The origin of classical predictability is investigated for the one dimensional harmonic chain considered as a closed quantum mechanical system. By comparing the properties of a family of coarse-grained descriptions of the chain, we conclude that local coarse-grainings in this family are more useful for prediction than nonlocal ones. A quantum mechanical system exhibits classical behavior when the probability is high for histories having the correlations in time implied by classical deterministic laws. But approximate classical determinism holds only for certain coarse-grainings and then only if the initial state of the system is suitably restricted. Coarse-grainings by the values of the hydrodynamic variables (integrals over suitable volumes of densities of approximately conserved quantities) define the histories usually used in classical physics. But what distinguishes this coarse-graining from others? This paper approaches this question by analyzing a family of coarse-grainings for the linear harmonic chain. At one extreme in the family the chain is divided into local groups of $N$ atoms. At the other extreme the $N$ atoms are distributed nonlocally over the whole chain. Each coarse-graining follows the average (center of mass) positions of the groups and ignores the ``internal'' coordinates within each group, these constituting a different environment for each coarse-graining. We conclude that noise, decoherence, and computational complexity favor locality over nonlocality for deterministic predictability.
A protocol of quantum energy teleportation is proposed for a one-dimensional harmonic chain. A coherent-state POVM measurement is performed to coupled oscillators of the chain in the ground state accompanied by energy infusion to the system. This measurement consumes a part of ground state entanglement. Depending on the measurement result, a displacement operation is performed on a distant oscillator accompanied by energy extraction from the zero-point fluctuation of the oscillator. We find that the amount of consumed entanglement is bounded from below by a positive value that is proportional …
We consider a quantum space with rotationally invariant noncommutative algebra of coordinates and momenta. The algebra contains tensors of noncommutativity constructed involving additional coordinates and momenta. In the rotationally invariant noncommutative phase space harmonic oscillator chain is studied. We obtain that noncommutativity affects on the frequencies of the system. In the case of a chain of particles with harmonic oscillator interaction we conclude that because of momentum noncommutativity the spectrum of the center-of-mass of the system is discrete and corresponds to the spectr…
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 provide an explicit method for solving general markovian master equations for quadratic bosonic Hamiltonians with linear bath operators. As an example we consider a one-dimensional quantum harmonic oscillator chain coupled to thermal reservoirs at both ends of the chain. We derive an analytic solution of the Redfield master equation for homogeneous harmonic chain and recover classical results, namely, vanishing temperature gradient and constant heat current in the thermodynamic limit. In the case of the disordered gapped chains we observe universal heat current scaling independent of the ba…
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.
We prove that temporal control of the strengths of springs connecting $N$ harmonic oscillators in a chain provides complete access to all Gaussian states of $N-1$ collective modes. The proof relies on the construction of a suitable basis of cradle modes for the system. An iterative algorithm to reach any desired Gaussian state requires at most $3N(N-1)/2$ operations. We illustrate this capability by engineering squeezed pseudo-phonon states - highly non-local, strongly correlated states that may result from various nonlinear processes. Tunable chains of coupled harmonic oscillators can be impl…
We investigate the mechanism of heat conduction in ordered and disordered harmonic onedimensional chains within the quantum mechanical Langevin method. In the case of the disordered chains we find indications for normal heat conduction which means that there is a finite temperature gradient but we cannot clearly decide whether the heat resistance increases linearly with the chain length. Furthermore we observe characteristic quantum mechanical features like Bose-Einstein statistics of the occupation numbers of the normal modes, freezing of the heat conductivity and the influence of the entangl…
We study the propagation of different types of correlations through a quantum bus formed by a chain of coupled harmonic oscillators. This includes steering, entanglement, mutual information, quantum discord, and Bell-like nonlocality. The whole system consists of the quantum bus (propagation medium) and other quantum harmonic oscillators (sources and receivers of quantum correlations) weakly coupled to the chain. We are particularly interested in using the point of view of transport to spot distinctive features displayed by different kinds of correlations. We found, for instance, that there ar…
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.
