorents-Drude cut-off. This work is an extension of the case reported in [Phys. Rev. A 80, 062324 (2009)] where the oscillators have the same free frequency.
We study the ground-state entanglement of one-dimensional harmonic chains that are coupled to each other by a collective interaction as realized e.g. in an anisotropic ion crystal. Due to the collective type of coupling, where each chain interacts with every other one in the same way,the total system shows critical behavior in the direction orthogonal to the chains while the isolated harmonic chains can be gapped and non-critical. We derive lower and most importantly upper bounds for the entanglement,quantified by the von Neumann entropy, between a compact block of oscillators and its environm…
We introduce a special class of truncated Weyl-Heisenberg algebra and discuss the corresponding Hilbertian and analytical representations. Subsequently, we study the effect of a quantum network of beam splitting on coherent states of this nonlinear class of harmonic oscillators. We particularly focus on quantum networks involving one and two beam splitters and examine the degree of bipartite as well as tripartite entanglement using the linear entropy.
The exact dynamics of the entanglement between two harmonic modes generated by an angular momentum coupling is examined. Such system arises when considering a particle in a rotating anisotropic harmonic trap or a charged particle in a fixed harmonic potential in a magnetic field, and exhibits a rich dynamical structure, with stable, unstable and critical regimes according to the values of the rotational frequency or field and trap parameters. Consequently, it is shown that the entanglement generated from an initially separable gaussian state can exhibit quite distinct evolutions, ranging from …
We examine the entanglement induced by an angular momentum coupling between two harmonic systems. The Hamiltonian corresponds to that of a charged particle in a uniform magnetic field in an anisotropic quadratic potential, or equivalently, to that of a particle in a rotating quadratic potential. We analyze both the vacuum and thermal entanglement, obtaining analytic expressions for the entanglement entropy and negativity through the gaussian state formalism. It is shown that vacuum entanglement diverges at the edges of the dynamically stable sectors, increasing with the angular momentum and sa…
A general and in principle exact approach for the continuous variable entanglement in a system of coupled harmonic oscillators in contact with a thermal bath is formulated. This allows a generalization to describe entanglement's existence between two sites in any system of this kind. It is employed a method of simultaneous diagonalization of two quadratic forms to obtain the uncoupled quantum Hamiltonian. Making use of the transformations that uncouple the system, the covariance matrix for two positions is constructed, and through the positive partial transpose criterion (PPT-criterion), the c…
For both cases with and without interactions, bipartite entanglement of two-fermions from a Fermi gas in a trap is investigated. We show how the entanglement depends on the locations of the two fermions and the total number of particles. Fermions at the verge of trap have longer entanglement distance (beyond it, the entanglement disappears) than those in the center. We derive a lower limitation to the average overlapping for two entangled fermions in the BCS ground state, it is shown to be $\sqrt{Q/2M}$, a function of Cooper pair number $Q$ and total number of occupied energy levels $M$.
