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arXiv:0912.4603

Entanglement between Atoms in a Diatomic Molecule

Nathan L. Harshman, William F. Flynn

Published Dec 23, 2009• 0 citations
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Abstract

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.

Related papers

Entanglement in massive coupled oscillators

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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.

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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.

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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 $\mathcal{H}={\mathcal{H}}_{\mathrm{el}}⨂{\mathcal{H}}_{\mathrm{vib}}$ 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 $\mathcal{H}$${}_{\mathrm{el}}$, and ${N}_{v}$ is the dimension of $\mathcal{H}$${}_{\mathrm{vib}}$. 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 ${1}_{g}(6s,6{p}_{3/2})$ states of the Cs${}_{2}$ 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 Cs${}_{2}$ 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.

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Entanglement in Quantum Harmonic Chains

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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.

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The stability conditions of diatomic molecules via analogy with the stability theory of lasers

J. Jahanpanah, M. Esmaeilzadeh

The vibrational motion equations of both homo- and hetero-nuclei diatomic molecules are here derived for the first time. A diatomic molecule is first considered as a quantum mechanics oscillator. The second- and third-order Hamiltonian operators are then formed by substituting the number operator for the quantum number in the corresponding energy eigenvalues. The temporal variations of relative position and linear momentum of two oscillating atoms are calculated by solving Heisenberg's equations of motion. The results indicate that the oscillatory behaviour of diatomic molecules are described by the wave packets which consisted of two different frequencies. The stability theory of optical oscillators (lasers) is then exploited to determine the stability conditions of an oscillating diatomic molecule in the different energy levels of Morse potential. It peculiarly turned out that the diatomic molecules are exactly dissociated at the energy level in which their equations of motion become unstable. We also determine the minimum oscillation frequency (cut-off frequency) of a diatomic molecule at the dissociation level of Morse potential.

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Beyond Gaussian Entanglement: Practical Criteria for Polynomially Excited Quantum States

Gábor Homa, Richárd Balka, Nikolett Német +3 more

Polynomial excitations consisting of creation and annihilation operators are studied. Subsystem entanglement is proven produced and preserved by such manipulations. NPT entanglement already present between any two subsystems of a general pure or mixed multipartite Gaussian state is preserved under all polynomial excitations. Nevertheless, these excitations can create NPT entanglement between initially separable subsystems, implying that the number of NPT entangled pairs cannot decrease. Polynomial excitations of a wide class of initially separable pure states are considered. For these states, the method reduces the infinite-dimensional positivity problem to a finite-dimensional one: after partial transposition, NPT entanglement is equivalent to a corresponding bilinear representation matrix not being positive semi-definite. Furthermore, using symplectic diagonalization and spectral decomposition, an explicit transformation is constructed between the Fock-space and position representations. It is applied to quadratic excitations of a three-mode mixed Gaussian state. The behavior of the NPT entanglement is tracked between subsystems as a function of a parameter $κ$, which tunes the non-locality of the excitation, and quantum negativity is evaluated. Under a chosen quadratic Hamiltonian, the resulting unitary time evolution maintains NPT entanglement for long intervals, interrupted only by short periods of PPT states. Finally, a closed analytical expression is derived for the covariance matrix of a general quadratic operator for an arbitrary number of degrees of freedom in phase space. Our framework directly connects mathematical foundations, theoretical modeling, and experimentally feasible implementations.

quant-pharXiv:2610.18365•Oct 14, 2026•0 citations
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.
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\mathcal{H}={\mathcal{H}}_{\mathrm{el}}⨂{\mathcal{H}}_{\mathrm{vib}}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 H\mathcal{H}Hel{}_{\mathrm{el}}el​, and Nv{N}_{v}Nv​ is the dimension of H\mathcal{H}Hvib{}_{\mathrm{vib}}vib​. 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){1}_{g}(6s,6{p}_{3/2})1g​(6s,6p3/2​) states of the Cs2{}_{2}2​ 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{}_{2}2​ 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.
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.
The vibrational motion equations of both homo- and hetero-nuclei diatomic molecules are here derived for the first time. A diatomic molecule is first considered as a quantum mechanics oscillator. The second- and third-order Hamiltonian operators are then formed by substituting the number operator for the quantum number in the corresponding energy eigenvalues. The temporal variations of relative position and linear momentum of two oscillating atoms are calculated by solving Heisenberg's equations of motion. The results indicate that the oscillatory behaviour of diatomic molecules are described by the wave packets which consisted of two different frequencies. The stability theory of optical oscillators (lasers) is then exploited to determine the stability conditions of an oscillating diatomic molecule in the different energy levels of Morse potential. It peculiarly turned out that the diatomic molecules are exactly dissociated at the energy level in which their equations of motion become unstable. We also determine the minimum oscillation frequency (cut-off frequency) of a diatomic molecule at the dissociation level of Morse potential.
Polynomial excitations consisting of creation and annihilation operators are studied. Subsystem entanglement is proven produced and preserved by such manipulations. NPT entanglement already present between any two subsystems of a general pure or mixed multipartite Gaussian state is preserved under all polynomial excitations. Nevertheless, these excitations can create NPT entanglement between initially separable subsystems, implying that the number of NPT entangled pairs cannot decrease. Polynomial excitations of a wide class of initially separable pure states are considered. For these states, the method reduces the infinite-dimensional positivity problem to a finite-dimensional one: after partial transposition, NPT entanglement is equivalent to a corresponding bilinear representation matrix not being positive semi-definite. Furthermore, using symplectic diagonalization and spectral decomposition, an explicit transformation is constructed between the Fock-space and position representations. It is applied to quadratic excitations of a three-mode mixed Gaussian state. The behavior of the NPT entanglement is tracked between subsystems as a function of a parameter κκκ, which tunes the non-locality of the excitation, and quantum negativity is evaluated. Under a chosen quadratic Hamiltonian, the resulting unitary time evolution maintains NPT entanglement for long intervals, interrupted only by short periods of PPT states. Finally, a closed analytical expression is derived for the covariance matrix of a general quadratic operator for an arbitrary number of degrees of freedom in phase space. Our framework directly connects mathematical foundations, theoretical modeling, and experimentally feasible implementations.