We show that a squeezed bath, that acts on the central element of a harmonic chain, can drive the whole system to a steady state featuring a series of nested entangled pairs of oscillators that, ideally, covers the whole chain regardless of its size. We study how to realize this effect in various physical implementations, including optomechanical and superconducting devices, using currently available technologies. In these cases no squeezed fields are actually needed, and the squeezed bath is, instead, simulated by quantum reservoir engineering with bichromatic drives.
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 have simulated the ground states of quantum harmonic oscillators driven either by constant forces of different magnitudes or time-dependent driving forces. The expectation values of position for various combinations of mass, natural angular frequency, and the coupling constant $λ$ were calculated for both driving modes. For constant forcing, coherent states were obtained. The results for both forcing scenarios match the theoretically expected values almost exactly. For the simulations, the Metropolis algorithm was implemented on a discrete time lattice to evaluate the imaginary time path in…
Harmonic oscillator chains connecting two harmonic reservoirs at different constant temperatures cannot act as thermal diodes, irrespective of structural asymmetry. However, here we prove that perfectly harmonic junctions can rectify heat once the reservoirs (described by white Langevin noise) are placed under temperature gradients, which are asymmetric at the two sides, an effect that we term "temperature-gradient harmonic oscillator diodes". This nonlinear diode effect results from the additional constraint -- the imposed thermal gradient at the boundaries. We demonstrate the rectification b…
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…
The nonzero ground-state energy of the quantum mechanical harmonic oscillator implies quantum fluctuations around the minimum of the potential with the mean square value proportional to Planck's constant. In classical mechanics thermal fluctuations occur when the oscillator is coupled to a heat bath of temperature $T$. At finite temperature quantum statistical mechanics allows the description of the transition from pure quantum fluctuations at $T=0$ to classical thermal fluctuations in the high temperature limit. It was early pointed out by Peierls that the mean square thermal fluctuations in …
The quantum dynamics of isolated systems under quench condition exhibits a variety of interesting features depending on the integrable/chaotic nature of system. We study the exact dynamics of trivially integrable system of harmonic chains under a multiple quench protocol. Out of time ordered correlator of two Hermitian operators at large time displays scrambling in the thermodynamic limit. In this limit, the entanglement entropy and the central component of momentum distribution both saturate to a steady state value. We also show that reduced density matrix assumes the diagonal form long after…
The density of ions trapped in a harmonic potential in one dimension is not uniform. Consequently the eigenmodes are not phonons. We calculate the long wavelength modes in the continuum limit, and evaluate the density of states in the short wavelength limit for chains of $N\gg 1$ ions. Remarkably, the results that are found analytically in the thermodynamic limit provide a good estimate of the spectrum of excitations of small chains down to few tens of ions. The spectra are used to compute the thermodynamic functions of the chain. Deviations from extensivity of the thermodynamic quantities are…
We study the entanglement in a chain of harmonic oscillators driven out of equilibrium by preparing the two sides of the system at different temperatures, and subsequently joining them together. The steady state is constructed explicitly and the logarithmic negativity is calculated between two adjacent segments of the chain. We find that, for low temperatures, the steady-state entanglement is a sum of contributions pertaining to left- and right-moving excitations emitted from the two reservoirs. In turn, the steady-state entanglement is a simple average of the Gibbs-state values and thus its s…
We investigate the application of the density-matrix renormalization group (DMRG) algorithm to a one-dimensional harmonic oscillator chain and compare the results with exact solutions, aiming to improve the algorithm efficiency. It has been demonstrated that the algorithm can give quite accurate results if the procedure is proper organized, for example, by using the optimized bases. The errors of calculated ground state energy and the energy gap between the ground state and the first excited state are analyzed, which are found to be critically dependent upon the size of the system or the energ…
We present a study that addresses both the stationary properties of the energy current and quantum correlations in a three-mode chain subjected to Ohmic and super-Ohmic dissipa- tions. An extensive numerical analysis shows that the mean value and the fluctuations of the energy current remain insensitive to the emergence of a rich variety of quantum correlations: such as, two-mode discord and entanglement, bipartite three-mode and genuine tripartite entanglement. The discussion of the numerical results is based on the derived expressions for the stationary properties in terms of the two-time co…