We study the structure of vacuum entanglement for two complimentary segments of a linear harmonic chain, applying the modewise decomposition of entangled gaussian states discussed in \cite {modewise}. We find that the resulting entangled mode shape hierarchy shows a distinctive layered structure with well defined relations between the depth of the modes, their characteristic wavelength, and their entanglement contribution. We re-derive in the strong coupling (diverging correlation length) regime, the logarithmic dependence of entanglement on the segment size predicted by conformal field theory…
We investigate a simple arrangement of coupled harmonic oscillators which brings out some interesting effects concerning creation of entanglement. It is well known that if each member in a linear chain of coupled harmonic oscillators is prepared in a ``classical state'', such as a pure coherent state or a mixed thermal state, no entanglement is created in the rotating wave approximation. On the other hand, if one of the oscillators is prepared in a nonclassical state (pure squeezed state, for instance), entanglement may be created between members of the chain. In the setup considered here, we …
We discuss the relation between entanglement and criticality in translationally invariant harmonic lattice systems with non-randon, finite-range interactions. We show that the criticality of the system as well as validity or break-down of the entanglement area law are solely determined by the analytic properties of the spectral function of the oscillator system, which can easily be computed. In particular for finite-range couplings we find a one-to-one correspondence between an area-law scaling of the bi-partite entanglement and a finite correlation length. This relation is strict in the one-d…
In the present paper we study the entanglement properties of thermal (a.k.a. Gibbs) states of quantum harmonic oscillator systems as functions of the Hamiltonian and the temperature. We prove the physical intuition that at sufficiently high temperatures the thermal state becomes fully separable and we deduce bounds on the critical temperature at which this happens. We show that the bound becomes tight for a wide class of Hamiltonians with sufficient translation symmetry. We find, that at the crossover the thermal energy is of the order of the energy of the strongest normal mode of the system a…
We study the entanglement dynamics of the two two-level atoms coupling with a single-mode polarized cavity field after incorporating the atomic centers of mass classical harmonic vibrations with micro amplitudes and low frequencies. We propose a quantitative vibrant factor to modify the concurrence of the two atoms states. When the vibrant frequencies are very low, we obtain that: (i) the factor depends on the relative vibrant displacements and the initial phases rather than the absolute amplitudes, and reduces the concurrence to three orders of magnitude; (ii) the concurrence increases with t…
We propose a photonics-based, continuous-variable (CV) form of remote entanglement utilizing strictly second-order nonlinear optical interactions that does not require the implementation of a state-projective measurement (i.e. remote entanglement without conditioning). This scheme makes use of two separate down-converters, wherein the corresponding nonlinear crystals are driven by strong classical fields as prescribed by the parametric approximation, as well as a fully quantum mechanical model of nondegenerate second harmonic generation (SHG) whose evolution is described by the trilinear Hamil…
The dynamics of mixedness and entanglement is examined by solving the time-dependent Schrödinger equation for three coupled harmonic oscillator system with arbitrary time-dependent frequency and coupling constants parameters. We assume that part of oscillators is inaccessible and remaining oscillators accessible. We compute the dynamics of entanglement between inaccessible and accessible oscillators. In order to show the dynamics pictorially we introduce three quenched models. In the quenched models both mixedness and entanglement exhibit oscillatory behavior in time with multi-frequencies. It…
We study the dynamics and redistribution of entanglement and coherence in three time-dependent coupled harmonic oscillators. We resolve the Schrödinger equation by using time-dependent Euler rotation together with a linear quench model to obtain the state of vacuum solution. Such state can be translated to the phase space picture to determine the Wigner distribution. We show that its Gaussian matrix $\mathbb{G}(t)$ can be used to directly cast the covariance matrix $σ(t)$. To quantify the mixedness and entanglement of the state one uses respectively linear and von Neumann entropies for three c…
The dynamics of entanglement and uncertainty relation is explored by solving the time-dependent Schrödinger equation for coupled harmonic oscillator system analytically when the angular frequencies and coupling constant are arbitrarily time-dependent. We derive the spectral and Schmidt decompositions for vacuum solution. Using the decompositions we derive the analytical expressions for von Neumann and Rényi entropies. Making use of Wigner distribution function defined in phase space, we derive the time-dependence of position-momentum uncertainty relations. In order to show the dynamics of enta…
Quantum fluctuation and quantum entanglement of the pump field reflected from an optical cavity for type II second harmonic generation are theoretically analyzed. The correlation spectra between the quadratures of the reflected subharmonic fields are interpreted in terms of pump parameter, intracavity losses and normalized frequency. Large correlation degrees of both amplitude and phase quadratures can be accessed in a triple resonant cavity before the pitchfork bifurcation occurs. The two reflected subharmonic fields are in an entangled state with the quantum correlation of phase quadratures …
We derive explicitly the thermal state of the two coupled harmonic oscillator system when the spring and coupling constants are arbitrarily time-dependent. In particular, we focus on the case of sudden change of frequencies. In this case we compute purity function, Rényi and von Neumann entropies, and mutual information analytically and examine their temperature-dependence. We also discuss on the thermal entanglement phase transition by making use of the negativity-like quantity. Our calculation shows that the critical temperature $T_c$ increases with increasing the difference between the init…
Multi-mode entanglement is investigated in the system composed of $N$ coupled identical harmonic oscillators interacting with a common environment. We treat the problem very general by working with the Hamiltonian without the rotating-wave approximation and by considering the environment as a non-Markovian reservoir to the oscillators. We invoke an $N$-mode unitary transformation of the position and momentum operators and find that in the transformed basis the system is represented by a set of independent harmonic oscillators with only one of them coupled to the environment. Working in the Wig…
We investigate entanglement between collective operators of two blocks of oscillators in an infinite linear harmonic chain. These operators are defined as averages over local operators (individual oscillators) in the blocks. On the one hand, this approach of "physical blocks" meets realistic experimental conditions, where measurement apparatuses do not interact with single oscillators but rather with a whole bunch of them, i.e., where in contrast to usually studied "mathematical blocks" not every possible measurement is allowed. On the other, this formalism naturally allows the generalization …
We revisist the issue of entanglement of thermal equilibrium states in composite quantum systems. The possible scenarios are exemplified in bipartite qubit/qubit and qubit/qutrit systems.
