This article investigates entanglement of the motional states of massive coupled oscillators. The specific realization of an idealized diatomic molecule in one-dimension is considered, but the techniques developed apply to any massive particles with two degrees of freedom and a quadratic Hamiltonian. We present two methods, one analytic and one approximate, to calculate the interatomic entanglement for Gaussian and non-Gaussian pure states as measured by the purity of the reduced density matrix. The cases of free and trapped molecules and hetero- and homonuclear molecules are treated. In general, when the trap frequency and the molecular frequency are very different, and when the atomic masses are equal, the atoms are highly-entangled for molecular coherent states and number states. Surprisingly, while the interatomic entanglement can be quite large even for molecular coherent states, the covariance of atomic position and momentum observables can be entirely explained by a classical model with appropriately chosen statistical uncertainty.
This article investigates entanglement of the motional states of massive coupled oscillators. The specific realization of an idealized diatomic molecule in one-dimension is considered, but the techniques developed apply to any massive particles with two degrees of freedom and a quadratic Hamiltonian. We present two methods, one analytic and one approximate, to calculate the interatomic entanglement for Gaussian and non-Gaussian pure states as measured by the purity of the reduced density matrix. The cases of free and trapped molecules and hetero- and homonuclear molecules are treated. In general, when the trap frequency and the molecular frequency are very different, and when the atomic masses are equal, the atoms are highly-entangled for molecular coherent states and number states. Surprisingly, while the interatomic entanglement can be quite large even for molecular coherent states, the covariance of atomic position and momentum observables can be entirely explained by a classical model with appropriately chosen statistical uncertainty.
This article investigates entanglement of the motional states of massive coupled oscillators. The specific realization of an idealized diatomic molecule in one-dimension is considered, but the techniques developed apply to any massive particles with two degrees of freedom and a quadratic Hamiltonian. We present two methods, one analytic and one approximate, to calculate the interatomic entanglement for Gaussian and non-Gaussian pure states as measured by the purity of the reduced density matrix. The cases of free and trapped molecules and hetero- and homonuclear molecules are treated. In general, when the trap frequency and the molecular frequency are very different, and when the atomic masses are equal, the atoms are highly-entangled for molecular coherent states and number states. Surprisingly, while the interatomic entanglement can be quite large even for molecular coherent states, the covariance of atomic position and momentum observables can be entirely explained by a classical model with appropriately chosen statistical uncertainty.
Harmonic chains are used as a model in quantum information theory for ion traps and simple solid state systems. Entanglement is a quantum effect in which observations made on one object instantaneously affect measurement outcomes on the other. This study investigates interparticle entanglement in finite chains of coupled harmonic oscillators as a function of the vibrational modes, mode excitations, and bipartition of oscillators. The methods used and results extend previous work for the Gaussian ground state to excited states. Entanglement is analyzed by calculating the purity of the reduced density matrix of the combined wavefunctions of the oscillators in the chain, tracing over subensembles. Analytic and numerical results for a varying effective spring constants between the particles and number of particles in the chain are presented. Interesting correlations between the symmetries of the vibrational modes and their excitations, and the symmetries of the partitions were discovered. Results show that higher excitation numbers result in higher entanglement. Adding energy to the modes will never decrease entanglement, however, the relationship between energy and entanglement is not simple. Different combinations of mode energy excitations will produce the same entanglement, even if the total energy of the system differs. Patterns indicate that the combination of symmetry in mode shapes and partition determines excitation number purity equivalence classes.
We consider the quantum entanglement of the electronic and vibrational degrees of freedom in molecules with tendencies towards double welled potentials. In these bipartite systems, the von Neumann entropy of the reduced density matrix is used to quantify the electron-vibration entanglement for the lowest two vibronic wavefunctions obtained from a model Hamiltonian based on coupled harmonic diabatic potential-energy surfaces. Significant entanglement is found only in the region in which the ground vibronic state contains a density profile that is bimodal (i.e., contains two separate local maxima). However, in this region two distinct types of density and entanglement profiles are found: one type arises purely from the degeneracy of energy levels in the two potential wells and is destroyed by slight asymmetry, while the other arises through strong interactions between the diabatic levels of each well and is relatively insensitive to asymmetry. These two distinct types are termed fragile degeneracy-induced entanglement and persistent entanglement, respectively. Six classic molecular systems describable by two diabatic states are considered: ammonia, benzene, BNB, pyridine excited triplet states, the Creutz-Taube ion, and the radical cation of the "special pair" of chlorophylls involved in photosynthesis. These chemically diverse systems are all treated using the same general formalism and the nature of the entanglement that they embody is elucidated.