This preprint contains the English translation of the paper "Relaxation dynamics of a quantum chain of harmonic oscillators", which was published by the author in 1980 in the Ukrainian Physical Journal. Comments describing its motivations, background ideas and results are presented as well. This paper addressed to the problem of approach to the thermal equilibrium in an isolated macroscopic quantum system, which was studied on the example of the quantum chain of weakly interacting harmonic oscillators. In the initial state, macroscopic energy was supplied to one oscillator (atom) in the chain.…
The thermodynamic implications for the out-of-equilibrium dynamics of quantum systems are to date largely unexplored, especially for quantum many-body systems. In this paper we investigate the paradigmatic case of an array of nearest-neighbor coupled quantum harmonic oscillators interacting with a thermal bath and subjected to a quench of the inter-oscillator coupling strength. We study the work done on the system and its irreversible counterpart, and characterize analytically the fluctuation relations of the ensuing out-of-equilibrium dynamics. Finally, we showcase an interesting functional l…
We address the problem of heat transport in a chain of coupled quantum harmonic oscillators, exposed to the influences of local environments of various nature, stressing the effects that the specific nature of the environment has on the phenomenology of the transport process. We study in detail the behavior of thermodynamically relevant quantities such as heat currents and mean energies of the oscillators, establishing rigorous analytical conditions for the existence of a steady state, whose features we analyse carefully. In particular we assess the conditions that should be faced to recover t…
We consider two chains, each made of $N$ independent oscillators, immersed in a common thermal bath and study the dynamics of their mutual quantum correlations in the thermodynamic, large-$N$ limit. We show that dissipation and noise due to the presence of the external environment are able to generate collective quantum correlations between the two chains at the mesoscopic level. The created collective quantum entanglement between the two many-body systems turns out to be rather robust, surviving for asymptotically long times even for non vanishing bath temperatures.
Cold ions in anisotropic harmonic potentials can form ion chains of various sizes. Here, the density of ions is not uniform, thus the eigenmodes are not phononic-like waves. We study chains of N>>1 ions and evaluate analytically the long wavelength modes and the density of states in the short wavelength limit. These results reproduce with good approximation the dynamics of chains consisting of dozens of ions. Moreover, they allow to determine the critical transverse frequency required for the stability of the linear structure, which is found in agreement with results obtained by different theo…
We inverse engineer fast rotations of a linear trap with two ions for a predetermined rotation angle and time, avoiding final excitation. Different approaches are analyzed and compared when the ions are of the same species or of different species. The separability into dynamical normal modes for equal ions in a common harmonic trap, or for different ions in non-harmonic traps with up to quartic terms allows for simpler computations of the rotation protocols. For non-separable scenarios, in particular for different ions in a harmonic trap, rotation protocols are also found using more costly num…
We address the problem of spring-like coupling between bosons in an open chain configuration where the counter-rotating terms are explicitly included. We show that fruitful insight can be gained by decomposing the time-evolution operator of this problem into a pattern of linear-optics elements. This allows us to provide a clear picture of the effects of the counter-rotating terms in the important problem of long-haul entanglement distribution. The analytic control over the variance matrix of the state of the bosonic register allows us to track the dynamics of the entanglement. This helps in de…
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 …
As known all physical properties of solids are described well by the system of quantum linear harmonic oscillators. It is shown in the present paper that the system consisting of classical linear harmonic oscillators having temperature dependent masses or (and) frequencies has the same partition function as the system consisting of quantum linear harmonic oscillators having temperature independent masses and frequencies while the means of the square displacements of the positions of the oscillators from their mean positions for the system consisting of classical linear harmonic oscillators hav…
This paper has been withdrawn by the authors due to a problem with *efficiently* predicting the large fourier coefficients. It is being reworked and will be resubmitted in the near future.
