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arXiv:quant-ph/0308165 · Physical Review A

Quantum entanglement and fixed-point bifurcations

Andrew P. Hines, Ross H. McKenzie, Gerard J Milburn

Published Apr 4, 2005• 68 citations
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Abstract

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.

DOI
10.1103/physreva.71.042303

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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.
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Interacting quantum systems evolving from an uncorrelated composite initial state generically develop quantum correlations---entanglement. As a consequence, a local description of interacting quantum systems is impossible as a rule. A unitarily evolving (isolated) quantum system generically develops extensive entanglement: the magnitude of the generated entanglement will increase without bounds with the effective Hilbert space dimension of the system. It is conceivable that coupling of the interacting subsystems to local dephasing environments will restrict the generation of entanglement to such extent that the evolving composite system may be considered as approximately disentangled. This conjecture is addressed in the context of some common models of a bipartite system with linear and nonlinear interactions and local coupling to dephasing environments. Analytical and numerical results obtained imply that the conjecture is generally false. Open dynamics of the quantum correlations is compared to the corresponding evolution of the classical correlations and a qualitative difference is found.
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.
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.