We propose the definition of the geometric measure of entanglement for continuous variable states. On the basis of this definition we examine entanglement of the graph states obtained as a result of action of a unitary operator on the ground state of a system of $N$ noninteracting harmonic oscillators. We find that the entanglement of a harmonic oscillator with other ones is defined by the value of its vertex degree.
We study the change of entanglement under general linear transformation of modes in a bosonic system and determine the conditions under which entanglement can be generated under such transformation. As an example we consider the thermal entanglement between the vibrational modes of two coupled oscillators and determine the temperature above which quantum correlations are destroyed by thermal fluctuations.
We study pairwise thermal entanglement in three-qubit Heisenberg models and obtain analytic expressions for the concurrence. We find that thermal entanglement is absent from both the antiferromagnetic $XXZ$ model, and the ferromagnetic $XXZ$ model with anisotropy parameter $Δ\ge 1$. Conditions for the existence of thermal entanglement are discussed in detail, as is the role of degeneracy and the effects of magnetic fields on thermal entanglement and the quantum phase transition. Specifically, we find that the magnetic field can induce entanglement in the antiferromagnetic $XXX$ model, but cann…
We study the entanglement properties of a closed chain of harmonic oscillators that are coupled via a translationally invariant Hamiltonian, where the coupling acts only on the position operators. We consider the ground state and thermal states of this system, which are Gaussian states. The entanglement properties of these states can be completely characterized analytically when one uses the logarithmic negativity as a measure of entanglement.
The correct form of the Schmidt decomposition of the stationary wave functions for a system of two interacting particles trapped in a two-dimensional harmonic potential is given
The thermal entanglement of two spin-1 atoms with nonlinear couplings in optical lattices is investigated in this paper. It is found that the nonlinear couplings favor the thermal entanglement creating. The dependence of the thermal entanglement in this system on the linear coupling, the nonlinear coupling, the magnetic field and temperature is also presented. The results show that the nonlinear couplings really change the feature of the thermal entanglement in the system, increasing the nonlinear coupling constant increases the critical magnetic field and the threshold temperature.
The upper and lower bounds of entanglement of formation are given for two mode squeezed thermal state. The bounds are compared with other entanglement measure or bounds. The entanglement distillation and the relative entropy of entanglement of infinitive squeezed state are obtained at the postulation of hashing inequality.
We study the long-time evolution of the bipartite entanglement in translationally invariant gapped harmonic lattice systems with finite-range interactions. A lower bound for the von Neumann entropy is derived in terms of the purity of the reduced density matrix. It is shown that starting from an initially Gaussian state the entanglement entropy increases at least linearly in time. This implies that the dynamics of gapped (non-critical) harmonic lattice systems cannot be efficiently simulated by algorithms based on matrix-product decompositions of the quantum state.
