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arXiv:1402.6959 · Physical Review A

Entanglement between electronic and vibrational degrees of freedom in a laser-driven molecular system

Mihaela Vatasescu

Published Dec 23, 2013• 28 citations
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Abstract

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.

DOI
10.1103/physreva.88.063415

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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 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×\else\texttimes\fiNv2\ifmmode\times\else\texttimes\fi{}{N}_{v}2\ifmmode×\else\texttimes\fiNv​ and 3\ifmmode×\else\texttimes\fiNv3\ifmmode\times\else\texttimes\fi{}{N}_{v}3\ifmmode×\else\texttimes\fiNv​ cases of the bipartite entanglement, where 2 and 3 are the dimensions of the electronic Hilbert space \mathcal{H}$${}_{\mathrm{el}}, and Nv{N}_{v}Nv​ is the dimension of \mathcal{H}$${}_{\mathrm{vib}}. We study the entanglement dynamics for two electronic states coupled by a laser pulse (a 2\ifmmode×\else\texttimes\fiNv2\ifmmode\times\else\texttimes\fi{}{N}_{v}2\ifmmode×\else\texttimes\fiNv​ case), taking as an example a coupling between the a3\ensuremathΣu+(6s,6s){a}^{3}{{\ensuremath{\Sigma}}_{u}}^{+}(6s,6s)a3\ensuremathΣu​+(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×\else\texttimes\fiNv3\ifmmode\times\else\texttimes\fi{}{N}_{v}3\ifmmode×\else\texttimes\fiNv​ 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 a3\ensuremathΣu+(6s,6s){a}^{3}{{\ensuremath{\Sigma}}_{u}}^{+}(6s,6s)a3\ensuremathΣu​+(6s,6s), 0g\ensuremath−(6s,6p3/2){0}_{g}{}^{\ensuremath{-}}(6s,6{p}_{3/2})0g​\ensuremath−(6s,6p3/2​), and 0g\ensuremath−(6s,5d){0}_{g}{}^{\ensuremath{-}}(6s,5d)0g​\ensuremath−(6s,5d) electronic states, which are coupled by a nonadiabatic radial coupling and a sequence of chirped laser pulses.
We characterize both entanglement and quantum coherence in a molecular system by connecting the linear entropy of electronic-nuclear entanglement with Wigner-Yanase skew information measuring vibronic coherence and local quantum uncertainty on electronic energy. Linear entropy of entanglement and quantifiers of quantum coherence are derived for a molecular system described in a bipartite Hilbert space H=Hel⨂Hvib\mathcal{H}={\mathcal{H}}_{el}⨂{\mathcal{H}}_{vib}H=Hel​⨂Hvib​ of finite dimension ${N}_{el}\ifmmode\times\else\texttimes\fi{}{N}_{v}$, and relations between them are established. For the specific case of the electronic-vibrational entanglement, we find the linear entropy of entanglement as having a more complex informational content than the von Neumann entropy. By keeping the information carried by the vibronic coherences in a molecule, linear entropy seizes vibrational motion in the electronic potentials as entanglement dynamics. We analyze entanglement oscillations in an isolated molecule, and show examples for the control of entanglement dynamics in a molecule through the creation of coherent vibrational wave packets in several electronic potentials by using chirped laser pulses.
Non-Markovian quantum evolution of the electronic subsystem in a laser-driven molecule is characterized through the appearance of negative decoherence rates in the canonical form of the electronic master equation. For a driven molecular system described in a bipartite Hilbert space H=Hel⨂Hvib\mathcal{H}={\mathcal{H}}_{el}⨂{\mathcal{H}}_{\mathrm{vib}}H=Hel​⨂Hvib​ of dimension $2\ifmmode\times\else\texttimes\fi{}{N}_{v}$, we derive the canonical form of the electronic master equation, deducing the canonical measures of non-Markovianity and the Bloch volume of accessible states. We find that one of the decoherence rates is always negative, accounting for the inherent non-Markovian character of the electronic evolution in the vibrational environment. Enhanced non-Markovian behavior, characterized by two negative decoherence rates, appears if there is a coupling between the electronic states g,eg,\phantom{\rule{0.16em}{0ex}}eg,e, such that the evolution of the electronic populations obeys d(PgPe)/dt>0d({P}_{g}{P}_{e})/dt>0d(Pg​Pe​)/dt>0. Non-Markovianity of the electronic evolution is analyzed in relation to temporal behaviors of the electronic-vibrational entanglement and electronic coherence, showing that enhanced non-Markovian behavior accompanies entanglement increase. Taking as an example the coupling of two electronic states by a laser pulse