Ladder operators for the simplest version of a rationally extended quantum harmonic oscillator (REQHO) are constructed by applying a Darboux transformation to the quantum harmonic oscillator system. It is shown that the physical spectrum of the REQHO carries a direct sum of a trivial and an infinite-dimensional irreducible representation of the polynomially deformed bosonized osp(1|2) superalgebra. In correspondence with this the ground state of the system is isolated from other physical states but can be reached by ladder operators via non-physical energy eigenstates, which belong to either a…
The balance of gain and loss in an open system may maintain certain Hermitian dynamical behaviors, which can be hardly observed in a popular Hermitian system. In this paper, we systematically study a 1D PT -symmetry non-Hermitian SSH model with open boundary condition based on exact approximate solution. We show that the long-wave length standing-wave modes can be achieved within the linear dispersion region when the system is tuned at the exceptional point (EP). The whole Hilbert space can be decomposed into two quasi-Hermitian subspaces, which are consisted of positive and negative energy le…
We consider a linear chain of quantum harmonic oscillators, in which the number of the individual oscillators is given by an arbitrary number N, and each oscillator is coupled at an arbitrary strength kappa to its nearest neighbors ("intra-coupling"), as well as the two end oscillators of the chain are coupled at an arbitrary strength c_nu to two separate baths at arbitrarily different temperatures, respectively. We derive an exact closed expression for the steady-state heat current flowing from a hot bath through the chain to a cold bath, in the Drude-Ullersma damping model going beyond the M…
We report that upon excitation by a single pulse, the classical harmonic oscillator immersed in classical electromagnetic zero-point radiation, as described by random electrodynamics, exhibits a quantized excitation spectrum in agreement to that of the quantum harmonic oscillator. This numerical result is interesting in view of the generally accepted idea that classical theories do not support quantized energy spectra.
In a previous paper a formalism to analyze the dynamical evolution of classical and quantum probability distributions in terms of their moments was presented. Here the application of this formalism to the system of a particle moving on a potential is considered in order to derive physical implications about the classical limit of a quantum system. The complete set of harmonic potentials is considered, which includes the particle under a uniform force, as well as the harmonic and the inverse harmonic oscillators. In addition, as an example of anharmonic system, the pure quartic oscillator is an…
The Wigner function is a quantum analogue of the classical joined distribution of position and momentum. As such is should be a good tool to study quantum-classical correspondence. In this paper, the classical limit of the Wigner function is shown using the quantum harmonic oscillator as an example. The Wigner function is found exactly for all states. The semi-classical wavefunctions for highly excited states are used as the approach to the classical limit. Therefore, one can found the classical limit of the Wigner function for highly excited states and shown that it gives the classical microc…
We propose a simplest detector of harmonic vibrations with micro amplitudes and low frequencies, i.e. the detector consisting of one atomic beam. Here the atomic beam is induced by a plane harmonic wave and has a classical collective harmonic vibrations, which vibrant directions are perpendicular to the wave vectors of atomic beam. Compared with the detector consisting of atomic Mach-Zehnder interferometer, the new detector has two advantages: (1) it is suitable for the detection of the harmonic vibrations induced either by a longitudinal plane harmonic wave or by a transverse plane harmonic w…
We construct a contour function for the entanglement entropies in generic harmonic lattices. In one spatial dimension, numerical analysis are performed by considering harmonic chains with either periodic or Dirichlet boundary conditions. In the massless regime and for some configurations where the subsystem is a single interval, the numerical results for the contour function are compared to the inverse of the local weight function which multiplies the energy-momentum tensor in the corresponding entanglement hamiltonian, found through conformal field theory methods, and a good agreement is obse…
In this work we study 2- and 3-body oscillators with quadratic and sextic pairwise potentials which depend on relative distances, $|{\bf r}_i - {\bf r}_j |$, between particles. The two-body harmonic oscillator is two-parametric and can be reduced to a one-dimensional radial Jacobi oscillator, while in the 3-body case such a reduction is not possible in general. Our study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only ($S$-states). We pay special attention to the cases where the masses of the particles and spring constants are …
The classical and quantum solutions of a nonlinear model describing harmonic oscillators on the sphere and the hyperbolic plane, derived in polar coordinates in a recent paper [Phys.\ Lett.\ A 379 (2015) 1589], are extended by the inclusion of an isotonic term.
A simple derivation of the classical solutions of a nonlinear model describing a harmonic oscillator on the sphere and the hyperbolic plane is presented in polar coordinates. These solutions are then related to those in cartesian coordinates, whose form was previously guessed. In addition, the nature of the classical orthogonal polynomials entering the bound-state radial wavefunctions of the corresponding quantum model is identified.