We investigate the entanglement between electronic and vibrational degrees of freedom produced by a vibronic coupling in a molecular system described in the Born-Oppenheimer approximation. Entanglement in a pure state of the Hilbert space H=Hel⨂Hvib is quantified using the von Neumann entropy of the reduced density matrix and the reduced linear entropy. Expressions for these entanglement measures are derived for the $2\ifmmode\times\else\texttimes\fi{}{N}_{v}$ and $3\ifmmode\times\else\texttimes\fi{}{N}_{v}$ cases of the bipartite entanglement, where 2 and 3 are the dimensions of the electronic Hilbert space Hel, and Nv is the dimension of Hvib. We study the entanglement dynamics for two electronic states coupled by a laser pulse (a $2\ifmmode\times\else\texttimes\fi{}{N}_{v}$ case), taking as an example a coupling between the ${a}^{3}{{\ensuremath{\Sigma}}_{u}}^{+}(6s,6s)$ and 1g(6s,6p3/2) states of the Cs2 molecule. The reduced linear entropy expression obtained for the $3\ifmmode\times\else\texttimes\fi{}{N}_{v}$ case is used to follow the entanglement evolution in a scheme proposed for the control of the vibronic dynamics in a Cs2 cold molecule, implying the ${a}^{3}{{\ensuremath{\Sigma}}_{u}}^{+}(6s,6s)$, ${0}_{g}{}^{\ensuremath{-}}(6s,6{p}_{3/2})$, and ${0}_{g}{}^{\ensuremath{-}}(6s,5d)$ electronic states, which are coupled by a nonadiabatic radial coupling and a sequence of chirped laser pulses.
Some of the most enduring questions in physics--including the quantum measurement problem and the quantization of gravity--involve the interaction of a quantum system with a classical environment. Two linearly coupled harmonic oscillators provide a simple, exactly soluble model for exploring such interaction. Even the ground state of a pair of identical oscillators exhibits effects on the quantum nature of one oscillator, e.g., a diminution of position uncertainty, and an increase in momentum uncertainty and uncertainty product, from their unperturbed values. Interaction between quantum and classical oscillators is simulated by constructing a quantum state with one oscillator initially in its ground state, the other in a coherent or Glauber state. The subsequent wave function for this state is calculated exactly, both for identical and distinct oscillators. The reduced probability distribution for the quantum oscillator, and its position and momentum expectation values and uncertainties, are obtained from this wave function. The oscillator acquires an oscillation amplitude corresponding to a beating between the normal modes of the system; the behavior of the position and momentum uncertainties can become quite complicated. For oscillators with equal unperturbed frequencies, i.e., at resonance, the uncertainties exhibit a time-dependent quantum squeezing which can be extreme.
We investigate gravitationally generated entanglement in two quantum harmonic oscillators induced by quadratic (Stelle) gravity, including corrections up to 1.5 post-Newtonian order. Starting from the quadratic action, we derive the effective two-body Hamiltonian for two harmonically trapped masses, incorporating the contributions of the massive spin-2 ghost (m2) and massive spin-0 (m0) degrees of freedom of the gravitational field. For two quantised oscillators prepared in their ground state, we compute the von Neumann and Rényi entropies of the reduced state and identify a frequency at which the gravitationally-induced entanglement vanishes due to cancellation between relativistic momentum squeezing and quantum-delocalisation-induced position squeezing. We further analyze the cancellation frequency and derive the approximate constraint m0<34m2 for the spin-2 and spin-0 modes. This relation follows from demanding a stable harmonic oscillator description. Finally, we study gravitationally-induced concurrence in a non-Gaussian setup and show how quadratic gravity modifies the entanglement generated between two spatial superpositions. The concurrence can approach O(1) for certain choices of mass, spatial superposition, particle distance, and spin-2 and spin-0 modes. The concurrence will deviate from Newtonian gravity at certain particle separations, depending on the energy of the spin-2 and spin-0 modes. For example, spin modes as low as 0.0197 eV become distinguishable from Newtonian gravity at a distance d∼40μm. This allows us to constrain the spin-2 and spin-0 masses in experiments.