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arXiv:quant-ph/0402016 · Physical Review A

Entanglement and bifurcations in Jahn-Teller models

Andrew P. Hines, Christopher M. Dawson, Ross H. McKenzie, Gerard J Milburn

Published Aug 5, 2004• 85 citations
arXivHTMLPDF

Abstract

We compare and contrast the entanglement in the ground state of two Jahn-Teller models. The $E\ensuremath{\bigotimes}\ensuremath{\beta}$ system models the coupling of a two-level electronic system, or qubit, to a single-oscillator mode, while the $E\ensuremath{\bigotimes}\ensuremath{\epsilon}$ models the qubit coupled to two independent, degenerate oscillator modes. In the absence of a transverse magnetic field applied to the qubit, both systems exhibit a degenerate ground state. Whereas there always exists a completely separable ground state in the $E\ensuremath{\bigotimes}\ensuremath{\beta}$ system, the ground states of the $E\ensuremath{\bigotimes}\ensuremath{\epsilon}$ model always exhibit entanglement. For the $E\ensuremath{\bigotimes}\ensuremath{\beta}$ case we aim to clarify results from previous work, alluding to a link between the ground-state entanglement characteristics and a bifurcation of a fixed point in the classical analog. In the $E\ensuremath{\bigotimes}\ensuremath{\epsilon}$ case we make use of an ansatz for the ground state. We compare this ansatz to exact numerical calculations and use it to investigate how the entanglement is shared between the three system degrees of freedom.

DOI
10.1103/physreva.70.022303

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We compare and contrast the entanglement in the ground state of two Jahn-Teller models. The E\ensuremath⨂\ensuremathβE\ensuremath{\bigotimes}\ensuremath{\beta}E\ensuremath⨂\ensuremathβ system models the coupling of a two-level electronic system, or qubit, to a single-oscillator mode, while the E\ensuremath⨂\ensuremathϵE\ensuremath{\bigotimes}\ensuremath{\epsilon}E\ensuremath⨂\ensuremathϵ models the qubit coupled to two independent, degenerate oscillator modes. In the absence of a transverse magnetic field applied to the qubit, both systems exhibit a degenerate ground state. Whereas there always exists a completely separable ground state in the E\ensuremath⨂\ensuremathβE\ensuremath{\bigotimes}\ensuremath{\beta}E\ensuremath⨂\ensuremathβ system, the ground states of the E\ensuremath⨂\ensuremathϵE\ensuremath{\bigotimes}\ensuremath{\epsilon}E\ensuremath⨂\ensuremathϵ model always exhibit entanglement. For the E\ensuremath⨂\ensuremathβE\ensuremath{\bigotimes}\ensuremath{\beta}E\ensuremath⨂\ensuremathβ case we aim to clarify results from previous work, alluding to a link between the ground-state entanglement characteristics and a bifurcation of a fixed point in the classical analog. In the E\ensuremath⨂\ensuremathϵE\ensuremath{\bigotimes}\ensuremath{\epsilon}E\ensuremath⨂\ensuremathϵ case we make use of an ansatz for the ground state. We compare this ansatz to exact numerical calculations and use it to investigate how the entanglement is shared between the three system degrees of freedom.
We consider the steady states of a harmonic oscillator coupled so strongly to a two-level system (a qubit) that the rotating wave approximation cannot be made. The Hamiltonian version of this model is known as the $E\ensuremath{\bigotimes}\ensuremath{\beta}$ Jahn-Teller model. The semiclassical version of this system exhibits a fixed-point bifurcation, which in the quantum model leads to a ground state with substantial entanglement between the oscillator and the qubit. We show that the dynamical bifurcation survives in a dissipative quantum description of the system, amidst an even richer bifurcation structure. We propose an experimental implementation of this model based on a superconducting cavity: a superconducting junction in the central conductor of a coplanar waveguide.
We discuss the ground state entanglement of the $E\ensuremath{\bigotimes}ϵ$ Jahn-Teller model in the presence of a strong transverse magnetic field as a function of the vibronic coupling strength. A complete characterization is given of the phenomenon of entanglement sharing in a system composed by a qubit coupled to two bosonic modes. Using the residual III tangle, we find that three-partite entanglement is significantly present in the system in the parameter region near the bifurcation point of the corresponding classical model.
We discuss the ground state entanglement of the E⊗ϵE\otimes\epsilonE⊗ϵ Jahn-Teller model in the presence of a strong transverse magnetic field as a function of the vibronic coupling strength. A complete characterization is given of the phenomenon of entanglement sharing in a system composed by a qubit coupled to two bosonic modes. Using the residual III-tangle, we find that three-partite entanglement is significantly present in the system in the parameter region near the bifurcation point of the corresponding classical model.
Jahn-Teller systems exhibit strong coupling between the electronic and nuclear degrees of freedom. We analyse how linear and quadratic coupling affect the electron-nuclear entanglement in the E⊗ϵE\otimes \epsilonE⊗ϵ system. A measure of quantum entanglement, recently developed by Shimony [Ann. New York Acad. Sci. 755, 675 (1995)], is used. States in the adiabatic regime with quadratic coupling are analysed. The electron-nuclear entanglement in vibronic states of the linear Exe model is calculated numerically. The physical significance of electron-nuclear entanglement is discussed in terms of reduction factors and magnetic vibrational circular dichroism.
Abstract We explore quantum correlations, in particular, quantum entanglement, among vibrational phonon modes as well as between electronic and vibrational degrees of freedom in molecular systems, described by Jahn-Teller mechanism. Specifically, to isolate and simplify the phonon- electron interactions in a complex molecular system, the basis of our discussions is taken to be the proposal of simulating two-frequency Jahn- Teller systems using superconducting circuit quantum electrodynamics systems (circuit QED) by Tekin Dereli and co-workers in 2012. We evaluate the quantum correlations, in particular entanglement between the vibrational phonon modes, and present analytical explanations using a single privileged Jahn-Teller mode picture. Furthermore, spin-orbit entanglement or quantum correlations between electronic and vibrational degrees of freedom are examined. We conclude by discussing experimental feasibility to detect such quantum correlations, considering the dephasing and decoherence in state-of-the-art superconducting two-level systems (qubits).
How does the classical phase-space structure for a composite system relate to the entanglement characteristics of the corresponding quantum system? We demonstrate how the entanglement in nonlinear bipartite systems can be associated with a fixed-point bifurcation in the classical dynamics. Using the example of coupled giant spins we show that when a fixed point undergoes a supercritical pitchfork bifurcation, the corresponding quantum state---the ground state---achieves its maximum amount of entanglement near the critical point. We conjecture that this will be a generic feature of systems whose classical limit exhibits such a bifurcation.