in the Cs2{\mathrm{Cs}}_{2}Cs2​ molecule, we analyze non-Markovian dynamics under laser pulses of various strengths, finding that the weaker pulse stimulates the bigger amount of non-Markovianity. Our results show that increase of the electronic-vibrational entanglement over a time interval is correlated to the growth of the total amount of non-Markovianity calculated over the same interval using canonical measures and connected with the increase of the Bloch volume. After the pulse, non-Markovian behavior is correlated to electronic coherence, such that vibrational motion in the electronic potentials which diminishes the nuclear overlap, implicitly increasing the linear entropy of entanglement, brings a memory character to dynamics.
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 compute the entanglement between the electronic and vibrational motions in the simplest molecular system, the hydrogen molecular ion, considering the molecule as a bipartite system, electron and vibrational motion. For that purpose we compute an accurate total non-Born-Oppenheimer wave function in terms of a huge expansion using nonorthogonal B-spline basis sets that expand separately the electronic and nuclear wave functions. According to the Schmidt decomposition theorem for bipartite systems, widely used in quantum-information theory, it is possible to find a much shorter but equivalent expansion in terms of the natural orbitals or Schmidt bases for the electronic and nuclear half spaces. Here we extend the Schmidt decomposition theorem to the case in which nonorthogonal bases are used to span the partitioned Hilbert spaces. This extension is first illustrated with two simple coupled systems, the former without an exact solution and the latter exactly solvable. In these model systems of distinguishable coupled particles it is shown that the entanglement content does not increase monotonically with the excitation energy, but only within the manifold of states that belong to an existing excitation mode, if any. In the hydrogen molecular ion the entanglement content for each non-Born-Oppenheimer vibronic state is quantified through the von Neumann and linear entropies and we show that entanglement serves as a witness to distinguish vibronic states related to different Born-Oppenheimer molecular energy curves or electronic excitation modes.
ABSTRACT: In the present work, we have computed the entanglement between the electronic and nuclear motions in two molecular model systems: the one-dimensional hydrogen molecular ion (H2+) and the Shin-Metiu model, considering the molecules as a bipartite systems: electron and nuclear motion. For that purpose, we have computed the Born-Oppenheimer and non-Born-Oppenheimer (Born-Huang) wave function in terms of the Fourier Grid Hamiltonian basis that expands both the electronic and nuclear wave functions. Also, according to the Schmidt decomposition theorem for bipartite systems, widely used in quantum-information theory, there is a much shorter but equivalent expansion in terms of the Schmidt bases for the electronic and nuclear sub-spaces. In these models of distinguishable coupled particles we have shown that the entanglements contents do not increase monotonically with the excitation energy. In the hydrogen molecular ion and in the ShinMetiu model, the entanglements contents for each Born-Oppenheimer electro-nuclear state is quantified through the von-Neumann and linear entropies and we have shown that entanglement serves as a witness of distinguishability of nuclear states related to different Born-Oppenheimer molecular energy curves or electronic excitation modes.
Rydberg atoms held in optical tweezer arrays combine vibrational and electronic degrees of freedom that can be coupled and manipulated at a microscopic level. This opens opportunities for the quantum simulation of artificial molecular systems and offers in particular a platform for probing complex vibronic dynamics in controlled settings with increasing complexity. Tailored interatomic interactions and electron-phonon couplings yield handles for designing electronic state manifolds, for studying structural transitions, and for exploring nonclassical vibrational states near molecular instabilities. Furthermore, this quantum simulator opens opportunities for testing and quantifying the validity of fundamental concepts, such as the Born-Oppenheimer approximation and quantum corrections to it.