We show that the thermal entanglement in a spin system using only magnetic susceptibility measurements is restricted to the insulator materials. We develop a generalization of the thermal entanglement witness that allows us to get information about the system entanglement with variable local spin lengths that can be used experimentally in conductor or insulator materials. As an application, we study thermal entanglement for the half-filled Hubbard model for linear, square and cubic clusters. We note that it is the itinerancy of electrons that favors the entanglement. Our results suggest a weak…
I study the three parameters bipartite quantum Gaussian state called squeezed asymmetric thermal state, calculate Gaussian entanglement of formation analytically and the up bound of relative entropy of entanglement, compare them with coherent information of the state.Based on the result obtained, one can determine the relative entropy of entanglement of the state with infinitive squeezing.
The entanglement between two identical two-level atoms interacting with two mode thermal field through a nondegenerate two-photon process has been suggested. It has been shown that for some atomic initial state the entanglement induced by nondegenerate two-photon interaction is larger than that induced by one-photon and degenerate two-photon processes.
We investigate the thermal entanglement in two superconducting qubits for arbitrary interaction strength and ground state frequencies. We calculate the concurrence of the system to quantify the thermal entanglement. We suggest a scheme, where an external tunable coupler qubit sandwich between two superconducting qubits generates entanglement. The behavior of concurrence is analyzed for three different cases, in which we consider the effects of the temperature, the qubit-qubit effective coupling strength, and the qubit frequencies on the thermal entanglement. What deserves mentioning here is th…
We study the thermal entanglement in a two-spin-qutrit system with anisotropy in the exchange coupling between two spins. We use the realignment criterion to distinguish the entangled states, and the negativity for measuring the entanglement in this system. We see that the anisotropy can provide an additional parameter for enhancing the entanglement.
We explore spin-orbit thermal entanglement in rare-earth ions, based on a witness obtained from mean energies. The entanglement temperature $T_{E}$, below which entanglement emerges, is found to be thousands of kelvin above room temperature for all light rare earths. This demonstrate the robustness to environmental fluctuations of entanglement between internal degrees of freedom of a single ion.
We compute the pair entanglement between two interacting bosons in a two dimensional (2D)isotropic harmonic trap. The interaction potential is modeled by a 2D regularized pseudo-potential. By analytically decomposing the wave function into the single particle basis, we show the dependency of the pair entanglement on the scattering length. Our results turn out to be in good agreements with earlier results using a quasi-2D geometry.
We study the effect of inhomogeneities in the magnetic field on the thermal entanglement of a two spin system. We show that in the ferromagnetic case a very small inhomogeneity is capable to produce large values of thermal entanglement. This shows that the absence of entanglement in the ferromagnetic Heisenberg system is highly unstable against inhomogeneoity of magnetic fields which is inevitably present in any solid state realization of qubits.
We study the robustness of multipartite entanglement of the ground state of the one-dimensional spin 1/2 XY model with a transverse magnetic field in the presence of thermal excitations, by investigating a threshold temperature, below which the thermal state is guaranteed to be entangled. We obtain the threshold temperature based on the geometric measure of entanglement of the ground state. The threshold temperature reflects three characteristic lines in the phase diagram of the correlation function. Our approach reveals a region where multipartite entanglement at zero temperature is high but …
We study the Von Neumann and Rényi entanglement entropy of long-range harmonic oscillators (LRHO) by both theoretical and numerical means. We show that the entanglement entropy in massless harmonic oscillators increases logarithmically with the sub-system size as $S=\frac{c_{eff}}{3}\log l$. Although the entanglement entropy of LRHO's shares some similarities with the entanglement entropy at conformal critical points we show that the Rényi entanglement entropy presents some deviations from the expected conformal behavior. In the massive case we demonstrate that the behavior of the entanglement…
We consider a quantum harmonic oscillator linearly coupled to a bath of harmonic oscillators and evaluate the degree of entanglement between system and bath using the negativity as an exact entanglement measure. We establish the existence of a critical temperature above which the system-bath negativity vanishes. Our results imply that system-bath entanglement is not responsible for the violation of the Clausius inequality observed in the low temperature/strong coupling regime [Phys. Rev. Lett. 85, 1799 (2000)], as the latter still occurs well above the critical temperature.
